On the optimal control of circular membranes
This paper investigates the controllability problem for the boundary control of a circular membrane. The state equation is given by \[ \begin{matrix} z_{tt}=\Delta z \quad & \text{on } (0,T) \times \Omega, \\ {\partial z \over \partial \nu}=u & \text{on } (0,T) \times \partial \Omega, \\ z=z_ t=0 & \text{on } \{0\} \times \Omega, \end{matrix} \] where \(\Omega=\bigl\{ (x,y) | x^ 2+y^ 2<1\bigr\}\) and \(u \in L^ 2\bigl( \partial \Omega \times(0,T) \bigr)\) is the control. The main result in this paper is that the reachability set \(D(T)= \biggl\{ \bigr( z(T,\cdot,\cdot),z_ t(T,\cdot,\cdot)\bigr) | u \in L^ 2 \bigl( \partial \Omega \times(0,T)\bigr) \biggr\}\) obeys \(D(T_ 1)\subsetneqq D(T_ 2)\) for \(T_ 1<T_ 2 \leq 2\), and \(D(2) \subsetneqq D(T_ 1)=D(T_ 2)\) for \(2<T_ 1<T_ 2\). The results are proven solving corresponding moment problems. They complement the results for the two- dimensional case by \textit{K. D. Graham} and \textit{D. L. Russell} [SIAM J. Control 13, 174-196 (1975; Zbl 0315.93004)] who investigated the same problem for unit spheres \(\Omega \in \mathbb{R}^ n\), \(n \geq 2\).
scientific article; zbMATH DE number 148682
scientific article