Dynamical system with boundary control associated with symmetric semi-bounded operator

M. I. Belishev ; M. N. Demchenko

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 42, Tome 409 (2012), pp. 17-39

Résumé

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 Green system $\{\mathcal H,\mathcal B;L_0,\Gamma_1,\Gamma_2\}$, where $\mathcal B$ is a Hilbert space, and $\Gamma_i\colon\mathcal H\to\mathcal B$ are the operators related through the Green formula $$ (L_0^*u, v)_\mathcal H-(u,L_0^*v)_\mathcal H=(\Gamma_1u,\Gamma_2v)_\mathcal B-(\Gamma_2u,\Gamma_1v)_\mathcal B. $$ The boundary space $\mathcal B$ and boundary operators $\Gamma_i$ are chosen canonically in the framework of the Vishik theory. With the Green system one associates a dynamical system with boundary control (DSBC) \begin{align*} {tt}+L_0^*u=0,(t)\in\mathcal H,\,\,t>0,\\ |_{t=0}=u_t|_{t=0}=0,\\ \Gamma_1u=f,(t)\in\mathcal B,\,\,\,t\geqslant0. \end{align*} 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 a wave spectrum of $L_0$ is introduced. It is a topological space determined by $L_0$ and constructed from reachable sets of the DSBC.