Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition

Abstract

<p style='text-indent:20px;'>In the paper, the problems of controllability and approximate controllability are studied for the control system <inline-formula><tex-math id="M1">\begin{document}$ w_t = w_{xx} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ w_x(0,\cdot) = u $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ x&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ t\in(0,T) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ u\in L^\infty(0,T) $\end{document}</tex-math></inline-formula> is a control. It is proved that each initial state of the system is approximately controllable to each target state in a given time <inline-formula><tex-math id="M6">\begin{document}$ T $\end{document}</tex-math></inline-formula>. A necessary and sufficient condition for controllability in a given time <inline-formula><tex-math id="M7">\begin{document}$ T $\end{document}</tex-math></inline-formula> is obtained in terms of solvability of a Markov power moment problem. It is also shown that there is no initial state which is null-controllable in a given time <inline-formula><tex-math id="M8">\begin{document}$ T $\end{document}</tex-math></inline-formula>. Orthogonal bases are constructed in <inline-formula><tex-math id="M9">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ H_1 $\end{document}</tex-math></inline-formula>. Using these bases, numerical solutions to the approximate controllability problem are obtained. The results are illustrated by examples.