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In this study, the new traveling wave solutions resulting from the interaction of the long-short wave system were obtained by using the exp-function method. Two and three dimensional graphs of the obtained solutions were drawn by selecting the appropriate parameters. Density graphs of the solution functions were obtained and the interaction of the waves was observed.

Exact controllability for the wave equation on a metric graph consisting of a cycle and two attached edges is proven. One boundary and one internal control are used. At the internal vertices, delta-prime conditions are satisfied. As a second example, we examine a tripod controlled at the root and the junction, while the leaves are fixed. These examples are key to understanding controllability properties in general metric graphs.

In this paper, we study the null controllability of a coupled parabolic-hyperbolic system in one dimension with a single control using the moment method. More precisely, we consider a system coupling Kuramoto-Sivanshinsky-Korteweg-de Vries equation and transport equation through first order derivatives. We explore the null controllability of four different control systems with the control acting either on the periodic boundary or in some open subset of the interior of the domain with periodic boundary conditions. Depending on the position of the control, we get some regular periodic Sobolev space as the space of initial data for which the null controllability holds, provided the time is sufficiently large.

We present a new proof of the turnpike property for nonlinear optimal control problems, when the running target is a steady control-state pair of the underlying dynamics. Our strategy combines the construction of suboptimal quasi-turnpike trajectories via controllability, and a bootstrap argument, and does not rely on analyzing the optimality system or linearization techniques. This in turn allows us to address several optimal control problems for finite-dimensional, control-affine systems with globally Lipschitz (possibly nonsmooth) nonlinearities, without any smallness conditions on the initial data or the running target. These results are motivated by the large-layer regime of residual neural networks, commonly used in deep learning applications. We show that our methodology is applicable to controlled PDEs as well, such as the semilinear wave and heat equation with a globally Lipschitz nonlinearity, once again without any smallness assumptions.

In this paper, we investigate the problem for optimal control of a viscous generalized θ \theta -type dispersive equation (VG θ \theta -type DE) with weak dissipation. First, we prove the existence and uniqueness of weak solution to the equation. Then, we present the optimal control of a VG θ \theta -type DE with weak dissipation under boundary condition and prove the existence of optimal solution to the problem.

We explore the derivation of distributed parameter system evolution laws (and in particular, partial differential operators and associated partial differential equations, PDEs) from spatiotemporal data. This is, of course, a classical identification problem; our focus here is on the use of manifold learning techniques (and, in particular, variations of Diffusion Maps) in conjunction with neural network learning algorithms that allow us to attempt this task when the dependent variables, and even the independent variables of the PDE are not known a priori and must be themselves derived from the data. The similarity measure used in Diffusion Maps for dependent coarse variable detection involves distances between local particle distribution observations; for independent variable detection we use distances between local short-time dynamics. We demonstrate each approach through an illustrative established PDE example. Such variable-free, emergent space identification algorithms connect naturally with equation-free multiscale computation tools., Comment: specified the corresponding author

We consider a rod that is heated/cooled and sensed at multiple point locations. To stabilize it to a constant temperature we set up a Linear Quadratic Regulator that we explicitly solve by the method of completing the square to find the optimal linear state feedback for the point actuators. But we don’t assume that the whole state is measurable so we construct an infinite dimensional Kalman filter to estimate the whole state from a finite number of noisy point measurements. These two components yield a Linear Quadratic Gaussian (LQG) Synthesis for the heat equation under point actuation and point sensing. AFOSR FA9550-20-1-0318

In this article we study the null controllability of some abstract linear parabolic systems in tensor product spaces. This special structure allows us to reduce our controllability problem to a particular set of equations that looks like a moment problem, but that does not fall into the previous existing results of the literature. We transform this non standard moment problem into an infinite family of more usual moment problems, yet coupled one from each other. This reformulation is done with enough care to ensure that the resulting set of equations can be solved, with suitable estimates, by using the recent "block moment method". This is based on a careful analysis of the spectral structure of the underlying operator. We notably apply our abstract result to show how strong the influence of geometry can be: we provide an example of boundary controlled parabolic system on a rectangle domain which is null controllable in arbitrary small time if two perpendicular faces of the boundary are controlled, whereas it is never null controllable if the control acts on only one face.

