236 research outputs found

    Local null-controllability of a system coupling Kuramoto-Sivashinsky-KdV and elliptic equations

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    This paper deals with the null-controllability of a system of mixed parabolic-elliptic pdes at any given time T>0T>0. More precisely, we consider the Kuramoto-Sivashinsky--Korteweg-de Vries equation coupled with a second order elliptic equation posed in the interval (0,1)(0,1). We first show that the linearized system is globally null-controllable by means of a localized interior control acting on either the KS-KdV or the elliptic equation. Using the Carleman approach we provide the existence of a control with the explicit cost KeK/TKe^{K/T} with some constant K>0K>0 independent in TT. Then, applying the source term method and the Banach fixed point argument, we conclude the small-time local null-controllability result of the nonlinear systems. Besides, we also established a uniform null-controllability result for an asymptotic two-parabolic system (fourth and second order) that converges to the concerned parabolic-elliptic model when the control is acting on the second order pde

    Boundary approximate controllability of some linear parabolic systems

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    International audienceThis paper focuses on the boundary approximate controllability of two classes of linear parabolic systems, namely a system of n heat equations coupled through constant terms and a 2x2 cascade system coupled by means of a fi rst order partial diff erential operator with space-dependent coe fficients. For each system we prove a suffi cient condition in any space dimension and we show that this condition turns out to be also necessary in one dimension with only one control. For the system of coupled heat equations we also study the problem on rectangle, and we give characterizations depending on the position of the control domain. Finally, we exhibit a cascade system for which the distributed controllability holds whereas the boundary controllability does not. The method relies on a general characterization due to H.O. Fattorini

    New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence

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    We consider the null controllability problem for two coupled parabolic equations with a space-depending coupling term. We analyze both boundary and distributed null controllability. In each case, we exhibit a minimal time of control, that is to say, a time T0 ∈ [0, ∞] such that the corresponding system is null controllable at any time T > T0 and is not if T < T0. In the distributed case, this minimal time depends on the relative position of the control interval and the support of the coupling term. We also prove that, for a fixed control interval and a time τ0 ∈ [0, ∞], there exist coupling terms such that the associated minimal time is τ0.Ministerio de Economía y CompetitividadDirección General Asuntos del Personal Académico (Universidad Nacional Autónoma de México

    New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence

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    International audienceWe consider the null controllability problem for two coupled parabolic equations with a space-depending coupling term. We analyze both boundary and distributed null controllabil-ity. In each case, we exhibit a minimal time of control, that is to say, a time T0 ∈ [0, ∞] such that the corresponding system is null controllable at any time T > T0 and is not if T < T0. In the distributed case, this minimal time depends on the relative position of the control interval and the support of the coupling term. We also prove that, for a fixed control interval and a time τ0 ∈ [0, ∞], there exist coupling terms such that the associated minimal time is τ0

    Controllability of linear and semilinear non-diagonalizable parabolic systems

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    This paper is concerned with the controllability of some (linear and semilinear) nondiagonalizable parabolic systems of PDEs. We will show that the well known null controllability properties of the classical heat equation are also satisfied by these systems at least when there are as many scalar controls as equations and some (maybe technical) conditions are satisfied. We will also show that, in some particular situations, the number of controls can be reduced. The minimal amount is then determined by a Kalman rank condition.Ministerio de Ciencia e InnovaciónDirección General Asuntos del Personal Académico (Univeridad Nacional Autónoma de México

    Controlling linear and semilinear systems formed by one elliptic and two parabolic PDEs with one scalar control

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    In this paper, we prove controllability results for some linear and semilinear systems where we find two parabolic PDEs and one elliptic PDE and we act through one locally supported in space scalar control. The arguments rely on a careful analysis of the linear case and an application of an inverse function theorem. The facts that we act through a single scalar control and one of the PDEs has no time derivative are the main novelties and introduce several nontrivial difficulties.Ministerio de Economía y Competitivida

    Boundary controllability of some coupled parabolic systems with Robin or Kirchhoff conditions

