Search

Your search keyword '"Takéo Takahashi"' showing total 38 results
Start Over Author "Takéo Takahashi" Language undetermined
38 results on '"Takéo Takahashi"'

2. 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
Full Text
View/download PDF

3. 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
Full Text
View/download PDF

5. 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
Full Text
View/download PDF

7. Controllability Results for Cascade Systems of m Coupled N-Dimensional Stokes and Navier-Stokes Systems by N – 1 Scalar Controls

Author

Takéo Takahashi, Luz de Teresa, and Yingying Wu-Zhang

Subjects
Computational Mathematics, Control and Optimization, Control and Systems Engineering
Abstract

In this paper we deal with the controllability properties of a system of m coupled Stokes systems or m coupled Navier-Stokes systems. We show the null-controllability of such systems in the case where the coupling is in a cascade form and when the control acts only on one of the systems. Moreover, we impose that this control has a vanishing component so that we control a m × N state (corresponding to the velocities of the fluids) by N — 1 distributed scalar controls. The proof of the controllability of the coupled Stokes systems is based on a Carleman estimate for the adjoint system. The local null-controllability of the coupled Navier-Stokes systems is then obtained by means of the source term method and a Banach fixed point.

Published
2023
Full Text
View/download PDF

8. $$L^{p}$$ 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
Full Text
View/download PDF

9. 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
Full Text
View/download PDF

10. Boundary stabilization of a fluid-rigid body interaction system

Author

Takéo Takahashi and Mehdi Badra

Subjects
Physics, 0209 industrial biotechnology, Change of variables, 020208 electrical & electronic engineering, Boundary (topology), Newton's laws of motion, 02 engineering and technology, Mechanics, Rigid body, Domain (mathematical analysis), Physics::Fluid Dynamics, Nonlinear system, 020901 industrial engineering & automation, Control and Systems Engineering, Position (vector), 0202 electrical engineering, electronic engineering, information engineering, Compressibility
Abstract

Let us consider a fluid-rigid body interaction system. We are interested in the feedback stabilization of this system by using a finite-dimensional control localized on the interface between the structure and the fluid. The fluid is assumed to be viscous and incompressible and to follow the Navier-Stokes system and we consider for the rigid body the Newton laws. We follow a general method for the stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domain is moving with time. We prove that for small initial velocities and if the initial position and the final position are close, we can stabilize the position and the velocity of the rigid body and the velocity of the fluid.

Published
2019
Full Text
View/download PDF

11. 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
Full Text
View/download PDF

12. 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
Full Text
View/download PDF

13. Analysis of a system modelling the motion of a piston in a viscous gas

Author

Marius Tucsnak, Debayan Maity, Takéo Takahashi, 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), 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), 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), and Tata Institute of Fundamental Research [Bombay] (TIFR)

Subjects
Physics, fluid-particle interaction, Compressible Navier-Stokes System, Applied Mathematics, 010102 general mathematics, Motion (geometry), Context (language use), Mechanics, Condensed Matter Physics, 01 natural sciences, Cylinder (engine), law.invention, 010101 applied mathematics, Computational Mathematics, Piston, law, Free boundary problem, Compressibility, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, Uniqueness, 0101 mathematics, Mathematical Physics
Abstract

International audience; We study a free boundary problem modelling the motion of a piston in a viscous gas. The gas-piston system fills a cylinder with fixed extremities, which possibly allow gas from the exterior to penetrate inside the cylinder. The gas is modeled by the 1D compressible Navier-Stokes system and the piston motion is described by the second Newton's law. We prove the existence and uniqueness of global in time strong solutions. The main novelty brought in by our results is that they include the case on nonhomogeneous boundary conditions which, as far as we know, have not been studied in this context. Moreover, even for homogeneous boundary conditions, our results require less regularity of the initial data than those obtained in previous works.