In this work, we address the local controllability of a one-dimensional free boundary problem for a fluid governed by the viscous Burgers equation. The free boundary manifests itself as one moving end of the interval, and its evolution is given by the value of the fluid velocity at this endpoint. We prove that, by means of a control actuating along the fixed boundary, we may steer the fluid to constant velocity in addition to prescribing the free boundary's position, provided the initial velocities and interface positions are close enough, This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex. E.Z. has received funding from the Alexander von Humboldt-Professorship program, the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement NO. 694126-DyCon), the Transregio 154 Project “Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks” of the German DFG, grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain) and by the Air Force Office of Scientific Research (AFOSR) under Award NO. FA9550-18-1-0242

In these notes we collect some results from several of the authors' works in order to make available a single source and show how the approximate geometric methods for regulation have been developed, and how the control design strategy has evolved from the theoretical methods, involving the regulator equations, to what we now call the regularized controller. In between these two extremes we developed, in a series of works, a fairly rigorous analysis of the regularization scheme leading to the regularized dynamic regulator equations and an iterative scheme that produces very accurate tracking and disturbance rejection control laws. In our most recent work we have extended dynamic regulator equations to what we now refer to as the regularized controller. This new formulation has only recently being applied to examples including linear and nonlinear delay equations., 27 pages, 1 figure

We study the controllability of a coupled system of linear parabolic equations, with non-negativity constraint on the state. We establish two results of controllability to trajectories in large time: one for diagonal diffusion matrices with an "approximate" nonnegativity constraint, and a another stronger one, with "exact" nonnegativity constraint, when all the diffusion coefficients are equal and the eigenvalues of the coupling matrix have nonnegative real part. The proofs are based on a "staircase" method. Finally, we show that state-constrained controllability admits a positive minimal time, even with weaker unilateral constraint on the state., Comment: 20 pages

In this paper, we study the controllability of a nonlinear system of coupled second-and fourth-order parabolic equations. This system can be regarded as a simplification of the wellknown stabilized Kuramoto-Sivashinsky system. Using only one control applied on the boundary of the second-order equation, we prove that the local-null controllability of the system holds if the diffusion coefficient of the second-order equation is a quadratic irrational number.

International audience; Several engineering problems result in a PDE-constrained optimization problem that aims at finding the shape of a solid inside a fluid which minimizes a given cost function. These problems are categorized as Topology Optimization (TO) problems. In order to tackle these problems, the solid may be located with a penalization term added in the constraints equations that vanishes in fluid regions and becomes large in solid regions. This paper addresses a TO problem for anisothermal flows modelled by the steady-state incompressible Navier-Stokessystem coupled to an energy equation, with mixed boundary conditions, under the Boussinesq approximation. We first prove the existence and uniqueness of a solution to these equations as well as the convergence of its finite element discretization. Next, we show that our TO problem has at least one optimal solution for cost functions that satisfy general assumptions. The convergence of discrete optimum toward the continuous one is then proved as well as necessary first order optimality conditions. Eventually, all these results let us design a numerical algorithm to solve a TO problem approximating solids with piecewise constant thermal diffusivities also refered as multi-materials. A physical problem solved numericallyfor varying parameters concludes this paper.

We consider a thermoelastic laminated beam with a finite memory acting on the effective rotation angle. We establish an explicit and optimal stability estimate for the solution energy with minimal conditions on the relaxation function, from which the exponential and polynomial stability are just particular cases. This new result improves substantially many earlier results in the literature.

The Schlögl system is governed by a nonlinear reaction-diffusion partial differential equation with a cubic nonlinearity. In this paper, feedback laws of Pyragas-type are presented that stabilize the system in a periodic state with a given period and given boundary traces. We consider the system both with boundary feedback laws of Pyragas type and distributed feedback laws of Pyragas and classical type. Stabilization to periodic orbits is important for medical applications that concern Parkinson’s disease. The exponential stability of the closed loop system with respect to the L2-norm is proved. Numerical examples are provided.

In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equations.