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    Dans cette thèse, on étudie la contrôlabilité à zéro par le bord de quelques systèmes paraboliques linéaires couplés par des termes de couplage intérieur et/ou au bord. Le premier chapitre est une introduction à l'ensemble du manuscrit. Dans le deuxième chapitre, on rappelle les principaux concepts et résultats autour des notions de contrôlabilité qui seront utilisés dans la suite. Dans le troisième chapitre, on étudie principalement la contrôlabilité par le bord d'un système couplé 2x2 de type cascade avec des conditions au bord de Robin. En particulier, on prouve que les contrôles associés satisfont des bornes uniformes par rapport aux paramètres de Robin et convergent vers un contrôle de Dirichlet lorsque les paramètres de Robin tendent vers l'infini. Cette étude fournit une justification, dans le contexte du contrôle, de la méthode de pénalisation qui est couramment utilisée pour prendre en compte des données de Dirichlet peu régulières en pratique. Dans le quatrième et dernier chapitre, on étudie d'abord la contrôlabilité à zéro d'un système 2x2 en dimension 1 contenant des termes de couplage à la fois à l'intérieur et au bord du domaine. Plus précisément, on considère une condition de type Kirchhoff sur l'un des bords du domaine et un contrôle de Dirichlet sur l'autre bord, dans l'une ou l'autre des équations. En particulier, on montre que les propriétés de contrôle du système diffèrent selon que le contrôle agisse sur la première ou sur la seconde équation, et selon les valeurs du coefficient de couplage intérieur et du paramètre de Kirchhoff. On étudie ensuite un modèle 3x3 avec un ou deux contrôle(s) aux limites de Dirichlet à une extrémité et une condition de type Kirchhoff à l'autre extrémité ; ici la troisième équation est couplée (couplage intérieur) avec la première. Dans ce cas, on obtient ce qui suit : en considérant le contrôle sur la première équation, on a contrôlabilité conditionnelle dépendant des choix du coefficient de couplage intérieur et du paramètre de Kirchhoff, et en considérant le contrôle sur la deuxième équation, on obtient toujours une contrôlabilité positive. En revanche, considérer un contrôle sur la troisième équation conduit à un résultat de contrôlabilité négative. Dans cette situation, on a besoin de deux contrôles aux limites sur deux des trois équations pour retrouver la contrôlabilité. Enfin, on expose quelques études numériques basées sur l'approche pénalisée HUM pour illustrer les résultats théoriques, ainsi que pour tester d'autres exemples.In this thesis, we study the boundary null-controllability of some linear parabolic systems coupled through interior and/or boundary. We begin by giving an overall introduction of the thesis in Chapter 1 and we discuss some essentials about the notion of parabolic controllability in the second chapter. In Chapter 3, we investigate the boundary null-controllability of some 2x2 coupled parabolic systems in the cascade form where the boundary conditions are of Robin type. This case is considered mainly in space dimension 1 and in the cylindrical geometry. We prove that the associated controls satisfy suitable uniform bounds with respect to the Robin parameters, which let us show that they converge towards a Dirichlet control when the Robin parameters go to infinity. This is a justification of the popular penalization method for dealing with Dirichlet boundary data in the framework of the controllability of coupled parabolic systems. Coming to the Chapter 4, we first discuss the boundary null-controllability of some 2x2 parabolic systems in 1-D that contains a linear interior coupling with real constant coefficient and a Kirchhoff-type condition through which the boundary coupling enters in the system. The control is exerted on a part of the boundary through a Dirichlet condition on either one of the two state components. We show that the controllability properties vary depending on which component the control is being applied; the choices of interior coupling coefficient and the Kirchhoff parameter play a crucial role to deduce positive or negative controllability results. Thereafter, we study a 3x3 model with one or two Dirichlet boundary control(s) at one end and a Kirchhoff-type boundary condition at the other; here the third equation is coupled (interior) through the first component. In this case we obtain the following: treating the control on the first component, we have conditional controllability depending on the choices of interior coupling coefficient and the Kirchhoff parameter, while considering a control on the second component always provides positive result. But in contrast, putting a control on the third entry yields a negative controllability result. In this situation, one must need two boundary controls on any two components to recover the controllability. Further in the thesis, we pursue some numerical studies based on the penalized Hilbert Uniqueness Method (HUM) to illustrate our theoretical results and test other examples in the framework of interior-boundary coupled systems

    Numerical controllability of the wave equation through primal methods and Carleman estimates

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    This paper deals with the numerical computation of boundary null controls for the 1D wave equation with a potential. The goal is to compute an approximation of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. We do not use in this work duality arguments but explore instead a direct approach in the framework of global Carleman estimates. More precisely, we consider the control that minimizes over the class of admissible null controls a functional involving weighted integrals of the state and of the control. The optimality conditions show that both the optimal control and the associated state are expressed in terms of a new variable, the solution of a fourth-order elliptic problem defined in the space-time domain. We first prove that, for some specific weights determined by the global Carleman inequalities for the wave equation, this problem is well-posed. Then, in the framework of the finite element method, we introduce a family of finite-dimensional approximate control problems and we prove a strong convergence result. Numerical experiments confirm the analysis. We complete our study with several comments

    Inverse problems for linear hyperbolic equations using mixed formulations

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    We introduce in this document a direct method allowing to solve numerically inverse type problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the wave equation posed in Ω×(0,T)\Omega\times (0,T) - Ω\Omega a bounded subset of RN\mathbb{R}^N - from a partial distributed observation. We employ a least-squares technique and minimize the L2L^2-norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We prove the strong convergence of the approximation and then discussed several examples for N=1N=1 and N=2N=2. The problem of the reconstruction of both the state and the source term is also addressed

    Numerical null controllability of the 1D heat equation: Carleman weights an duality

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    This paper deals with the numerical computation of distributed null controls for the 1D heat equation. The goal is to compute a control that drives (a numerical approximation of) the solution from a prescribed initial state at t = 0 exactly to zero at t = T. We extend the earlier contribution of Carthel, Glowinski and Lions [5], which is devoted to the computation of minimal L2-norm controls. We start from some constrained extremal problems introduced by Fursikov and Imanuvilov [15]) and we apply appropriate duality techniques. Then, we introduce numerical approximations of the associated dual problems and we apply conjugate gradient algorithms. Finally, we present several experiments, we highlight the in uence of the weights and we analyze this approach in terms of robustness and e fficiency
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