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55 results on '"Takéo Takahashi"'

1. On the null-controllability of a radiative heat transfer system

Author: Takéo Takahashi and Mohamed Ghattassi
Subjects: 0209 industrial biotechnology, Convective heat transfer, Banach fixed-point theorem, Mathematical analysis, Null (mathematics), General Engineering, 02 engineering and technology, Fixed point, Domain (mathematical analysis), Controllability, 020901 industrial engineering & automation, Bounded function, 0202 electrical engineering, electronic engineering, information engineering, Radiative transfer, 020201 artificial intelligence & image processing, Mathematics
Abstract: This paper studies the null controllability for radiative conductive convective heat transfer system in a grey, semi-transparent, scattering and bounded domain with control acting locally in a subset. The well-posedness of PDE is proved by the Banach fixed point theorem. Moreover, the null controllability proof is based on a fixed point argument and Carleman estimates in the presence of source terms, and it avoids tackling the linearized problems. Finally, a numerical experiment is included to validate the theoretical results.
Published: 2021
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2. Existence of strong solutions for a system of interaction between a compressible viscous fluid and a wave equation*

Author: Arnab Roy, Takéo Takahashi, and Debayan Maity
Subjects: Cuboid, Applied Mathematics, 010102 general mathematics, Mathematical analysis, General Physics and Astronomy, Statistical and Nonlinear Physics, Wave equation, 01 natural sciences, Physics::Fluid Dynamics, 010101 applied mathematics, Nonlinear system, Barotropic fluid, Fluid–structure interaction, Compressibility, 0101 mathematics, Convection–diffusion equation, Displacement (fluid), Mathematical Physics, Mathematics
Abstract: In this article, we consider a fluid-structure interaction system where the fluid is viscous and compressible and where the structure is a part of the boundary of the fluid domain and is deformable. The fluid is governed by the barotropic compressible Navier-Stokes system whereas the structure displacement is described by a wave equation. We show that the corresponding coupled system admits a unique strong solution for an initial fluid density and an initial fluid velocity in $H^3$ and for an initial deformation and an initial deformation velocity in $H^4$ and $H^3$ respectively. The reference configuration for the fluid domain is a rectangular cuboid with the elastic structure being the top face. We use a modified Lagrangian change of variables to transform the moving fluid domain into the rectangular cuboid and then analyze the corresponding linear system coupling a transport equation (for the density), a heat-type equation, and a wave equation. The corresponding results for this linear system and estimations of the coefficients coming from the change of variables allow us to perform a fixed point argument and to prove the existence and uniqueness of strong solutions for the nonlinear system, locally in time.
Published: 2021
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3. Stabilization of a rigid body moving in a compressible viscous fluid

Author: Takéo Takahashi, Arnab Roy, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Takahashi, Takéo
Subjects: Viscous liquid, 01 natural sciences, Damper, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), Position (vector), Fluid-structure interaction, Fluid–structure interaction, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Ball (mathematics), [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, 010102 general mathematics, Mathematical analysis, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Rigid body, MSC : 35Q35, 35D30, 35D35, 35R37, 76N10, 93D15, 93D20, compressible Navier-Stokes system, stabilization, 010101 applied mathematics, Spring (device), Compressibility, global solutions, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis of PDEs (math.AP)
Abstract: We consider the stabilizability of a fluid-structure interaction system where the fluid is viscous and compressible and the structure is a rigid ball. The feedback control of the system acts on the ball and corresponds to a force that would be produced by a spring and a damper connecting the center of the ball to a fixed point $$h_1$$ . We prove the global-in-time existence of strong solutions for the corresponding system under a smallness condition on the initial velocities and on the distance between the initial position of the center of the ball and $$h_1$$ . Then, we show with our feedback law, that the fluid and the structure velocities go to 0 and that the center of the ball goes to $$h_1$$ as $$t\rightarrow \infty $$ .
Published: 2020
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4. Existence and uniqueness of strong solutions for the system of interaction between a compressible Navier-Stokes-Fourier fluid and a damped plate equation

Author: Takéo Takahashi, Debayan Maity, Center for Applicable Mathematics [Bangalore] (TIFR-CAM), Tata Institute for Fundamental Research (TIFR), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Debayan Maity was partially supported by INSPIRE faculty fellowship (IFA18-MA128) and by Department of Atomic Energy, Government of India, under project no. 12-R & D-TFR-5.01-0520. Takéo Takahashi was partially supported by the ANR research project IFSMACS (ANR-15-CE40-0010)., and ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015)
Subjects: Change of variables, fluid-structure interaction, R-sectorial operators, Fixed point, 01 natural sciences, Physics::Fluid Dynamics, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Physics, Small data, Applied Mathematics, Operator (physics), 010102 general mathematics, Mathematical analysis, Linear system, maximal regularity, General Engineering, General Medicine, compressible NavierStokesFourier system, 010101 applied mathematics, Computational Mathematics, Compressibility, strong solutions, General Economics, Econometrics and Finance, Displacement (fluid), Analysis, Analysis of PDEs (math.AP)
Abstract: The article is devoted to the mathematical analysis of a fluid–structure interaction system where the fluid is compressible and heat conducting and where the structure is deformable and located on a part of the boundary of the fluid domain. The fluid motion is modeled by the compressible Navier–Stokes–Fourier system and the structure displacement is described by a structurally damped plate equation. Our main results are the existence of strong solutions in an L p − L q setting for small time or for small data. Through a change of variables and a fixed point argument, the proof of the main results is mainly based on the maximal regularity property of the corresponding linear systems. For small time existence, this property is obtained by decoupling the linear system into several standard linear systems whereas for global existence and for small data, the maximal regularity property is proved by showing that the corresponding linear coupled fluid–structure operator is R -sectorial.
Published: 2021
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5. Remark on the global null controllability for a viscous Burgers-particle system with particle supported control

Author: Arnab Roy, Mythily Ramaswamy, Takéo Takahashi, Chennai Mathematical Institute [Inde], Institute of Mathematics of the Czech Academy of Science (IM / CAS), Czech Academy of Sciences [Prague] (CAS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Arnab Roy and Takéo Takahashi were partially supported by the ANR research project IFSMACS (ANR-15-CE40-0010). The three authors were partially supported by the IFCAM project 'Analysis, Control and Homogenization of Complex Systems'., and ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015)
Subjects: 0209 industrial biotechnology, Work (thermodynamics), Point particle, 02 engineering and technology, 01 natural sciences, Physics::Fluid Dynamics, Mathematics - Analysis of PDEs, 020901 industrial engineering & automation, Position (vector), Fluid-structure interaction, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics - Optimization and Control, Mathematics, Particle system, Applied Mathematics, 010102 general mathematics, Null (mathematics), Mathematical analysis, AMS subject classifications. 35Q35, 35D30, 35D35, 35R37, 35L10, 93D15, 93D20, global controllability, Burgers' equation, Controllability, viscous Burgers equation, Optimization and Control (math.OC), Particle, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis of PDEs (math.AP)
Abstract: International audience; This paper is devoted to study the controllability of a one-dimensional fluid-particle interaction model where the fluid follows the viscous Burgers equation and the point mass obeys Newton's second law. We prove the null controllability for the velocity of the fluid and the particle and an approximate controllability for the position of the particle with a control variable acting only on the particle. One of the novelties of our work is the fact that we achieve this controllability result in a uniform time for all initial data and without any smallness assumptions on the initial data.
Published: 2020
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6. $$L^{p}$$ Theory for the Interaction Between the Incompressible Navier–Stokes System and a Damped Plate

Author: Takéo Takahashi, Debayan Maity, Center for Applicable Mathematics [Bangalore] (TIFR-CAM), Tata Institute for Fundamental Research (TIFR), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Incompressible Navier-Stokes System, Physics, Change of variables, AMS subject classifications. 35Q35, 76D03, 76D05, 74F1, Applied Mathematics, Operator (physics), 010102 general mathematics, Linear system, Mathematical analysis, Boundary (topology), Fixed point, Condensed Matter Physics, 01 natural sciences, Action (physics), Physics::Fluid Dynamics, 010101 applied mathematics, Computational Mathematics, Fluid-structure interaction, Fluid–structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Maximal Lp regularity, Uniqueness, 0101 mathematics, Strong solutions, Mathematical Physics
Abstract: International audience; We consider a viscous incompressible fluid governed by the Navier-Stokes system written in a domain where a part of the boundary is moving as a damped beam under the action of the fluid. We prove the existence and uniqueness of global strong solutions for the corresponding fluid-structure interaction system in an Lp-Lq setting. The main point in the proof consists in the study of a linear parabolic system coupling the non stationary Stokes system and a damped beam. We show that this linear system possesses the maximal regularity property by proving the R-sectoriality of the corresponding operator. The proof of the main results is then obtained by an appropriate change of variables to handle the free boundary and a fixed point argument to treat the nonlinearities of this system.
Published: 2021
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7. Well-posedness for the coupling between a viscous incompressible fluid and an elastic structure