This paper is concerned with the fault diagnosis problem for general linear heterodirectional hyperbolic ODE-PDE systems. A systematic solution is presented for additive time-varying actuator, process and sensor faults in the presence of disturbances. The faults and disturbances are represented by the solutions of finite-dimensional signal models, which allow to take a large class of signals into account. For disturbances, that are only bounded, a threshold for secured fault diagnosis is derived. By applying integral transformations to the system an algebraic fault detection equation to detect faults in finite time is obtained. The corresponding integral kernels result from the realization of a finite-time transition between a non-equilibrium initial state and a vanishing final state of a hyperbolic ODE-PDE system. For this new challenging problem, a systematic trajectory planning approach is presented. In particular, this problem is facilitated by mapping the kernel equations into backstepping coordinates and tracing the solution of the transition problem back to a simple trajectory planning. The fault diagnosis for a $4\times 4$ heterodirectional hyperbolic system coupled with a second order ODE demonstrates the results of the paper., 14 pages, 6 figures, submitted to Automatica

In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order necessary conditions relying on the concept of alternative costates and quasi-radial critical directions. The appendix provides an example illustrating the applicability of our results., Comment: This is the first part of a work on optimality conditions for a control problem of a semilinear heat equation. More precisely, the full version, available at arXiv:1906.00237v1, has been divided in two, resulting in the current manuscript (that corresponds to Part I) and arXiv:1909.05056 (which is Part II)

International audience; Given a linear control system in a Hilbert space with a bounded control operator, we establish a characterization of exponential stabilizability in terms of an observability inequality. Such dual characterizations are well known for exact (null) controllability. Our approach exploits classical Fenchel duality arguments and, in turn, leads to characterizations in terms of observability inequalities of approximately null controllability and of α-null controllability. We comment on the relationships between those various concepts, at the light of the observability inequalities that characterize them.

In previous work [23], Mustafa considered a viscoelastic laminated beam system with structural damping in the case of equal-speed wave propagations, and established explicit energy decay formula which gives the best decay rates. In this paper, we continue to consider the similar problems and establish the general decay result for the energy, to system with structural damping in the case of non-equal wave speeds and to system without structural damping in the case of equal wave speeds, respectively. For the first case, we use the second-order energy method to overcome the difficulty of estimating the non-equal speeds term. For the second case, we construct an appropriated perturbed functional to estimate $\|w_{t}\|^{2}_{2}$ so as to overcome the absence of structural damping.

We consider the motion of a rigid body immersed in a two-dimensional irrotational perfect incompressible fluid. The fluid is governed by the Euler equation, while the trajectory of the solid is given by Newton’s equation, the force term corresponding to the fluid pressure on the body’s boundary only. The system is assumed to be confined in a bounded domain with an impermeable condition on a part of the external boundary. The issue considered here is the following: is there an appropriate boundary condition on the remaining part of the external boundary (allowing some fluid going in and out the domain) such that the immersed rigid body is driven from some given initial position and velocity to some final position (in the same connected component of the set of possible positions as the initial position) and velocity in a given positive time, without touching the external boundary? In this paper we provide a positive answer to this question thanks to an impulsive control strategy. To that purpose we make use of a reformulation of the solid equation into an ODE of geodesic form, with some force terms due to the circulation around the body, as used by Glass, Munnier and Sueur (Invent. Math. 214:1 (2018), 171–287), and some extra terms here due to the external boundary control.

In this paper we survey the various implementations of a new data assimilation (downscaling) algorithm based on spatial coarse mesh measurements. As a paradigm, we demonstrate the application of this algorithm to the 3D Leray-α subgrid scale turbulence model. Most importantly, we use this paradigm to show that it is not always necessary to collect coarse mesh measurements of all the state variables that are involved in the underlying evolutionary system, in order to recover the corresponding exact reference solution. Specifically, we show that in the case of the 3D Leray-α model of turbulence, the solutions of the algorithm, constructed using only coarse mesh observations of any two components of the three-dimensional velocity field, and without any information on the third component, converge, at an exponential rate in time, to the corresponding exact reference solution of the 3D Leray-α model. This study serves as an addendum to our recent work on abridged continuous data assimilation for the 2D Navier-Stokes equations. Notably, similar results have also been recently established for the 3D viscous Planetary Geostrophic circulation model in which we show that coarse mesh measurements of the temperature alone are sufficient for recovering, through our data assimilation algorithm, the full solution; i.e. the three components of velocity vector field and the temperature. Consequently, this proves the Charney conjecture for the 3D Planetary Geostrophic model; namely, that the history of the large spatial scales of temperature is sufficient for determining all the other quantities (state variables) of the model.

