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

The aim of this paper is to study the problem of constrained controllability for distributed parabolic linear system evolving in spatial domain \begin{document}$ \Omega $\end{document} using the Reverse Hilbert Uniqueness Method (RHUM approach) introduced by Lions in 1988. It consists in finding the control \begin{document}$ u $\end{document} that steers the system from an initial state \begin{document}$ y_{_{0}} $\end{document} to a state between two prescribed functions. We give some definitions and properties concerning this concept and then we resolve the problem that relays on computing a control with minimum cost in the case of \begin{document}$ \omega = \Omega $\end{document} and in the regional case where \begin{document}$ \omega $\end{document} is a part of \begin{document}$ \Omega $\end{document} .