Null controllability results for parabolic equations in unbounded domains

In this talk we present some results concerning the null controllability for a heat equation in unbounded domains. We characterize the conditions that must satisfy the weight function, introduced by Fursikov and Imanuvilov, in order to prove a global Carleman inequality for the adjoint problem and then to get a null controllability result. We give some examples of unbounded domains (Ω, ω) that satisfy these sufficient conditions. Finally, when Ω \ ω is bounded, we prove the null controllability of the semi-linear heat equation when the nonlinearity is slightly superlinear

Controllability results for cascade systems of m coupled parabolic PDEs by one control force

In this paper we will analyze the controllability properties of a linear coupled parabolic system of m equations when a unique distributed control is exerted on the system. We will see that, when a cascade system is considered, we can prove a global Carleman inequality for the adjoint system which bounds the global integrals of the variable ϕ = (ϕ1, . . . , ϕm)∗ in terms of a unique localized variable. As a consequence, we will obtain the null controllability property for the system with one control force.Dirección General de InvestigaciónConsejo Nacional de Ciencia y Tecnología (México)Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológico (Universidad Nacional Autónoma de México

Controllability of linear and semilinear non-diagonalizable parabolic systems

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

Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations

This paper is concerned with the null-exact controllability of a cascade system formed by a semilinear heat and a semilinear wave equation in a cylinder Ω×(0, T). More precisely, we intend to drive the solution of the heat equation (resp. the wave equation) exactly to zero (resp. exactly to a prescribed but arbitrary final state). The control acts only on the heat equation and is supported by a set of the form ω × (0, T), where ω ⊂ Ω. In the wave equation, the restriction of the solution to the heat equation to another set O × (0, T) appears. The nonlinear terms are assumed to be globally Lipschitz-continuous. In the main result in this paper, we show that, under appropriate assumptions on T, ω and O, the equations are simultaneously controllable.Dirección General de Enseñanza SuperiorDirección General Asuntos del Personal Académico (Universidad Nacional Autónoma de México

Boundary controllability of parabolic coupled equations

This paper is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional parabolic equations are considered. We show that, in this framework, boundary controllability is not equivalent and is more complex than distributed controllability. In our main result, we provide necessary and sufficient conditions for the null controllability.Ministerio de Ciencia e InnovaciónDirección General Asuntos del Personal Académico (Univeridad Nacional Autónoma de México

Sur la contrôlabilité frontière des systèmes paraboliques non scalaires

This Note is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional linear parabolic equations are considered. We show that, with boundary controls, the situation is much more complex than for similar distributed control systems. In our main result, we provide necessary and sufficient conditions for null controllability.Cette Note concerne la contrôlabilité frontière des systèmes paraboliques linéaires non scalaires. Plus précisement, on considère un système de deux équations paraboliques linéaires de dimension 1 en espace. Nous montrons qu'il est beaucoup plus compliqué de contrôler sur une partie du bord que de le faire avec des contrôles distribués. Dans notre résultat principal, on donne des conditions nécessaires et suffisantes pour la contrôlabilité exacte à zéro

Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences

Let (A, D(A)) a diagonalizable generator of a C0−semigroup of contractions on a complex Hilbert space X, B2L(C, Y ), Y being some suitable extrapolation space of X, and u 2 L2 (0, T; C). Under some assumptions on the sequence of eigenvalues Λ = {λk}k≥1 ⇢ C of (A, D(A)), we prove the existence of a minimal time T0 2 [0, 1] depending on Bernstein’s condensation index of Λ and on B such that y 0 = Ay+Bu is null-controllable at any time T >T0 and not null-controllable for T <T0. As a consequence, we solve controllability problems of various systems of coupled parabolic equations. In particular, new results on the boundary controllability of one-dimensional parabolic systems are derived. These seem to be difficult to achieve using other classical tools.Ministerio de Ciencia e InnovaciónPrograma de Apoyo a Proyectos de Investigación e Innovación Tecnológica (Universidad Nacional Autónoma de México

A new relation between the condensation index of complex sequences and the null controllability of parabolic systems

In this note we present a new result that relates the condensation index of a sequence of complex numbers with the null controllability of parabolic systems. We show that a minimal time is required for controllability. The results are used to prove the boundary controllability of some coupled parabolic equations.Une nouvelle rélation entre l’indice de condensation de séquences complexes et la nulle contrôlabilitée des systèmes paraboliques.On annonce un résultat qui connecte l’indice de condensation des suites complexes et la nulle controlabilitée des systèmes paraboliques. On montre qu’un temps minimal est nécessaire pour controller, puis on voit le controle a zéro sur le bord de quelques équations paraboliques couplées.Agence Nationale de la Recherche (France)Ministerio de Ciencia e InnovaciónPrograma de Apoyo a Proyectos de Investigación e Innovación Tecnológica (Universidad Nacional Autónoma de México

Minimal time of controllability for some parabolic systems

The general aim of this talk is to show a phenomenon which arise when we deal with the null controllability properties of parabolic coupled systems: minimal time of controllability: 1 Boundary control: The condensation index of the complex sequence of eigenvalues of the corresponding matrix elliptic operator. 2 Distributed control: The action and the geometric position of the support of the coupling term when this support does not intersect the control domain ω

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

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