International audience; In this paper, we study the insensitizing control problem in the discrete setting of finite-differences. We prove the existence of a control that insensitizes the norm of the observed solution of a 1-D semi-discrete parabolic equation. We derive a (relaxed) observability estimate that yields a controllability result for the cascade system arising in the insensitizing control formulation. Moreover, we deal with the problem of computing numerical approximations of insensitizing controls for the heat equation by using the Hilbert Uniqueness Method (HUM). We present various numerical illustrations.

In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is believed to be accurate when the flow velocity is too large for Darcy's law to be valid, and additionally the porosity is not too small. The algorithm is inspired by ideas developed for designing finite-parameters feedback control for dissipative systems. It aims to obtaining improved estimates of the state of the physical system by incorporating deterministic or noisy measurements and observations. Specifically, the algorithm involves a feedback control that nudges the large scales of the approximate solution toward those of the reference solution associated with the spatial measurements. In the first part of the paper, we present few results of existence and uniqueness of weak and strong solutions of the 3D BFeD system. The second part is devoted to the setting and convergence analysis of the data assimilation algorithm.

Let the matrix operator $L = D\partial_{xx} + q(x)A_0$, with $D = diag(1, \nu)$, $\nu \neq 1$, $q \in L^{\infty} (0, $\pi$)$, and $A_0$ is a Jordan block of order 1. We analyze the boundary null controllability for system $y_t - Ly = 0$. When $ \nu \notin \mathbb{Q} ^*_+ $ and $q(x) = 1$, $x $\in$ (0, $\pi$)$, there exists a family of root vectors of $(L * , D(L *)) $ forming a Riesz basis, moreover, F. Ammar Khodja, A.Benabdallah, M.Gonzalez-Burgos, L.Teresa, show the existence of a minimal time of control depending on condensation of eigenvalues of $(L^* , D(L^*))$. But there exists $q \in L^{\infty} (0, $\pi$)$ such that the family of eigenfunctions of $(L^* , D(L^*))$ is complete but it is not a Riesz basis. In this framework new phenomena arise : simultaneous condensation of eigenvalues and eigenfunctions. We prove the existence of a minimal time $T_0 \in [0, +\infty]$ depending on the condensation of eigenvalues and associated eigenfunctions of $(L^* , D(L^*))$, such that the corresponding system is null controllable at any time $T > T_0$ and is not if $T < T_0$., Comment: ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, In press, 27, pp.S29

This is a post-peer-review, pre-copyedit version of an article published in Numerische Mathematik. The final authenticated version is available online at: https://doi.org/10.1007/s00211-018-1005-z, We adapt and apply greedy methods to approximate in an efficient way the optimal controls for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling the parameter-space to compute controls for each parameter value. The same method can be adapted for parabolic control problems, but this leads to greedy selections of the realizations of the parameters that depend on the initial datum under consideration. The turnpike property (which ensures that parabolic optimal control problems behave nearly in a static manner when the control horizon is long enough) allows using the elliptic greedy choice of the parameters in the parabolic setting too. We present various numerical experiments and an extensive discussion of the efficiency of our methodology for parabolic control and indicate a number of open problems arising when analyzing the convergence of the proposed algorithms, This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694126-DyCon). Part of this research was done while the second author visited DeustoTech and Univesity of Deusto with the support of the DyCon project. The second author was also partially supported by Croatian Science Foundation under ConDyS Project, IP-2016-06-2468. The work of the third author was partially supported by the Grants MTM2014-52347, MTM2017-92996 of MINECO (Spain) and ICON of the French ANR

In previous work, Lo and Tatar studied the exponential decay for a laminated beam with viscoelastic damping acting on the effective rotation angle in the case of equal-speed wave propagations. In this paper, we continue consideration of the same problem in the case of non-equal wave speeds. In this case, the main difficulty is how to estimate the non-equal speed term. To overcome this difficulty, the second-order energy method introduced in Guesmia and Messaoudi seems to be the best choice for our problem. For a wide class of relaxation functions, we establish the general decay result for the energy without any kind of internal or boundary control.