Author: Takéo Takahashi, Muriel Boulakia, Sergio Guerrero, Sorbonne Université (SU), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), COmputational Mathematics for bio-MEDIcal Applications (COMMEDIA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Coupling, Elastic structure, Applied Mathematics, 010102 general mathematics, Linear elasticity, Mathematical analysis, 2010 Mathematics Subject Classification. 76D03, 76D05, 35Q74, 76D27, Structure (category theory), General Physics and Astronomy, Statistical and Nonlinear Physics, Viscous incompressible fluid, 01 natural sciences, Domain (mathematical analysis), Navier-Stokes system, Physics::Fluid Dynamics, 010101 applied mathematics, Bounded function, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Fluid-structure, Displacement (fluid), Mathematical Physics, Mathematics
Abstract: International audience; In this paper, we consider a system modeling the interaction between a viscous incompressible fluid and an elastic structure. The fluid motion is represented by the classical Navier-Stokes equations while the elastic displacement is described by the linearized elasticity equation. The elastic structure is immersed in the fluid and the whole system is confined into a general bounded smooth domain of R3. Our main result is the local in time existence and uniqueness of a strong solution of the corresponding system.
Published: 2019
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8. Approximate controllability and stabilizability of a linearized system for the interaction between a viscoelastic fluid and a rigid body

Author: Arnab Roy, Takéo Takahashi, Debanjana Mitra, Department of Mathematics, Indian Institute of Technology Bombay, Powai, Maharashtra 400076, India, Institute of Mathematics of the Czech Academy of Science (IM / CAS), Czech Academy of Sciences [Prague] (CAS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Subjects: 0209 industrial biotechnology, Control and Optimization, Structure (category theory), 02 engineering and technology, 93D15, 01 natural sciences, controllability, Domain (mathematical analysis), Viscoelasticity, finite dimensional controls 2010 Mathematics Subject Classification 76A10, 020901 industrial engineering & automation, Position (vector), 0101 mathematics, [MATH]Mathematics [math], Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Order (ring theory), stabilizability, Rigid body, Exponential function, viscoelastic fluids, Controllability, 74F10, Control and Systems Engineering, Fluid-structure interaction systems, 93B52, Signal Processing, 35Q35
Abstract: International audience; We study control properties of a linearized fluid-structure interaction system, where the structure is a rigid body and where the fluid is a viscoelastic material. We establish the approximate controllability and the exponential stabilizability for the velocities of the fluid and of the rigid body and for the position of the rigid body. In order to prove this, we prove a general result for this kind of systems that generalizes in particular the case without structure. The exponential stabilization of the system is obtained with a finite-dimensional feedback control acting only on the momentum equation on a subset of the fluid domain and up to some rate that depends on the coefficients of the system. We also show that, as in the case without structure, the system is not exactly null-controllable in finite time.
Published: 2020

9. Feedback stabilization of parabolic systems with input delay

Author: Imene Aicha Djebour, Takéo Takahashi, Julie Valein, Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), The two first authors were partially supported by the ANR research project IFSMACS (ANR-15-CE40-0010). The third author was partially supported by the ANR research projects ISDEEC (ANR-16-CE40-0013) and ANR ODISSE (ANR-19-CE48-0004-01)., ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015), ANR-16-CE40-0013,ISDEEC,Interactions entre Systèmes Dynamiques, Equations d'Evolution et Contrôle(2016), and ANR-19-CE48-0004,ODISSE,Synthèse d'observateur pour des systèmes de dimension infinie(2019)
Subjects: 0209 industrial biotechnology, Work (thermodynamics), Control and Optimization, Feedback control, parabolic systems, 2010 Mathematics Subject Classification 93B52, 93D15, 35Q30, 76D05, 93C20, 02 engineering and technology, 01 natural sciences, Navier-Stokes system, 020901 industrial engineering & automation, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics - Optimization and Control, Mathematics, delay control, Applied Mathematics, 010102 general mathematics, Mathematical analysis, stabilizability, finite-dimensional control, Nonlinear system, Transformation (function), Optimization and Control (math.OC), [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Constant (mathematics), Stationary state, Analysis of PDEs (math.AP)
Abstract: This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay: the \begin{document}$ N $\end{document}-dimensional linear reaction-convection-diffusion equation with \begin{document}$ N\geq 1 $\end{document} and the Oseen system. We end the article by showing that this theory can be used to stabilize nonlinear parabolic systems with input delay by proving the local feedback distributed stabilization of the Navier-Stokes system around a stationary state.
Published: 2020
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10. Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity

Author: Sorin Micu and Takéo Takahashi
Subjects: 0209 industrial biotechnology, Applied Mathematics, 010102 general mathematics, Mathematical analysis, 02 engineering and technology, 01 natural sciences, Burgers' equation, Controllability, Nonlinear system, Viscosity, 020901 industrial engineering & automation, Fixed-point iteration, Linearization, Biorthogonal system, 0101 mathematics, Fourier series, Analysis, Mathematics
Abstract: This article studies the local controllability to trajectories of a Burgers equation with nonlocal viscosity. By linearization we are led to an equation with a non local term whose controllability properties are analyzed by using Fourier decomposition and biorthogonal techniques. Once the existence of controls is proved and the dependence of their norms with respect to the time is established for the linearized model, a fixed point method allows us to deduce the result for the nonlinear initial problem.
Published: 2018
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11. On the Existence of Strong Solutions to a Fluid Structure Interaction Problem with Navier Boundary Conditions

Author: Takéo Takahashi, Imene Aicha Djebour, Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Physics, Small data, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), Condensed Matter Physics, 01 natural sciences, Navier-Stokes system, Physics::Fluid Dynamics, 010101 applied mathematics, Strong solutions, Computational Mathematics, damped beam equation, Fluid–structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Fluid motion, strong solutions, Uniqueness, Boundary value problem, 0101 mathematics, fluid-structure interaction systems, Plate equation, Mathematical Physics
Abstract: International audience; We consider a fluid-structure interaction system composed by a three-dimensional viscous incompressible fluid and an elastic plate located on the upper part of the fluid boundary. The fluid motion is governed by the Navier-Stokes system whereas we add a damping in the plate equation. We use here Navier-slip boundary conditions instead of the standard no-slip boundary conditions. The main results are the local in time existence and uniqueness of strong solutions of the corresponding system and the global in time existence and uniqueness of strong solutions for small data and if we assume the presence of frictions in the boundary conditions.
Published: 2019
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12. An optimal control approach to ciliary locomotion

Author: Marius Tucsnak, Jorge San Martín, Takéo Takahashi, Departamento de Ingeniería Matemática, Facultad de Ciencias Fisicas y Matemáticas, Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), and ANR-11-BS03-0002,HAMECMOPSYS,Approche Hamiltonienne pour l'analyse et la commande des systèmes multiphysiques à paramètres distribués(2011)
Subjects: 0209 industrial biotechnology, Control and Optimization, media_common.quotation_subject, Boundary (topology), 02 engineering and technology, controllability, 01 natural sciences, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], optimal control, symbols.namesake, 020901 industrial engineering & automation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Sensitivity (control systems), 0101 mathematics, Eccentricity (behavior), Astrophysics::Galaxy Astrophysics, Mathematics, media_common, Applied Mathematics, Mathematical analysis, Ode, Scalar (physics), Stokes equations, Reynolds number, Optimal control, 010101 applied mathematics, Controllability, Gegenbauer functions, symbols, ciliates, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: We consider a class of low Reynolds number swimmers, of prolate spheroidal shape, which can be seen as simplified models of ciliated microorganisms. Within this model, the form of the swimmer does not change, the propelling mechanism consisting in tangential displacements of the material points of swimmer's boundary. Using explicit formulas for the solution of the Stokes equations at the exterior of a translating prolate spheroid the governing equations reduce to a system of ODE's with the control acting in some of its coefficients (bilinear control system). The main theoretical result asserts the exact controllability of the prolate spheroidal swimmer. In the same geometrical situation, we consider the optimal control problem of maximizing the efficiency during a stroke and we prove the existence of a maximum. We also provide a method to compute an approximation of the efficiency by using explicit formulas for the Stokes system at the exterior of a prolate spheroid, with some particular tangential velocities at the fluid-solid interface. We analyze the sensitivity of this efficiency with respect to the eccentricity of the considered spheroid and show that for small positive eccentricity, the efficiency of a prolate spheroid is better than the efficiency of a sphere. Finally, we use numerical optimization tools to investigate the dependence of the efficiency on the number of inputs and on the eccentricity of the spheroid. The ``best'' numerical result obtained yields an efficiency of $30.66\%$ with $13$ scalar inputs. In the limiting case of a sphere our best numerically obtained efficiency is of $30.4\%$, whereas the best computed efficiency previously reported in the literature is of $22\%$.
Published: 2016
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13. Local null controllability of a rigid body moving into a Boussinesq flow

