Dynamical System with Boundary Control Associated with Symmetric Semi-Bounded Operator

Abstract

Let $L_0$ be a closed densely defined symmetric semi-bounded operator with nonzero defect indexes in a separable Hilbert space $\cal H$. It determines a {\it Green system} $\{{\cal H}, {\cal B}; L_0, \Gamma_1, \Gamma_2\}$, where ${\cal B}$ is a Hilbert space, and $\Gamma_i: {\cal H} \to \cal B$ are the operators related through the Green formula $$(L_0^*u, v)_{\cal H}-(u,L_0^*v)_{\cal H}=(\Gamma_1 u, \Gamma_2 v)_{\cal B} - (\Gamma_2 u, \Gamma_1 v)_{\cal B}.$$ The {\it boundary operators} $\Gamma_i$ are chosen canonically in the framework of the Vishik theory. With the Green system one associates a {\it dynamical system with boundary control} (DSBC) {align*} & u_{tt}+L_0^*u = 0 && {\rm in}\,\,\,{\cal H}, \,\,\,t>0 & u|_{t=0}=u_t|_{t=0}=0 && {\rm in}\,\,\,{\cal H} & \Gamma_1 u = f && {\rm in}\,\,\,{\cal B},\,\,\,t \geqslant 0. {align*} We show that this system is {\it controllable} if and only if the operator $L_0$ is completely non-self-adjoint. A version of the notion of a {\it wave spectrum} of $L_0$ is introduced. It is a topological space determined by $L_0$ and constructed from reachable sets of the DSBC.