Null controllability for distributed systems with time-varying constraint and applications to parabolic-like equations

Abstract

<p style='text-indent:20px;'>We consider the null controllability problem fo linear systems of the form <inline-formula><tex-math id="M1">\begin{document}$ y'(t) = Ay(t)+Bu(t) $\end{document}</tex-math></inline-formula> on a Hilbert space <inline-formula><tex-math id="M2">\begin{document}$ Y $\end{document}</tex-math></inline-formula>. We suppose that the control operator <inline-formula><tex-math id="M3">\begin{document}$ B $\end{document}</tex-math></inline-formula> is bounded from the control space <inline-formula><tex-math id="M4">\begin{document}$ U $\end{document}</tex-math></inline-formula> to a larger extrapolation space containing <inline-formula><tex-math id="M5">\begin{document}$ Y $\end{document}</tex-math></inline-formula>. The control <inline-formula><tex-math id="M6">\begin{document}$ u $\end{document}</tex-math></inline-formula> is constrained to lie in a time-varying bounded subset <inline-formula><tex-math id="M7">\begin{document}$ \Gamma(t) \subset U $\end{document}</tex-math></inline-formula>. From a general existence result based on a selection theorem, we obtain various properties on local and global constrained null controllability. The existence of the time optimal control is established in a general framework. When the constraint set <inline-formula><tex-math id="M8">\begin{document}$ \Gamma (t) $\end{document}</tex-math></inline-formula> contains the origin in its interior at each <inline-formula><tex-math id="M9">\begin{document}$ t&gt;0 $\end{document}</tex-math></inline-formula>, the local constrained property turns out to be equivalent to a weighted dual observability inequality of <inline-formula><tex-math id="M10">\begin{document}$ L^{1} $\end{document}</tex-math></inline-formula> type with respect to the time variable. We treat also the problem of determining a steering control for general constraint sets <inline-formula><tex-math id="M11">\begin{document}$ \Gamma (t) $\end{document}</tex-math></inline-formula> in nonsmooth convex analysis context. Applications to the heat equation are treated for distributed and boundary controls under the assumptions that <inline-formula><tex-math id="M12">\begin{document}$ \Gamma (t) $\end{document}</tex-math></inline-formula> is a closed ball centered at the origin and its radius is time-varying.