Author: Takéo Takahashi, Arnab Roy, Center for Applicable Mathematics [Bangalore] (TIFR-CAM), Tata Institute of Fundamental Research [Bombay] (TIFR), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015), and Tata Institute for Fundamental Research (TIFR)
Subjects: Controllability, 0209 industrial biotechnology, Control and Optimization, Carleman inequality, Newton's laws of motion, 02 engineering and technology, system, 01 natural sciences, Physics::Fluid Dynamics, Rigid body, 020901 industrial engineering & automation, Position (vector), Fluid–structure interaction, Fluid-structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boussinesq, 0101 mathematics, Navier–Stokes equations, Physics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Null (mathematics), Flow (mathematics), AMS subject classifications 35Q30, 93C20, 76D05, 93B05, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Navier-Stokes equations
Abstract: International audience; In this paper, we study the controllability of a fluid-structure interaction system. We consider a viscous and incompressible fluid modeled by the Boussinesq system and the structure is a rigid body with arbitrary shape which satisfies Newton's laws of motion. We assume that the motion of this system is bidimensional in space. We prove the local null controllability for the velocity and temperature of the fluid and for the position and velocity of rigid body for a control acting only on the temperature equation on a fixed subset of the fluid domain.
Published: 2019
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14. Mathematical analysis of the motion of a rigid body in a compressible Navier-Stokes-Fourier fluid

Author: Debayan Maity, Marius Tucsnak, Takéo Takahashi, Bernhard H. Haak, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
Subjects: General Mathematics, Motion (geometry), R-sectorial operators, 76N10, strong, 01 natural sciences, AMS subject classifications 35Q30, symbols.namesake, Mathematics - Analysis of PDEs, Thermal insulation, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, solutions, Mathematics, fluid-particle interaction, Small data, business.industry, 010102 general mathematics, Mathematical analysis, Linear system, maximal regularity, Rigid body, 76D05, 010101 applied mathematics, Fourier transform, Compressible Navier-Stokes-Fourier System, symbols, Compressibility, business, Analysis of PDEs (math.AP)
Abstract: International audience; We study an initial and boundary value problem modelling the motion of a rigid body in a heat conducting gas. The solid is supposed to be a perfect thermal insulator. The gas is described by the compressible Navier-Stokes-Fourier equations, whereas the motion of the solid is governed by Newton's laws. The main results assert the existence of strong solutions, in an L p-L q setting, both locally in time and globally in time for small data. The proof is essentially using the maximal regularity property of associated linear systems. This property is checked by proving the R-sectoriality of the corresponding operators, which in turn is obtained by a perturbation method.
Published: 2019
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15. Gevrey regularity for a system coupling the Navier-Stokes system with a beam equation

Author: Takéo Takahashi, Mehdi Badra, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)
Subjects: Coupling, Gevrey class semigroups, Semigroup, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), Fixed point, 01 natural sciences, Domain (mathematical analysis), fluid-structure, 010101 applied mathematics, Physics::Fluid Dynamics, Navier-Stokes system, Computational Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Gevrey class, 2010 Mathematics Subject Classification : 76D03, 76D05, 35Q74, 76D27, Analysis, Beam (structure), Mathematics
Abstract: International audience; We analyse a bi-dimensional fluid-structure interaction system composed by a viscous incompressible fluid and a beam located at the boundary of the fluid domain. Our main result is the existence and uniqueness of strong solutions for the corresponding coupled system. The proof is based on a the study of the linearized system and a fixed point procedure. In particular, we show that the linearized system can be written with a Gevrey class semigroup. The main novelty with respect to previous results is that we do not consider any approximation in the beam equation.
Published: 2019
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16. Analysis of a simplified model of rigid structure floating in a viscous fluid

Author: Takéo Takahashi, Jorge San Martín, Marius Tucsnak, Debayan Maity, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Centre de modélisation mathématique (CMM), Universitad de Chile-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-16-CE92-0028,INFIDHEM,Systèmes interconnectés de dimension infinie pour les milieux hétérogènes(2016), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Function space, Differential equation, Applied Mathematics, Mathematical analysis, return to equilibrium, General Engineering, Viscous shallow water equations, fluid-structure interaction, floating structure, Viscous liquid, 01 natural sciences, 010305 fluids & plasmas, 010101 applied mathematics, Viscosity, AMS subject classifications 35Q35 74F10, Flow (mathematics), Linearization, Modeling and Simulation, 0103 physical sciences, Fluid–structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], strong solutions, 0101 mathematics, Shallow water equations, Mathematics
Abstract: International audience; We study the interaction of surface water waves with a floating solid constraint to move only in the vertical direction. The first novelty we bring in is that we propose a new model for this interaction, taking into consideration the viscosity of the fluid. This is done supposing that the flow obeys a shallow water regime (modeled by the viscous Saint-Venant equations in one space dimension) and using a Hamiltonian formalism. Another contribution of this work is establishing the well-posedness of the obtained PDEs/ODEs system in function spaces similar to the standard ones for strong solutions of viscous shallow water equations. Our well-posedness results are local in time for every initial data and global in time if the initial data are close (in appropriate norms) to an equilibrium state. Moreover, we show that the linearization of our system around an equilibrium state can be described, at least for some initial data, by an integro-fractional differential equation related to the classical Cummins equation and which reduces to the Cummins equation when the viscosity vanishes and the fluid is supposed to fill the whole space. Finally, we describe some numerical tests, performed on the original nonlinear system, which illustrate the return to equilibrium and the influence of the viscosity coefficient.
Published: 2019
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17. Asymptotic analysis of an optimal control problem for a viscous incompressible fluid with Navier slip boundary conditions

Author: Takéo Takahashi, Claudia Gariboldi, National University of Río Cuarto = Universidad Nacional de Río Cuarto (UNRC), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Subjects: 0209 industrial biotechnology, Asymptotic analysis, General Mathematics, Mathematics::Analysis of PDEs, 02 engineering and technology, Slip (materials science), 01 natural sciences, Navier-Stokes system, Physics::Fluid Dynamics, symbols.namesake, optimal control, 020901 industrial engineering & automation, Mathematics - Analysis of PDEs, Convergence (routing), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Mathematics, Sequence, 010102 general mathematics, Mathematical analysis, Viscous incompressible fluid, Optimal control, Dirichlet boundary condition, symbols, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Navier slip boundary condition, Analysis of PDEs (math.AP)
Abstract: International audience; We consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by $\alpha$ the friction coefficient and we analyze the asymptotic behavior of such a problem as $\alpha\to\infty$. More precisely, we prove that if we take an optimal control for each $\alpha$, then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier–Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.
Published: 2019
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18. Existence of weak solutions for a Bingham fluid-rigid body system

Author: Benjamin Obando, Takéo Takahashi, Departamento de Ingeniería Matemática [Santiago] (DIM), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE)
Subjects: Physics, Viscoplasticity, Applied Mathematics, Weak solution, 010102 general mathematics, Mathematical analysis, Newton's laws of motion, Weak formulation, Rigid body, 01 natural sciences, Displacement (vector), 010101 applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Penalty method, 0101 mathematics, Bingham plastic, Mathematical Physics, Analysis
Abstract: International audience; We consider the motion of a rigid body in a viscoplastic material. This material is modeled by the 3D Bingham equations, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is an inequality (due to the plasticity of the fluid), and it involves a free boundary (due to the motion of the rigid body). We approximate it by regularizing the convex terms in the Bingham fluid and by using a penalty method to take into account the presence of the rigid body.
Published: 2019
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19. Stabilization of a fluid–rigid body system