In this paper, we study initial-boundary value problems for the Cahn--Hilliard system with convection and nonconvex potential, where dynamic boundary conditions are assumed for both the associated order parameter and the corresponding chemical potential. While recent works addressed the case of viscous Cahn--Hilliard systems, the `pure' nonviscous case is investigated here. In its first part, the paper deals with the asymptotic behavior of the solutions as time approaches infinity. It is shown that the $\omega$-limit of any trajectory can be characterized in terms of stationary solutions, provided the initial data are sufficiently smooth. The second part of the paper deals with the optimal control of the system by the fluid velocity. Results concerning existence and first-order necessary optimality conditions are proved. Here, we have to restrict ourselves to the case of everywhere defined smooth potentials. In both parts of the paper, we start from corresponding known results for the viscous case, derive sufficiently strong estimates that are uniform with respect to the (positive) viscosity parameter, and then let the viscosity tend to zero to establish the sought results for the nonviscous case., Comment: arXiv admin note: text overlap with arXiv:1803.04318

We consider the problem of boundary control for a vibrating string with $N$ interior point masses. We assume the control is at the left end, and the string is fixed at the right end. Singularities in waves are "smoothed" out to one order as they cross a point mass. We characterize the reachable set for a $L^2$ control. The control problem is reduced to a moment problem, which is then solved using the theory of exponential divided differences in tandem with unique shape and velocity controllability results.

Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems arerequired. We present preconditioned iterative techniques for solving a number of these problems using Krylov subspace methods, considering in what circumstances one may predict rapid convergence of the solvers in theory, as well as the solutions observed from practical computations.

The paper, authored by N. Anantharaman and M. Léautaud, includes an appendix by S.Nonnenmacher; We address the decay rates of the energy for the damped wave equation when the damping coefficient $b$ does not satisfy the geometric control condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove in an abstract setting that the observability of the Schrödinger equation implies that the solutions of the damped wave equation decay at least like 1$\sqrt{t}$ (which is a stronger rate than the general logarithmic one predicted by the Lebeau theorem). Second, we focus on the 2-dimensional torus. We prove that the best decay one can expect is 1/$t$, as soon as the damping region does not satisfy GCC. Conversely, for smooth damping coefficients $b$ vanishing flatly enough, we show that the semigroup decays at least like 1/$t^{1−\epsilon}$ , for all $\epsilon$ > 0. The proof relies on a second microlocalization around trapped directions, and resolvent estimates. In the case where the damping coefficient is a characteristic function of a strip (hence discontinuous), Stéphane Nonnenmacher computes in an appendix part of the spectrum of the associated damped wave operator, proving that the semigroup cannot decay faster than 1/$t^{2/3}$. In particular, our study emphasizes that the decay rate highly depends on the way $b$ vanishes

We introduce here a simple finite-dimensional feedback control scheme for stabilizing solutions of infinite-dimensional dissipative evolution equations, such as reaction-diffusion systems, the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. The designed feedback control scheme takes advantage of the fact that such systems possess finite number of determining parameters (degrees of freedom), namely, finite number of determining Fourier modes, determining nodes, and determining interpolants and projections. In particular, the feedback control scheme uses finitely many of such observables and controllers. This observation is of a particular interest since it implies that our approach has far more reaching applications, in particular, in data assimilation. Moreover, we emphasize that our scheme treats all kinds of the determining projections, as well as, the various dissipative equations with one unified approach. However, for the sake of simplicity we demonstrate our approach in this paper to a one-dimensional reaction-diffusion equation paradigm.

In this paper, we study the local stabilization of one dimensional compressible Navier-Stokes equations around a constant steady solution $(\rho_s, u_s)$, where $\rho_s>0, u_s\neq 0$. In the case of periodic boundary conditions, we determine a distributed control acting only in the velocity equation, able to stabilize the system, locally around $(\rho_s, u_s)$, with an arbitrary exponential decay rate. In the case of Dirichlet boundary conditions, we determine boundary controls for the velocity and for the density at the inflow boundary, able to stabilize the system, locally around $(\rho_s, u_s)$, with an arbitrary exponential decay rate.

Author(s): Farhat, Aseel; Lunasin, Evelyn; Titi, Edriss S | Abstract: Analyzing the validity and success of a data assimilation algorithm when some state variable observations are not available is an important problem in meteorology and engineering. We present an improved data assimilation algorithm for recovering the exact full reference solution (i.e. the velocity and temperature) of the 3D Planetary Geostrophic model, at an exponential rate in time, by employing coarse spatial mesh observations of the temperature alone. This provides, in the case of this paradigm, a rigorous justification to an earlier conjecture of Charney which states that temperature history of the atmosphere, for certain simple atmospheric models, determines all other state variables.

