Dynamical system with boundary control associated with a symmetric semibounded operator

Abstract

Let L 0 be a closed densely defined symmetric semibounded operator with nonzero defect indices in a separable Hilbert space \(\mathcal H\). It determines a Green system \(\{{\mathcal H}, {\mathcal B}; L_0, \Gamma_1, \Gamma_2\}\), where \({\mathcal B}\) is a Hilbert space, and the \(\Gamma_i: {\mathcal H} \to \mathcal B\) are operators connected by the Green formula $$ (L_0^*u, v)_{\mathcal H}-(u,L_0^*v)_{\mathcal H} =(\Gamma_1 u, \Gamma_2 v)_{\mathcal B} - (\Gamma_2 u, \Gamma_1 v)_{\mathcal B}. $$ The boundary space \(\mathcal B\) and the boundary operators Γ i are chosen canonically in the framework of the Vishik theory. With the Green system one associates a dynamical system with boundary control (DSBC): $$ \begin{array}{lll} && u_{tt}+L_0^*u = 0, \quad u(t) \in {\mathcal H}, \quad t>0,\\ && u\big|_{t=0}=u_t\big|_{t=0}=0, \\ && \Gamma_1 u = f, \quad f(t) \in {\mathcal B},\quad t \geq 0. \end{array} $$ We show that this system is controllable if and only if the operator L 0 is completely non-self-adjoint. A version of the notion of wave spectrum of L 0 is introduced. It is a topological space determined by L 0 and constructed from reachable sets of the DSBC. Bibliography: 15 titles.