Author: Takéo Takahashi, George Weiss, Marius Tucsnak, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and Tel Aviv University [Tel Aviv]
Subjects: asymptotic stability, Applied Mathematics, Mathematical analysis, PD controller, switching feedback, Fixed point, Rigid body, Damper, Exponential stability, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Initial value problem, global solutions, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Uniqueness, Ball (mathematics), Navier-Stokes equations, fluid-structure interactions, Navier–Stokes equations, Analysis, Mathematics
Abstract: We consider the mathematical model of a rigid ball moving in a viscous incompressible fluid occupying a bounded domain Ω, with an external force acting on the ball. We investigate in particular the case when the external force is what would be produced by a spring and a damper connecting the center of the ball h to a fixed point h 1 ∈ Ω . If the initial fluid velocity is sufficiently small, and the initial h is sufficiently close to h 1 , then we prove the existence and uniqueness of global (in time) solutions for the model. Moreover, in this case, we show that h converges to h 1 , and all the velocities (of the fluid and of the ball) converge to zero. Based on this result, we derive a control law that will bring the ball asymptotically to the desired position h 1 even if the initial value of h is far from h 1 , and the path leading to h 1 is winding and complicated. Now, the idea is to use the force as described above, with one end of the spring and damper at h , while other end is jumping between a finite number of points in Ω, that depend on h (a switching feedback law).
Published: 2015
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20. Elastic energy of a convex body

Author: Antoine Henrot, Chiara Bianchini, and Takéo Takahashi
Subjects: 010101 applied mathematics, General Mathematics, 010102 general mathematics, Mathematical analysis, Elastic energy, Convex body, 0101 mathematics, 01 natural sciences, Mathematics
Published: 2015
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21. Optimal boundary control for steady motions of a self-propelled body in a Navier-Stokes liquid

Author: Toshiaki Hishida, Takéo Takahashi, Ana L. Silvestre, Nagoya University, Center for Computacional and Stochastic Mathematics (CEMAT), Instituto Superior Técnico, Universidade Técnica de Lisboa (IST), Departamento de Matemática [Lisbonne] (DM/IST), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), T. Hishida is partially supported by Grant-in-Aid for Scientific Research, 18K03363, from JSPS., A. L. Silvestre acknowledges the financial support of the Portuguese FCT - Fundação para a Ciência e a Tecnologia, through the projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT/IST-ID., T. Takahashi is partially supported by the project IFSMACS ANR-15-CE40-0010, financed by the French Agence Nationale de la Recherche, and ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015)
Subjects: Control and Optimization, Gâteaux derivative, 2010 Mathematics Subject Classification : 76D05, 49K21, 76D55, 49J21, Motion (geometry), Boundary (topology), 01 natural sciences, Domain (mathematical analysis), Rotating body, Mathematics - Analysis of PDEs, Drag reduction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 3-D Navier-Stokes equations, Self-propelled motion, 0101 mathematics, Mathematics - Optimization and Control, Mathematics, Boundary control, 010102 general mathematics, Mathematical analysis, Rigid body, 010101 applied mathematics, Computational Mathematics, Control and Systems Engineering, Drag, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Unit (ring theory), Exterior domain, Reference frame
Abstract: Consider a rigid body 𝒮 ⊂ ℝ3 immersed in an infinitely extended Navier-Stokes liquid and the motion of the body-fluid interaction system described from a reference frame attached to 𝒮. We are interested in steady motions of this coupled system, where the region occupied by the fluid is the exterior domain Ω = ℝ3 \ 𝒮. This paper deals with the problem of using boundary controls v*, acting on the whole ∂Ω or just on a portion Γ of ∂Ω, to generate a self-propelled motion of 𝒮 with a target velocity V (x) := ξ + ω × x and to minimize the drag about 𝒮. Firstly, an appropriate drag functional is derived from the energy equation of the fluid and the problem is formulated as an optimal boundary control problem. Then the minimization problem is solved for localized controls, such that supp v*⊂ Γ, and for tangential controls, i.e, v*⋅ n|∂Ω = 0, where n is the outward unit normal to ∂Ω. We prove the existence of optimal solutions, justify the Gâteaux derivative of the control-to-state map, establish the well-posedness of the corresponding adjoint equations and, finally, derive the first order optimality conditions. The results are obtained under smallness restrictions on the objectives |ξ| and |ω| and on the boundary controls.
Published: 2020
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22. Numerical observers with vanishing viscosity for the 1d wave equation

Author: Takéo Takahashi, Galina C. García, Departamento de Matemática y Ciencia de la Computación [Santiago de Chile] (DMCC), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), CORIDA, Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), and Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est
Subjects: 35R30, 35L05, 65M06, Reconstruction algorithm, Discretization, Iterative method, Generalization, Applied Mathematics, Numerical observers, Mathematical analysis, Inverse problem, Wave equation, Space (mathematics), Computational Mathematics, Viscosity (programming), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: We consider a numerical scheme associated with the iterative method developed in Ramdani et al. (ESAIM Control Optim. Calc. Var. 13(3):503---527, 2007) to recover initial conditions of conservative systems. In this method, the initial conditions are reconstructed by using observers. Here we use a finite-difference discretization in space of these observers and our aim is to prove estimates of the errors with respect to the mesh size and to the number of steps in the iterative method. This is done in the particular example of the 1d wave equation. In order to avoid restrictions of the number of steps with respect to the mesh size, we add a numerical viscosity in the numerical observers. A generalization for other equations is also given.
Published: 2013
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23. Boundary local null-controllability of the Kuramoto-Sivashinsky equation

Author: Takéo Takahashi, Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015)
Subjects: 0209 industrial biotechnology, Control and Optimization, Boundary (topology), 02 engineering and technology, 01 natural sciences, 020901 industrial engineering & automation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Nonlinear Sciences::Pattern Formation and Solitons, Mathematics, Physics::Computational Physics, Kuramoto-Sivashinsky equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Null (mathematics), null controllability 2010 Mathematics Subject Classification: 93B05, Kuramoto–Sivashinsky equation, Controllability, Nonlinear system, Control and Systems Engineering, Signal Processing, 35K55, 93C10, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Beam (structure)
Abstract: International audience; We prove that the Kuramoto-Sivashinsky equation is locally controllable in 1D and in 2D with one boundary control. Our method consists in combining several general results in order to reduce the null-controllability of this nonlinear parabolic equation to the exact controllability of a linear beam or plate system. This improves known results on the controllability of Kuramoto-Sivashinsky equation and gives a general strategy to handle the null-controllability of nonlinear parabolic systems.
Published: 2017
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24. On the reconstruction of obstacles and of rigid bodies immersed in a viscous incompressible fluid

Author: Erica L. Schwindt, Takéo Takahashi, Jorge San Martín, Departamento de Ingeniería Matemática [Santiago] (DIM), Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE), Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), and Université d'Orléans (UO)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Convex hull, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mathematics::Analysis of PDEs, Boundary (topology), Inverse problem, Rigid body, 01 natural sciences, 010101 applied mathematics, Physics::Fluid Dynamics, Nonlinear system, Classical mechanics, complex geometrical solutions, Geometrical inverse problems, fluid-structure interaction, Navier–Stokes system, enclosure method, Obstacle, Fluid–structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Point (geometry), 0101 mathematics, Mathematics
Abstract: We consider the geometrical inverse problem consisting in recovering an unknown obstacle in a viscous incompressible fluid by measurements of the Cauchy force on the exterior boundary. We deal with the case where the fluid equations are the nonstationary Stokes system and using the enclosure method, we can recover the convex hull of the obstacle and the distance from a point to the obstacle. With the same method, we can obtain the same result in the case of a linear fluid-structure system composed by a rigid body and a viscous incompressible fluid. We also tackle the corresponding nonlinear systems: the Navier–Stokes system and a fluid-structure system with free boundary. Using complex spherical waves, we obtain some partial information on the distance from a point to the obstacle.
Published: 2017
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25. A boundary control problem for the steady self-propelled motion of a rigid body in a Navier-Stokes fluid