Motivated by the presence of a finite number of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages for dissipative dynamical systems, we present here a continuous data assimilation algorithm for three-dimensional viscous hydrodynamic models. However, to validate the convergence of this algorithm our proofs require the existence of uniform global bounds on the gradients of the solutions of the underlying system in terms of certain combinations of the physical parameters (such as kinematic viscosity, the size of the domain and the forcing term). Therefore our proofs cannot be applied to the three-dimensional Navier-Stokes equations; instead we demonstrate the implementation of this algorithm, for instance, in the context of the three-dimensional Navier-Stokes-α equations. This algorithm consists of introducing a nudging process through a general type of approximation interpolation operator (which is constructed from observational measurements) that synchronizes the large spatial scales of the approximate solutions with those of unknown solutions of the Navier-Stokes-α equations corresponding to these measurements. Our main result provides conditions on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, which is obtained from this collected data, converges to the unknown reference solution over time. These conditions are given in terms of the physical parameters.

Motivated by the presence of a finite number of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages for dissipative dynamical systems, we present here a continuous data assimilation algorithm for three-dimensional viscous hydrodynamic models. However, to validate the convergence of this algorithm our proofs require the existence of uniform global bounds on the gradients of the solutions of the underlying system in terms of certain combinations of the physical parameters (such as kinematic viscosity, the size of the domain and the forcing term). Therefore our proofs cannot be applied to the three-dimensional Navier-Stokes equations; instead we demonstrate the implementation of this algorithm, for instance, in the context of the three-dimensional Navier-Stokes-α equations. This algorithm consists of introducing a nudging process through a general type of approximation interpolation operator (which is constructed from observational measurements) that synchronizes the large spatial scales of the approximate solutions with those of unknown solutions of the Navier-Stokes-α equations corresponding to these measurements. Our main result provides conditions on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, which is obtained from this collected data, converges to the unknown reference solution over time. These conditions are given in terms of the physical parameters.

International audience; We investigate the nonhomogeneous initial boundary value problem for the Camassa-Holm equation on an interval. We provide a local in time existence theorem and a weak strong uniqueness result. Next we establish a result on the global asymptotic stabilization problem by means of a boundary feedback law.

The general theory on exact boundary controllability for general first order quasilinear hyperbolic systems requires that the characteristic speeds of system do not vanish. This paper deals with exact boundary controllability, when this is not the case. Some important models are also shown as applications of the main result. The strategy uses the return method, which allows in certain situations to recover non zero characteristic speeds., Comment: 20 pages

This paper studies the exact boundary controllability of the semi-linear Schrodinger equation posed on a bounded domain Ω ⊂ R n with either the Dirichlet boundary conditions or the Neumann boundary conditions. It is shown that if s > n 2 , or 0 ⩽ s n 2 with 1 ⩽ n 2 + 2 s , or s = 0 , 1 with n = 2 , then the systems are locally exactly controllable in the classical Sobolev space H s ( Ω ) around any smooth solution of the cubic Schrodinger equation.

Dynamics of the hybrid systems of aerial cable-ways is investigated. The eigenvalue problems are considered for such hybrid systems with different assumptions. An overview of different methods for eigenvalue problems is given. In the research, the method of the normal fundamental systems is applied, which turns out to be very effective for the considered problems. Changes of dynamical characteristics of the systems depending on the controlled parameter are studied., Accepted (15-May-2006) to the Proceedings of the "International Conference of Hybrid Systems and Applications", The University of Louisiana, Lafayette, LA, USA, May 22-26 2006, to be published in the journal "Nonlinear Analysis: Hybrid Systems and Applications"

In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this boundary, a scalar unknown (buoyancy or surface temperature of the fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging is done on this two-dimensional model, yet ultimately synchronizes the streamfunction of the three-dimensional flow. The main analytical difficulties are due to the presence of a nonlocal dissipative operator in the surface quasi-geostrophic equation. This is overcome by exploiting a suitable partition of unity, the modulus of continuity characterization of Sobolev space norms, and the Littlewood-Paley decomposition to ultimately establish various boundedness and approximation-of-identity properties for the observation operators., Comment: 28 pages, referee comments incorporated, references added, abstract and introduction modified, main theorems cover full subcritical range of dissipation, certain boundedness properties of observation operators extended

We adapt a previously introduced continuous in time data assimilation (downscaling) algorithm for the two-dimensional Navier--Stokes equations to the more realistic case when the measurements are obtained discretely in time and may be contaminated by systematic errors. Our algorithm is designed to work with a general class of observables, such as low Fourier modes and local spatial averages over finite volume elements. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution, and the time step between successive measurements, we obtain an asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors. A stationary statistical analysis of our discrete data assimilation algorithm is also provided.

