Wave propagation in abstract dynamical system with boundary control

M. I. Belishev

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 53, Tome 521 (2023), pp. 8-32

Résumé

Let $L_0$ be a positive definite operator in a Hilbert space $\mathscr H$ with the defect indexes $n_\pm\geqslant 1$ and let $\{{\rm Ker }L^*_0;\Gamma_1,\Gamma_2\}$ 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}\quad {\mathscr H}, t>0;\\ u\big|_{t=0}=u_t\big|_{t=0}=0 {\rm in }\quad {\mathscr H};\\ \Gamma_1 u=f(t), t\geqslant 0, \end{align*} where $f$ is a boundary control (a ${\rm Ker }L^*_0$-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.