Author: Ana L. Silvestre, Toshiaki Hishida, Takéo Takahashi, Nagoya University, Center for Computacional and Stochastic Mathematics (CEMAT), Instituto Superior Técnico, Universidade Técnica de Lisboa (IST), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015)
Subjects: Boundary (topology), Motion (geometry), 01 natural sciences, Rotating body, Physics::Fluid Dynamics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Self-propelled motion, Navier stokes, 0101 mathematics, Mathematical Physics, Mathematics, 3-D Navier–Stokes equations, Steady state, Boundary control, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Rigid body, Asymptotic behavior, 010101 applied mathematics, Distribution (mathematics), Classical mechanics, Flow velocity, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Exterior domain, Analysis
Abstract: International audience; We study the self-propelled motions of a rigid body immersed in a viscous incompressible fluid which fills the exterior domain of the rigid body.The mechanism used by the body to reach the desired motion is modeled through a distribution of velocities at its boundary. The fluid motion is modeled by the stationary Navier-Stokes system. These equations are coupled with two relations for the balance of forces and torques.We prove that there exists a control allowing the rigid body to move with a prescribed rigid velocity provided the velocity is small enough. We also show that since the net force exerted by the fluid to the rigid body vanishes, we have a better summability of the fluid velocity than the classical summability result for the solutions of the stationary Navier-Stokes system in exterior domains.
Published: 2017
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26. Feedback boundary stabilization of 2d fluid-structure interaction systems

Author: Mehdi Badra, Takéo Takahashi, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015)
Subjects: Applied Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), fluid-structure interaction, feedback stabilization, 01 natural sciences, Domain (mathematical analysis), 010101 applied mathematics, Physics::Fluid Dynamics, Nonlinear system, Classical mechanics, Position (vector), Fluid–structure interaction, No-slip condition, beam equation, Discrete Mathematics and Combinatorics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Navier-Stokes equations, Navier–Stokes equations, Analysis, Stationary state, Mathematics
Abstract: International audience; We study the feedback stabilization of a system composed by an incompressible viscous fluid and a deformable structure located at the boundary of the fluid domain. We stabilize the position and the velocity of the structure and the velocity of the fluid around a stationary state by means of a Dirichlet control, localized on the exterior boundary of the fluid domain and with values in a finite dimensional space. Our result concerns weak solutions for initial data close to the stationary state. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domain of the stationary state and of the stabilized solution are different. We prove that for initial data close to the stationary state, we can stabilize the position and the velocity of the deformable structure and the velocity of the fluid.
Published: 2017
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27. On the Navier–Stokes system with the Coulomb friction law boundary condition

Author: Takéo Takahashi, Jorge San Martín, Loredana Bălilescu, Departamento de Matemática = Mathematics Department [Florianópolis] (MTM), Centro de Ciências Físicas e Matemáticas [Florianópolis] (CFM), Universidade Federal de Santa Catarina = Federal University of Santa Catarina [Florianópolis] (UFSC)-Universidade Federal de Santa Catarina = Federal University of Santa Catarina [Florianópolis] (UFSC), Department of Mathematics [Pitesti], Faculty of Mathematics, Universitatea din Pitesti [Roumanie] (UPIT)-Universitatea din Pitesti [Roumanie] (UPIT), Center for Mathematical Modeling (CMM), Universidad de Chile = University of Chile [Santiago] (UCHILE), Departamento de Ingeniería Matemática [Santiago] (DIM), Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015), and Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Cauchy stress tensor, Applied Mathematics, General Mathematics, Weak solution, 010102 general mathematics, Mathematical analysis, General Physics and Astronomy, 010103 numerical & computational mathematics, Mixed boundary condition, 16. Peace & justice, 01 natural sciences, Coulomb's law, Physics::Fluid Dynamics, symbols.namesake, Classical mechanics, Law, symbols, No-slip condition, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Cauchy boundary condition, Uniqueness, Boundary value problem, 0101 mathematics, Mathematics
Abstract: International audience; We propose a new model for the motion of a viscous incompressible fluid. More precisely, we consider the Navier–Stokes system with a boundary condition governed by the Coulomb friction law. With this boundary condition, the fluid can slip on the boundary if the tangential component of the stress tensor is too large. We prove the existence and uniqueness of weak solution in the two–dimensional problem and the existence of at least one solution in the three–dimensional case, together with regularity properties and an energy estimate. We also propose a fully discrete scheme of our problem using the characteristic method and we present numerical simulations in two physical examples.
Published: 2017
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28. Fluid-rigid structure interaction system with Coulomb's law

Author: Takéo Takahashi, Loredana Bălilescu, Jorge San Martín, Department of Mathematics [Pitesti], Faculty of Mathematics, Universitatea din Pitesti [Roumanie] (UPIT)-Universitatea din Pitesti [Roumanie] (UPIT), Departamento de Matemática = Mathematics Department [Florianópolis] (MTM), Centro de Ciências Físicas e Matemáticas [Florianópolis] (CFM), Universidade Federal de Santa Catarina = Federal University of Santa Catarina [Florianópolis] (UFSC)-Universidade Federal de Santa Catarina = Federal University of Santa Catarina [Florianópolis] (UFSC), Center for Mathematical Modeling (CMM), Universidad de Chile = University of Chile [Santiago] (UCHILE), Departamento de Ingeniería Matemática [Santiago] (DIM), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015), and Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE)
Subjects: Coulomb's law 2010 Mathematics Subject Classification: 35Q30, 01 natural sciences, Coulomb's law, Physics::Fluid Dynamics, Navier-Stokes system, symbols.namesake, 76D03, Neumann boundary condition, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Fluid-structure system, Boundary value problem, 0101 mathematics, Physics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mixed boundary condition, 16. Peace & justice, Robin boundary condition, 76D27, 010101 applied mathematics, Computational Mathematics, Classical mechanics, Dirichlet boundary condition, No-slip condition, symbols, Cauchy boundary condition, Cauchy theory, Analysis
Abstract: International audience; We propose a new model in a fluid-structure system composed by a rigid body and a viscous incompress-ible fluid using a boundary condition based on Coulomb's law. This boundary condition allows the fluid to slip on the boundary if the tangential component of the stress is too large. In the opposite case, we recover the standard Dirichlet boundary condition. The governing equations are the Navier-Stokes system for the fluid and the Newton laws for the body. The corresponding coupled system can be written as a variational inequality. We prove that there exists a weak solution of this system.
Published: 2017
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29. Single input controllability of a simplified fluid-structure interaction model

Author: Yuning Liu, Marius Tucsnak, Takéo Takahashi, Universität Regensburg (UR), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), CORIDA, Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM)
Subjects: 0209 industrial biotechnology, Control and Optimization, 010102 general mathematics, Mathematical analysis, Scalar (physics), Control variable, fluid-structure interaction, 35L10, 65M60, 93B05, 93B40, 93D15, 02 engineering and technology, Null-controllability, Rigid body, 01 natural sciences, Parabolic partial differential equation, Controllability, Computational Mathematics, Nonlinear system, viscous Burgers equation, 020901 industrial engineering & automation, Control and Systems Engineering, Fluid–structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Spectral method, Mathematics
Abstract: International audience; In this paper we study a controllability problem for a simpli ed one dimensional model for the motion of a rigid body in a viscous uid. The control variable is the velocity of the uid at one end. One of the novelties brought in with respect to the existing literature consists in the fact that we use a single scalar control. Moreover, we introduce a new methodology, which can be used for other nonlinear parabolic systems, independently of the techniques previously used for the linearized problem. This methodology is based on an abstract argument for the null controllability of parabolic equations in the presence of source terms and it avoids tackling linearized problems with time dependent coefficients.
Published: 2012
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30. Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid

Author: Olivier Glass, Takéo Takahashi, Franck Sueur, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), ANR-09-BLAN-0213,CISIFS,Controle et Identification pour les Systemes d'Interaction Fluide-Structure(2009), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), and Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est
Subjects: Smoothness (probability theory), General Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), Motion (geometry), Perfect fluid, Rigid body, 01 natural sciences, Domain (mathematical analysis), 010101 applied mathematics, Mathematics - Analysis of PDEs, Bounded function, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Initial value problem, 0101 mathematics, Analysis of PDEs (math.AP), Mathematics
Abstract: International audience; We consider the motion of a rigid body immersed in an incompressible perfect fluid which occupies a threedimensional bounded domain. For such a system the Cauchy problem is well-posed locally in time if the initial velocity of the fluid is in the H¨older space C1,r. In this paper we prove that the smoothness of the motion of the rigid body may be only limited by the smoothness of the boundaries (of the body and of the domain). In particular for analytic boundaries the motion of the rigid body is analytic (till the classical solution exists and till the solid does not hit the boundary). Moreover in this case this motion depends smoothly on the initial data.
Published: 2012
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31. Strong Solutions for a Phase Field Navier–Stokes Vesicle–Fluid Interaction Model