International audience; This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices.The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a “backstepping” method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control.Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.

We apply the Postprocessing Galerkin method to a recently introduced continuous data assimilation (downscaling) algorithm for obtaining a numerical approximation of the solution of the two-dimensional Navier-Stokes equations corresponding to given measurements from a coarse spatial mesh. Under suitable conditions on the relaxation (nudging) parameter, the resolution of the coarse spatial mesh and the resolution of the numerical scheme, we obtain uniform in time estimates for the error between the numerical approximation given by the Postprocessing Galerkin method and the reference solution corresponding to the measurements. Our results are valid for a large class of interpolant operators, including low Fourier modes and local averages over finite volume elements. Notably, we use here the 2D Navier-Stokes equations as a paradigm, but our results apply equally to other evolution equations, such as the Boussinesq system of Benard convection and other oceanic and atmospheric circulation models.

In this paper we propose a continuous data assimilation (downscaling) algorithm for the B\'enard convection in porous media using only coarse mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed (finite dimensional projection of) temperature data. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements., Comment: arXiv admin note: text overlap with arXiv:1504.05978

We present a computational study of a simple finite-dimensional feedback control algorithm for stabilizing solutions of infinite-dimensional dissipative evolution equations such as reaction-diffusion systems, the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. This feedback control scheme takes advantage of the fact that such systems possess finite number of determining parameters or degrees of freedom, namely, finite number of determining Fourier modes, determining nodes, and determining interpolants and projections. In particular, the feedback control scheme uses finitely many of such observables and controllers that are acting on the coarse spatial scales. We demonstrate our numerical results for the stabilization of the unstable zero solution of the 1D Chafee-Infante equation and 1D Kuramoto-Sivashinksky equation. We give rigorous stability analysis for the feedback control algorithm and derive sufficient conditions relating the control parameters and model parameter values to attune to the control objective., 32 pages

The speed of propagation of a premixed turbulent flame correlates with the intensity of the turbulence encountered by the flame. One consequence of this property is that premixed flames in both laboratory experiments and practical combustors require some type of stabilization mechanism to prevent blow-off and flashback. The stabilization devices often introduce a level of geometric complexity that is prohibitive for detailed computational studies of turbulent flame dynamics. Furthermore, the stabilization introduces additional fluid mechanical complexity into the overall combustion process that can complicate the analysis of fundamental flame properties. To circumvent these difficulties we introduce a simple, heuristic feedback control algorithm that allows us to computationally stabilize a turbulent premixed flame in a simple geometric configuration. For the simulations, we specify turbulent inflow conditions and dynamically adjust the integrated fueling rate to control the mean location of the flame in the domain. We outline the numerical procedure, and illustrate the behavior of the control algorithm on methane flames at various equivalence ratios in two dimensions. The simulation data are used to study the local variation in the speed of propagation due to flame surface curvature.

43 p.; International audience; We formulate and study robust control problems for a two-dimensional time-dependent Ginzburg-Landau model with Robin boundary conditions on phase-field parameter, which describes the phase transitions taking place in superconductor films with variable thickness. The objective of such study is to control the motion of vortices in the superconductor films by taking into account the influence of noises in data. Firstly, we introduce the perturbation problem of the nonlinear governing coupled system of equations (the deviation from the desired target). The existence and the uniqueness of the solution of the perturbation are proved as well as stability under mild assumptions. Afterwards, the robust control problems are formulated in the case when the control is in the external magnetic field and in the case when the control is in the initial condition of the vector potential. We show the existence of an optimal solution, and we also find necessary conditions for a saddle point optimality

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.

The paper deals with a distributed parameter system related to the so-called fixed-bed bioreactor. The original nonlinear partial differential system is linearized around the steady state. We find that the linearized system is not exactly controllable but it is approximatively controllable when certain algebraic equations hold. We apply frequency-domain methods (transfer function analysis) to consider a related output tracking problem. The input-output system can be formulated as a translation invariant pseudodifferential equation. A simulation shows that the calculation scheme is stable. An idea to use frequency-domain methods and certain pseudodifferential operators for parametrization of control systems of more general systems is pointed out.