Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach

Abstract

<p style='text-indent:20px;'>The aim of this paper is to study the problem of constrained controllability for distributed parabolic linear system evolving in spatial domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> using the <i>Reverse Hilbert Uniqueness Method</i> (RHUM approach) introduced by Lions in 1988. It consists in finding the control <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> that steers the system from an initial state <inline-formula><tex-math id="M3">\begin{document}$ y_{_{0}} $\end{document}</tex-math></inline-formula> to a state between two prescribed functions. We give some definitions and properties concerning this concept and then we resolve the problem that relays on computing a control with minimum cost in the case of <inline-formula><tex-math id="M4">\begin{document}$ \omega = \Omega $\end{document}</tex-math></inline-formula> and in the regional case where <inline-formula><tex-math id="M5">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> is a part of <inline-formula><tex-math id="M6">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>.</p>