Author: Takéo Takahashi, Yuning Liu, Marius Tucsnak, Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), CORIDA, and Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est
Subjects: Field (physics), Navier-Stokes, Vesicle membrane, Navier–Stokes existence and smoothness, 01 natural sciences, 010305 fluids & plasmas, External flow, Physics::Fluid Dynamics, symbols.namesake, Phase field, Phase (matter), 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Fluid vesicle interaction, Mathematical Physics, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Eulerian path, Non-dimensionalization and scaling of the Navier–Stokes equations, Condensed Matter Physics, Computational Mathematics, Classical mechanics, symbols, Compressibility, 35Q30, 35Q35, 35D35, 76D03, 76D05, 76T10
Abstract: International audience; In this paper we study a mathematical model for the dynamics of vesicle membranes in a 3D incompressible viscous fluid. The system is in the Eulerian formulation, involving the coupling of the incompressible Navier-Stokes system with a phase field equation. This equation models the vesicle deformations under external flow fields. We prove the local in time existence and uniqueness of strong solutions. Moreover, we show that, given T > 0, for initial data which are small (in terms of T), these solutions are defined on [0, T] (almost global existence).
Published: 2011
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32. Stabilization of Parabolic Nonlinear Systems with Finite Dimensional Feedback or Dynamical Controllers: Application to the Navier–Stokes System

Author: Mehdi Badra, Takéo Takahashi, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, and Institut National de Recherche en Informatique et en Automatique (Inria)
Subjects: Analytic semigroup, 0209 industrial biotechnology, Pure mathematics, Control and Optimization, Boundary (topology), 02 engineering and technology, 01 natural sciences, Riccati equation, 020901 industrial engineering & automation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Linear combination, Mathematics, Feedback stabilization, Generator (category theory), Applied Mathematics, Operator (physics), 010102 general mathematics, Mathematical analysis, Lyapunov functional, Nonlinear system, Finite dimensional control, Bounded function, Dirichlet boundary control, Navier-Stokes equations
Abstract: International audience; Let $A : \mathcal{D}(A)\to \mathcal{X}$ be the generator of an analytic semigroup and $B : \mathcal{U} \to [{\cal D}(A^*)]'$ a quasi-bounded operator. In this paper, we consider the stabilization of the system $y'=Ay+Bu$ where $u$ is the linear combination of a family $(v_1,\ldots,v_K)$. Our main result shows that if $(A^*,B^*)$ satisfies a unique continuation property and if $K$ is greater or equal to the maximum of the geometric multiplicities of the the unstable modes of $A$, then the system is generically stabilizable with respect to the family $(v_1,\ldots,v_K)$. With the same functional framework, we also prove the stabilizability of a class of nonlinear system when using feedback or dynamical controllers. We apply these results to stabilize the Navier--Stokes equations in 2D and in 3D by using boundary control.
Published: 2011
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33. Collisions in Three-Dimensional Fluid Structure Interaction Problems

Author: Matthieu Hillairet and Takéo Takahashi
Subjects: Partial differential equation, Applied Mathematics, Weak solution, 010102 general mathematics, Mathematical analysis, Equations of motion, Horizontal plane, 01 natural sciences, Physics::Fluid Dynamics, 010101 applied mathematics, Computational Mathematics, Fluid–structure interaction, Ball (mathematics), 0101 mathematics, Reynolds-averaged Navier–Stokes equations, Analysis, CFD-DEM, Mathematics
Abstract: This paper deals with a system composed of a rigid ball moving into a viscous incompressible fluid over a fixed horizontal plane. The equations of motion for the fluid are the Navier–Stokes equations, and the equations for the motion of the rigid ball are obtained by applying Newton's laws. We show that for any weak solution of the corresponding system satisfying the energy inequality, the rigid ball never touches the plane. This result is the extension of that obtained in [M. Hillairet, Comm. Partial Differential Equations, 32 (2007), pp. 1345–1371] in the two-dimensional setting.
Published: 2009
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34. Exact controllability of a fluid–rigid body system

Author: Oleg Yu. Imanuvilov, Takéo Takahashi, Department of Mathematics, Colorado State University [Fort Collins] (CSU), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), CORIDA, and Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
Subjects: Controllability, Mathematics(all), 0209 industrial biotechnology, Carleman inequality, Differential equation, General Mathematics, Mathematics::Analysis of PDEs, 02 engineering and technology, 01 natural sciences, Rigid body, Physics::Fluid Dynamics, Navier–Stokes equations, 020901 industrial engineering & automation, Fluid-structure interaction, Fluid–structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, 35Q30, 76D05, 93B05, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Equations of motion, Stokes flow, Ordinary differential equation, Navier-Stokes equations
Abstract: International audience; This paper is devoted to the controllability of a 2D fluid-structure system. The fluid is viscous and incompressible and its motion is modelled by the Navier-Stokes equations whereas the structure is a rigid ball which satisfies Newton's laws. We prove the local null controllability for the velocities of the fluid and of the rigid body and the exact controllability for the position of the rigid body. An important part of the proof relies on a new Carleman inequality for an auxiliary linear system coupling the Stokes equations with some ordinary differential equations.
Published: 2007
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35. A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator

Author: Takéo Takahashi, Gérald Tenenbaum, Marius Tucsnak, Karim Ramdani, Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
Subjects: 0209 industrial biotechnology, Schrödinger equation, Boundary exact controllability, 02 engineering and technology, 01 natural sciences, Poincaré–Steklov operator, Bounded operator, symbols.namesake, 020901 industrial engineering & automation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Observability, 0101 mathematics, Mathematics, 93C25, 93B07, 93C20, 11N36, Partial differential equation, Operator (physics), 010102 general mathematics, Mathematical analysis, Hilbert space, Hautus test, Exact differential equation, Plate equation, symbols, Wave equation, Boundary exact observability, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis
Abstract: Let A be a possibly unbounded skew-adjoint operator on the Hilbert space X with compact resolvent. Let C be a bounded operator from D ( A ) to another Hilbert space Y. We consider the system governed by the state equation z ˙ ( t ) = Az ( t ) with the output y ( t ) = Cz ( t ) . We characterize the exact observability of this system only in terms of C and of the spectral elements of the operator A. The starting point in the proof of this result is a Hautus-type test, recently obtained in Burq and Zworski (J. Amer. Soc. 17 (2004) 443–471) and Miller (J. Funct. Anal. 218 (2) (2005) 425–444). We then apply this result to various systems governed by partial differential equations with observation on the boundary of the domain. The Schrodinger equation, the Bernoulli–Euler plate equation and the wave equation in a square are considered. For the plate and Schrodinger equations, the main novelty brought in by our results is that we prove the exact boundary observability for an arbitrarily small observed part of the boundary. This is done by combining our spectral observability test to a theorem of Beurling on nonharmonic Fourier series and to a new number theoretic result on shifted squares.
Published: 2005
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36. Convergence of the Lagrange–Galerkin method for a fluid–rigid system

Author: Marius Tucsnak, Jean-François Scheid, Takéo Takahashi, Jorge San Martín, Departamento de Ingeniería Matemática [Santiago] (DIM), University of Chile [Santiago]-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Differential equation, Numerical analysis, 010102 general mathematics, Mathematical analysis, 010103 numerical & computational mathematics, General Medicine, Rigid body, 01 natural sciences, Finite element method, Physics::Fluid Dynamics, Pressure-correction method, Ordinary differential equation, Convergence (routing), 0101 mathematics, Galerkin method, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: In this Note, we consider a Lagrange–Galerkin scheme to approximate a two dimensional fluid–rigid body problem. The system is modelled by the incompressible Navier–Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the rigid body. In this problem, the equations of the fluid are written in a domain whose variation is one of the unknowns. We introduce a numerical method based on the use of characteristics and on finite elements with a fixed mesh. Our main result asserts the convergence of this scheme. To cite this article: J. San Martin et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).
Published: 2004
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37. Existence of strong solutions for the problem of a rigid-fluid system

Author: Takéo Takahashi
Subjects: Physics::Fluid Dynamics, Strong solutions, Bounded function, Mathematical analysis, Boundary (topology), Motion (geometry), Geometry, General Medicine, Uniqueness, Rigid body, Domain (mathematical analysis), Displacement (vector), Mathematics
Abstract: This Note is devoted to the study of a fluid–rigid body interaction problem. The motion of the fluid is modelled by the Navier–Stokes equations, written in an unknown bounded domain depending on the displacement of the rigid body. Our main result yields the existence and uniqueness of strong solutions, which are global provided that the rigid body does not touch the boundary. To cite this article: T. Takahashi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).
Published: 2003
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38. Small moving rigid body into a viscous incompressible fluid