Published
2016
Full Text
View/download PDF

14. 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
Full Text
View/download PDF

15. 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
Full Text
View/download PDF

16. 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
Full Text
View/download PDF

17. 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
Full Text
View/download PDF

18. 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
Full Text
View/download PDF

19. On the Fattorini criterion for approximate controllability and stabilizability of parabolic 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), 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), CORIDA, 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)-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, parabolic equation, 02 engineering and technology, Type (model theory), 01 natural sciences, Set (abstract data type), coupled, Continuation, Mathematics - Analysis of PDEs, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], 020901 industrial engineering & automation, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Stokes and MHD system, 0101 mathematics, Mathematics, Partial differential equation, 010102 general mathematics, Linear system, stabilizability, Approximate controllability, AMS subject classifications 93B05, 93D15, 35Q30, 76D05, 76D07, 76D55, 93B52, 93C20, Controllability, Computational Mathematics, Nonlinear system, Control and Systems Engineering, Bounded function, finite dimensional control, Analysis of PDEs (math.AP)
Abstract

International audience; In this paper, we consider the well-known Fattorini's criterion for approximate controllability of infinite dimensional linear systems of type $y'=A y+Bu$. We precise the result proved by H. O. Fattorini in \cite{Fattorini1966} for bounded input $B$, in the case where $B$ can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini's criterion is satisfied and if the set of geometric multiplicities of $A$ is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in using the Fattorini's criterion to obtain the feedback stabilizability of linear and nonlinear parabolic systems with feedback controls in a finite dimensional space. In particular, for systems described by partial differential equations, such a criterion reduces to a unique continuation theorem for a stationary system. We illustrate such a method by tackling some coupled Navier-Stokes type equations (MHD system and micropolar fluid system) and we sketch a systematic procedure relying on Fattorini's criterion for checking stabilizability of such nonlinear systems. In that case, the unique continuation theorems rely on local Carleman inequalities for stationary Stokes type systems.

Published
2014
Full Text
View/download PDF

20. The Motion of a Fluid-Rigid Ball System at the Zero Limit of the Rigid Ball Radius

Author

Takéo Takahashi, Ana L. Silvestre, Center for Computacional and Stochastic Mathematics (CEMAT), Instituto Superior Técnico, Universidade Técnica de Lisboa (IST), 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), CORIDA, 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)-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
Mechanical Engineering, Rotation, Rigid body, Physics::Fluid Dynamics, Fluid particle, symbols.namesake, Mathematics (miscellaneous), Classical mechanics, Flow velocity, symbols, Fluid dynamics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Ball (mathematics), Arch, Analysis, Lagrangian, Mathematics
Abstract

We study the limiting motion of a system of rigid ball moving in a Navier–Stokes fluid in \({\mathbb{R}^3}\) as the radius of the ball goes to zero. Recently, Dashti and Robinson solved this problem in the two-dimensional case, in the absence of rotation of the ball (Dashti and Robinson in Arch Rational Mech Anal 200:285–312, 2011). This restriction was caused by the difficulty in obtaining appropriate uniform bounds on the second order derivatives of the fluid velocity when the rigid body can rotate. In this paper, we show how to obtain the required uniform bounds on the velocity fields in the three- dimensional case. These estimates then allow us to pass to the zero limit of the ball radius and show that the solution of the coupled system converges to the solution of the Navier–Stokes equations describing the motion of only fluid in the whole space. The trajectory of the centre of the ball converges to a fluid particle trajectory, which justifies the use of rigid tracers for finding Lagrangian paths of fluid flow.

Published
2013
Full Text
View/download PDF

21. 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
Full Text
View/download PDF

22. 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
Full Text
View/download PDF

23. 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
Full Text
View/download PDF

24. 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
Full Text
View/download PDF

25. 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
Full Text
View/download PDF

26. 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
Full Text
View/download PDF

27. Towards the simulation of dense suspensions: a numerical tool

Author

Sébastien Martin, Bertrand Maury, Sylvain Faure, Takéo Takahashi, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-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), 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
Materials science, Interaction forces, Stokesian dynamics, Nanotechnology, Mechanics, Viscous liquid, 01 natural sciences, 010305 fluids & plasmas, Contact force, Physics::Fluid Dynamics, Condensed Matter::Soft Condensed Matter, Rheology, 0103 physical sciences, Lubrication, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], SPHERES, 010306 general physics, Suspension (vehicle)
Abstract

