Wave propagation in abstract dynamical system with boundary control

Abstract

Let $L_0$ be a positive definite operator in a Hilbert space $\mathcal{H}$ with the defect indexes $n_{\pm }⩾1$ and let $\{\mathrm{Ker}{L}_0^{\ast };{\Gamma }_1,{\Gamma }_2\}$ be its canonical (by M.I.Vishik) boundary triple. The paper deals with an evolutionary dynamical system of the form \begin{align*} & u_{tt}+{L_0^*} u=0 &&\text{in}\,\,{\mathscr H},\,\,\,t>0;\\ & u\big|_{t=0}=u_t\big|_{t=0}=0 && {\rm in}\,\,{\mathscr H};\\ & Γ_1 u=f(t), && t\geqslant 0, \end{align*} where $f$ is a boundary control (a $\mathrm{Ker}{L}_0^{\ast }$-valued function of time), $u=u^f(t)$ is a trajectory. Some of the general properties of such systems are considered. An abstract analog of the finiteness principle of wave propagation speed is revealed.