Author: Takéo Takahashi, Christophe Lacave, Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), C.L. is partially supported by the project 'Instabilities in Hydrodynamics' funded by the Paris city hall (program 'Emergences') and the Fondation Sciences Mathématiques de Paris., ANR-13-BS01-0003,DYFICOLTI,DYnamique des Fluides, Couches Limites, Tourbillons et Interfaces(2013), ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015), Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF), and Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)
Subjects: Semigroup, Mechanical Engineering, 010102 general mathematics, Mathematical analysis, Zero (complex analysis), Viscous liquid, Fixed point, Viscous incompressible fluid, Rigid body, 01 natural sciences, Physics::Fluid Dynamics, Mathematics - Analysis of PDEs, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Mathematics (miscellaneous), [INFO.INFO-IR]Computer Science [cs]/Information Retrieval [cs.IR], 0103 physical sciences, Convergence (routing), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010307 mathematical physics, 0101 mathematics, Analysis, Energy (signal processing), Mathematics, Analysis of PDEs (math.AP)
Abstract: International audience; We consider a single disk moving under the influence of a 2D viscous fluid and we study the asymptotic as the size of the solid tends to zero.If the density of the solid is independent of $\varepsilon$, the energy equality is not sufficient to obtain a uniform estimate for the solid velocity. This will be achieved thanks to the optimal $L^p-L^q$ decay estimates of the semigroup associated to the fluid-rigid body system and to a fixed point argument. Next, we will deduce the convergence to the solution of the Navier-Stokes equations in $\mathbb{R}^2$.
Published: 2015
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39. On the minimization of total mean curvature

Author: Simon Masnou, Antoine Henrot, Jeremy Dalphin, Takéo Takahashi, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-12-BS01-0007,OPTIFORM,Optimisation de Formes(2012), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Mathematics - Differential Geometry, Scalar (mathematics), Open set, 01 natural sciences, Omega, Combinatorics, Primary 49Q10, secondary 53A05, 58E35, 0103 physical sciences, Minkowski space, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, geometric inequality, Mathematics, Mean curvature, 010102 general mathematics, Mathematical analysis, Minkowski inequality, Regular polygon, Total mean curvature, Differential geometry, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], shape optimization, 010307 mathematical physics, Geometry and Topology
Abstract: In this paper we are interested in possible extensions of an inequality due to Minkowski: $\int_{\partial\Omega} H\,dA \geq \sqrt{4\pi A(\partial\Omega)}$ valid for any regular open set $\Omega\subset\mathbb{R}^3$, where $H$ denotes the scalar mean curvature and $A$ the area. We prove that this inequality holds true for axisymmetric domains which are convex in the direction orthogonal to the axis of symmetry. We also show that this inequality cannot be true in more general situations. However we prove that $\int_{\partial\Omega} |H|\,dA \geq \sqrt{4\pi A(\partial\Omega)}$ remains true for any axisymmetric domain., Comment: Equipe Equations aux derivees partielles
Published: 2014
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40. Well-posedness for a one-dimensional fluid-particle interaction model

Author: Frédéric Lagoutière, Takéo Takahashi, Boris Andreianov, Nicolas Seguin, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Numerical Analysis, Geophysics and Ecology (ANGE), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ANR-11-JS01-0006,CoToCoLa,Thématiques actuelles en lois de conservation(2011), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Seguin, Nicolas, and Jeunes Chercheuses et Jeunes Chercheurs - Thématiques actuelles en lois de conservation - - CoToCoLa2011 - ANR-11-JS01-0006 - JCJC - VALID
Subjects: Conservation law, Differential equation, Point particle, Applied Mathematics, Mathematical analysis, Ode, 35L65, 35L81, 35R06, 65M12, Fixed point, Non-conservative coupling, Burgers' equation, Burgers equation, Computational Mathematics, Splitting, Well-posedness, Wave-front tracking, Fixed-Point, Bounded variation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Entropy (arrow of time), BV estimates, Analysis, Fluid-particle interaction, Mathematics
Abstract: The fluid-particle interaction model introduced by the three last authors in [J. Differential Equations, 245 (2008), pp. 3503-3544] is the object of our study. This system consists of the Burgers equation with a singular source term (term that models the interaction via a drag force with a moving point particle) and of an ODE for the particle path. The notion of entropy solution for the singular Burgers equation is inspired by the theory of conservation laws with discontinuous flux developed by the first author, Kenneth Hvistendahl Karlsen and Nils Henrik Risebro in [Arch. Ration. Mech. Anal., 201 (2011), pp. 26-86]. In this paper, we prove well-posedness and justify an approximation strategy for the particle-in-Burgers system in the case of initial data of bounded variation. Existence result for L∞ data is also given.
Published: 2014

41. Feedback stabilization of a simplified 1d fluid- particle system

Author: Mehdi Badra, Takéo Takahashi, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), and Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
Subjects: Change of variables, Feedback stabilization, Applied Mathematics, Mathematical analysis, Boundary (topology), Vis cous Burgers equation, 74F10, 35Q35, 76D55, 93C20, 93D15, Burgers' equation, Nonlinear system, Control theory, Fluid–structure interaction, Fluid-structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Burgers vortex, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Exponential decay, Mathematical Physics, Analysis, Stationary state, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: We consider the feedback stabilization of a simplified 1d model for a fluid–structure interaction system. The fluid equation is the viscous Burgers equation whereas the motion of the particle is given by the Newton's laws. We stabilize this system around a stationary state by using feedbacks located at the exterior boundary of the fluid domain. With one input, we obtain a local stabilizability of the system with an exponential decay rate of order σ σ 0 . An arbitrary order for the exponential decay rate can be proved if a unique continuation result holds true or if two inputs are used to stabilize the system. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domains of the stationary state and of the stabilized solution are different.
Published: 2014
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42. Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid

Author: Takéo Takahashi, Muriel Boulakia, Erica L. Schwindt, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Numerical simulation of biological flows (REO), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centro de Modelamiento Matemático [Santiago] (CMM), Centre National de la Recherche Scientifique (CNRS)-Universidad de Santiago de Chile [Santiago] (USACH), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), Universidad de Santiago de Chile [Santiago] (USACH)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: existence and uniqueness of strong solutions, Applied Mathematics, deformable structure, 010102 general mathematics, Mathematical analysis, Linear elasticity, MSC: 74F10, 76D03, 35Q30, 35Q35, 37N15, 76D05, Structure (category theory), Motion (geometry), Fluid mechanics, 01 natural sciences, Domain (mathematical analysis), 010101 applied mathematics, Physics::Fluid Dynamics, Classical mechanics, incompressible Navier-Stokes equations, Fluid–structure interaction, Fluid-structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, Boundary value problem, 0101 mathematics, Mathematics
Abstract: International audience; In this paper we study a three-dimensional fluid-structure interaction problem. The motion of the fluid is modeled by the Navier-Stokes equations and we consider for the elastic structure a finite dimensional approximation of the equation of linear elasticity. The time variation of the fluid domain is not known a priori, so we deal with a free boundary value problem. Our main result yields the local in time existence and uniqueness of strong solutions for this system.
Published: 2012
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43. Inverse problem and null-controllability for parabolic systems

Author: Takéo Takahashi, Galina C. García, Departamento de Matemática y Ciencia de la Computación, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), and CORIDA
Subjects: Generalized inverse, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Null (mathematics), Boundary (topology), Inverse problem, controllability, 01 natural sciences, Parabolic partial differential equation, 010101 applied mathematics, Controllability, Inverse scattering problem, inverse problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Initial value problem, 0101 mathematics, 93C25, 93B07, 93B05, 93C20, 35R30, Mathematics
Abstract: In this paper, we present some abstract results giving a general connection between null-controllability and several inverse problems for a class of parabolic equations. We obtain some conditional stability estimates for the inverse problems consisting of determining the initial condition and the source term, from interior or boundary measurements. We apply this framework for Stokes system with interior and boundary observations, for a coupling of two Stokes system and a linear fluid-structure system.
Published: 2011
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44. Homogenization and singular limits for the complete Navier-Stokes-Fourier system

