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

Let $L_0$ be a closed densely defined symmetric semi-bounded operator with nonzero defect indexes in a separable Hilbert space $\mathcal{H}$. It determines a {\it Green system} $\{\mathcal{H},\mathcal{B};L_0,{\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2\}$, where $\mathcal{B}$ is a Hilbert space, and ${\mathrm{\Gamma }}_i:\mathcal{H}\to \mathcal{B}$ are the operators related through the Green formula $$(L_0^{\ast }u,v{)}_{\mathcal{H}}-(u,L_0^{\ast }v{)}_{\mathcal{H}}=({\mathrm{\Gamma }}_{1}u,{\mathrm{\Gamma }}_{2}v{)}_{\mathcal{B}}-({\mathrm{\Gamma }}_{2}u,{\mathrm{\Gamma }}_{1}v{)}_{\mathcal{B}}\mathrm{.}$$ The {\it boundary operators} ${\mathrm{\Gamma }}_i$ 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.