International audience; We present a numerical tool which aims at investigating the rheology of dense suspensions of entities such as spheres, red blood cells, polymer chains, or any kind of rigid or deformable bodies, in a viscous fluid. We shall pay a special attention to the short-range interactions between those entities (contact forces, lubrication forces). As for the fluid itself, our strategy consists in avoiding the direct and costly solution of the Stokes equations by integrating only the interaction forces which are likely to play a significant role in the overall behaviour of the suspension, in the spirit of Stokesian Dynamics. We present some preliminary results for suspensions of spheres, Red Blood Cells, and polymer-like chains.

Published
2009
Full Text
View/download PDF

28. 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
Full Text
View/download PDF

29. 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
Full Text
View/download PDF

30. 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
Full Text
View/download PDF

31. 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
Full Text
View/download PDF

32. 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
Full Text
View/download PDF

33. 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
Full Text
View/download PDF

35. Liquid jet generation and break-up

Author

Céline Baranger, G. Baudin, L. Boudin, Bruno Després, Frédéric Lagoutière, E. Lapébie, and Takéo Takahashi

Subjects
010101 applied mathematics, 010102 general mathematics, 0101 mathematics, 01 natural sciences
Published
2005
Full Text
View/download PDF

36. The electrostatic contribution in heterogeneous nucleation theory: Pure liquids

Author

Takéo Takahashi and W.A Tiller

Subjects
Materials science, General Engineering, Nucleation, Physical chemistry, Thermodynamics, Work function, Substrate (electronics), Electron, Electrical conductor, Surface energy, Catalysis, Conductor
Abstract

The electrostatic contribution, γe, to the total interfacial energy, γ, is negative and large in magnitude when two good conductors are in contact; when ¦ γ e ¦ (calculated) is large, the experimental value of the solid-liquid interfacial energy is small. When a conductor is in contact with a non-conductor, ¦ γ e ¦ is small. For metals that contract upon freezing, a good catalyst for nucleation must be a reasonable conductor and must have an electron work function that is greater than that of the average of the solid and liquid. For metals that expand upon freezing, the work function of the catalyst must be less than that of the average of the solid and liquid. These predictions seem to be in qualitative agreement with all the published experimental data on substrate effectiveness for heterogeneous nucleation.

Published
1969
Full Text
View/download PDF

37. Feedback stabilization of a fluid-rigid body interaction system

Author

Badra, M. and Takéo Takahashi

Subjects
74F10, 76D55, Applied Mathematics, 93D15, 35Q35, Analysis, 76D05
Abstract

We study the feedback stabilization of a system composed by an incompressible viscous fluid and a rigid body. We stabilize the position and the velocity of the rigid body 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 first result concerns weak solutions in the two-dimensional case, 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. This additional difficulty leads to the assumption that the initial position of the rigid body is the position associated to the stationary state. Without this hypothesis, we work with strong solutions, and to deal with compatibility conditions at the initial time, we use finite dimensional dynamical controls. We again prove that for initial data close to the stationary state, we can stabilize the position and the velocity of the rigid body and the velocity of the fluid. In the three dimensional case, we also obtain the local stabilization of strong solutions with finite dimensional dynamical controls.

38. Convergence results for a semilinear problem and for a Stokes problem in a periodic geometry

Author

Baillet, S., Henrot, A., Takéo Takahashi, 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), IFP Energies nouvelles (IFPEN), 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), Henrot, Antoine, 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
Mathematics::Analysis of PDEs, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]
Abstract

International audience; In this paper, we study the asymptotic behavior of the solution of a semilinear problem and of a Stokes problem, with periodic data, when the size of the domain increases. In particular, we prove exponential convergence to the solution of the corresponding problem with periodic boundary conditions.

Catalog

Books, media, physical & digital resources
See catalog results