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

Banff International Research Station for Mathematical Innovation and Discovery

One of the main goal in control theory is to drive the state of the system to a given configuration using a control that act through a source term located inside the domain or through a boundary condition. The reference works for the control of linear parabolic problems are due to H.O.~Fattorini and D.L.~Russell in the 70's, \cite{FR1} for the one dimensional case and to A.V.~Fursikov, O.Yu.~Imanuvilov, \cite{FI} on one side and G.~Lebeau, L.~Robbiano, \cite{LR} on the other side both in the 90's for the multi-dimensional case. They established null-controllability of heat equations with distributed or boundary controls in any time and for any control domain. The aim of this talk is to give an overview on the recent results on the controllability of parabolic {\bf{systems}}. Through simple examples, I will show that new phenomena appear as minimal time of control, dependance on the location of the control. \subsection{Results} In this talk, I will focus on controllability of two simple examples in order to illustrate theses unexpected behavior in control of parabolic systems. \subsection{Boundary control for parabolic systems} $$ \label{Pbi} \left\{ \begin{array}{ll} \displaystyle \partial_t y - \left( D \partial^2_{xx} + A\right) y=0 & \hbox{in }Q=(0,\pi )\times (0,T), \\ \noalign{\smallskip} y(0,\cdot ) = Bv, \quad y(\pi ,\cdot )=0 & \hbox{on }(0,T), \\ \noalign{\smallskip} y(\cdot ,0)=y^{0}\ & \hbox{in }(0,\pi ), \end{array}% \right. $$ where $T>0$ is a given time, \begin{equation*} D = \left( \begin{array}{cc} 1 & 0 \\ 0 & d \end{array} \right) \ (\hbox{with }d>0), \quad A=\left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right) ,\quad B=\left(\begin{array}{cc}0\\1\end{array}\right)\in \mathcal M_{2,1}(\R)\end{equation*} \begin{theorem}[F.~Ammar Khodja, A.~Benabdallah, M.~Gonz\'alez-Burgos, L.~de Teresa, \cite{JFA}] \vskip 0.3cm Let $d\neq 1$ \begin{enumerate} \item $ \forall T>0:$ Approximate controllability if and only if $\sqrt{d}\not\in \Q $. \item $\exists T_0=c(\Lambda)\in [0,+\infty]$ such that \begin{enumerate} \item The system is null controllable at time $T$ if $\sqrt{d}\not\in \Q $ and $T>T_0$. \item Even if $\sqrt{d}\not\in \Q $, if $T<T_0$, the system is not null controllable at time $T$. \end{enumerate} \end{enumerate} \end{theorem} $c(\Lambda)$ is the index of condensation of the sequence $\Lambda=(k^2,dk^2)_{k\geq 1}$. \subsection{Geometrical dependance on the location of the control in a distributed control of parabolic equations} Let $q\in L^\infty(0,\pi)$, $Q= (0,\pi)\times (0,T)$. % \begin{equation}\label{s2x2} \left\{ \begin{array}{ll} y_t - y_{xx} + q (x) A y = B v_{\omega} & \hbox{in } Q , \\ \noalign{\smallskip} y (0, \cdot) = 0 , \quad y (\pi , \cdot) = 0 & \hbox{on } ( 0,T) , \\ \noalign{\smallskip} y( \cdot , 0) =y_{0}, & \hbox{in }( 0,\pi ) , \end{array} \right. \end{equation} % \begin{equation*} D = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \ , \quad A=\left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right) ,\quad B=\left(\begin{array}{cc}0\\1\end{array}\right) \end{equation*} \begin{theorem}[ F.~Ammar Khodja, A.~Benabdallah, M.~Gonz\'alez-Burgos, L.~de Teresa,\cite{ABGT:16} ] Let us consider $q\in L^{\infty }(0,\pi )$, a function satisfying\begin{equation*} \hbox{\rm Supp}\, q \cap \omega =\emptyset .\end{equation*} Let us define % \begin{equation} \label{mtime} T_{0}(q) := \limsup_{k \to \infty} \frac{\min \{-\log |I_{1,k}(q)|, -\log |I_{k}(q)| \}}{k^{2}}. \end{equation} Then, given $T>0$, one has: \begin{enumerate} \item Assume that $T>T_0(q)$. Then, system~\eqref{s2x2} is null controllable at time $T$. \item If $T<T_{0}(q)$, then system~\eqref{s2x2} is not null controllable at time $T$. \end{enumerate} \end{theorem}

Mathematics; Partial differential equations; Systems theory; control; Control/Optimization/Operation Research

video/mp4

Author affiliation: Aix Marseille University

2020-12-06

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/76666

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/