Author: Eduard Feireisl, Takéo Takahashi, Antonin Novotny, Mathematical Institute [Praha] - Academy of Sciences, Czech Academy of Sciences [Prague] (CAS), Institut de Mathématiques de Toulon - EA 2134 (IMATH), Université de Toulon (UTLN), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), Robust control of infinite dimensional systems and applications (CORIDA), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), CORIDA, Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Czech Academy of Sciences [Prague] ( ASCR ), Institut de Mathématiques de Toulon - EA 2134 ( IMATH ), Université de Toulon ( UTLN ), Institut Élie Cartan de Nancy ( IECN ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université Henri Poincaré - Nancy 1 ( UHP ) -Université Nancy 2-Institut National Polytechnique de Lorraine ( INPL ) -Centre National de la Recherche Scientifique ( CNRS ), Robust control of infinite dimensional systems and applications ( CORIDA ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université Henri Poincaré - Nancy 1 ( UHP ) -Université Nancy 2-Institut National Polytechnique de Lorraine ( INPL ) -Centre National de la Recherche Scientifique ( CNRS ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université Henri Poincaré - Nancy 1 ( UHP ) -Université Nancy 2-Institut National Polytechnique de Lorraine ( INPL ) -Centre National de la Recherche Scientifique ( CNRS ) -Laboratoire de Mathématiques et Applications de Metz ( LMAM ), Université Paul Verlaine - Metz ( UPVM ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Paul Verlaine - Metz ( UPVM ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Nancy - Grand Est, and Institut National de Recherche en Informatique et en Automatique ( Inria )
Subjects: Mathematics(all), Compressible Navier-Stokes-Fourier equations, General Mathematics, Viscous liquid, 01 natural sciences, Homogenization (chemistry), Compressible flow, Physics::Fluid Dynamics, symbols.namesake, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, Homogenization, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Porous medium equation, 010101 applied mathematics, Nonlinear system, Fourier transform, Fourier analysis, symbols, Compressibility, Porous medium, Compressible Navier–Stokes–Fourier equations
Abstract: International audience; In this paper we apply the methods of homogenization to the full Navier-Stokes-Fourier system describing the motion of a general viscous, compressible, and heat conducting fluid. We study the asymptotic behavior of solutions in perforated domains with tiny holes, where the diameter of the holes is proportional to their mutual distance. As a limit system, we identify a porous medium type equation with a nonlinear Darcy's law.
Published: 2010
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45. Small solids in an inviscid fluid

Author: Takéo Takahashi, Nicolas Seguin, Frédéric Lagoutière, Boris Andreianov, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), Laboratoire de Recherche en Informatique (LRI), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), CORIDA, Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), and Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM)
Subjects: Statistics and Probability, Solid-fluid interaction, 01 natural sciences, symbols.namesake, Quadratic equation, 35F25, 35L80, 65M99, Inviscid flow, singular source term, well-balanced scheme, random-choice method, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, adapted entropy, Mathematics, Partial differential equation, Finite volume method, Applied Mathematics, 010102 general mathematics, Mathematical analysis, General Engineering, Riemann solver, Computer Science Applications, Burgers' equation, Burgers equation, 010101 applied mathematics, Drag, Ordinary differential equation, symbols
Abstract: We present in this paper several results concerning a simple model of interaction between an inviscid fluid, modeled by the Burgers equation, and a particle, assumed to be point-wise. It is composed by a first-order partial differential equation which involves a singular source term and by an ordinary differential equation. The coupling is ensured through a drag force that can be linear or quadratic. Though this model can be considered as a simple one, its mathematical analysis is involved. We put forward a notion of entropy solution to our model, define a Riemann solver and make first steps towards well-posedness results. The main goal is to construct easy-to-implement and yet reliable numerical approximation methods; we design several finite volume schemes, which are analyzed and tested.
Published: 2010
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46. Solving inverse source problems using observability. Applications to the Euler-Bernoulli plate equation

Author: Ana L. Silvestre, Carlos J.S. Alves, Takéo Takahashi, Marius Tucsnak, Instituto Superior Técnico, Universidade Técnica de Lisboa (IST), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), and Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
Subjects: 0209 industrial biotechnology, Control and Optimization, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), Inverse, Observable, 02 engineering and technology, 01 natural sciences, Bernoulli's principle, symbols.namesake, 020901 industrial engineering & automation, Euler's formula, symbols, Identifiability, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Point (geometry), Observability, 0101 mathematics, Mathematics
Abstract: The aim of this paper is to provide a general framework for solving a class of inverse source problems by using exact observability of infinite dimensional systems. More precisely, we show that if a system is exactly observable, then a source term in this system can be identified by knowing its intensity and appropriate observations which often correspond to measurements of some boundary traces. This abstract theory is then applied to obtain new identifiability results for a system governed by the Euler-Bernoulli plate equation. Using a different methodology, we show that exact observability can be used to identify both the locations and the intensities of combinations of point sources in the plate equation.
Published: 2009
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47. Semi-discrétisation en espace du problème de la stabilisation interne de l'équation des poutres

Author: Karim Ramdani, Takéo Takahashi, Marius Tucsnak, Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
Subjects: 0209 industrial biotechnology, 010102 general mathematics, Mathematical analysis, Boundary (topology), 02 engineering and technology, Space (mathematics), 01 natural sciences, 020901 industrial engineering & automation, Frequency domain, Calculus, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Exponential decay, Beam (structure), Mathematics
Abstract: International audience; We consider the finite-difference space semi-discretization of the beam equation with an internal stabilization. By using a frequency domain approach, we prove that this scheme yields a uniform exponential decay rate (with respect to the mesh size). This positive result is in contrast with previous negative ones for the case of boundary stabilization or for other evolution equations.
Published: 2007
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48. Uniformly exponentially stable approximations for a class of second order evolution equations. Application to LQR optimization problems

Author: Karim Ramdani, Takéo Takahashi, Marius Tucsnak, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), and Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est
Subjects: 0209 industrial biotechnology, Control and Optimization, Discretization, hp-FEM, 02 engineering and technology, Linear-quadratic regulator, 01 natural sciences, Algebraic Riccati equation, 020901 industrial engineering & automation, Exponential stability, Riccati equation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, finite difference, 010102 general mathematics, Mathematical analysis, Finite difference, 93D15, 65M60, 65M12, Finite element method, uniform exponential stability, Computational Mathematics, Control and Systems Engineering, finite element, LQR optimal control problem, wave equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: A paraître dans ESAIM COCV; International audience; We consider the approximation of a class of exponentially stable infinite dimensional linear systems modelling the damped vibrations of one dimensional vibrating systems or of square plates. It is by now well known that the approximating systems obtained by usual finite element or finite difference are not, in general, uniformly stable with respect to the discretization parameter. Our main result shows that, by adding a suitable numerical viscosity term in the numerical scheme, our approximations are uniformly exponentially stable. This result is then applied to obtain strongly convergent approximations of the solutions of the algebraic Riccati equations associated to an LQR optimal control problem. We next give an application to a non-homogeneous string equation. Finally we apply similar techniques for approximating the equations of a damped square plate.
Published: 2007
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49. Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid

Author: Patricio Cumsille, Takéo Takahashi, Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Centro de Modelamiento Matemático [Santiago] (CMM), and Universidad de Santiago de Chile [Santiago] (USACH)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Conservation law, General Mathematics, 010102 general mathematics, Mathematical analysis, Geometry, Fluid mechanics, Reynolds stress, Viscous liquid, Rigid body, 01 natural sciences, Physics::Fluid Dynamics, 010101 applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Navier–Stokes equations, Poinsot's ellipsoid, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: In this paper, we consider the interaction between a rigid body and an incompressible, homogeneous, viscous fluid. This fluid-solid system is assumed to fill the whole space ℝd, d = 2 or 3. The equations for the fluid are the classical Navier-Stokes equations whereas the motion of the rigid body is governed by the standard conservation laws of linear and angular momentum. The time variation of the fluid domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem.
Published: 2007

50. A uniformly stable finite difference space semi-discretization for the internal stabilization of the plate equation in a square

Author: Marius Tucsnak, Takéo Takahashi, Karim Ramdani, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), and Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est
Subjects: Discretization, 010102 general mathematics, Mathematical analysis, Finite difference, Finite difference coefficient, 010103 numerical & computational mathematics, Space (mathematics), 01 natural sciences, Square (algebra), Scheme (mathematics), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Exponential decay, Plate equation, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: International audience; We propose a finite difference space semi-discretization of the stabilized Bernoulli-Euler plate equation in a square. The scheme studied yields a uniform exponential decay rate with respect to the mesh size.
Published: 2006
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