Wave propagation in abstract dynamical system with boundary control

TL;DR

This paper studies wave propagation in an abstract dynamical system with boundary control, revealing a principle analogous to finite wave speed in a general operator-theoretic framework.

Contribution

It introduces an abstract model of wave dynamics with boundary control and uncovers a generalization of the finite propagation speed principle.

Findings

Identifies properties of wave propagation in abstract boundary-controlled systems

Establishes an analog of the finiteness principle for wave speed

Provides a framework for analyzing boundary control in Hilbert space operators

Abstract

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}\,\,{\mathscr H},\,\,\,t>0;\\ & u\big|_{t=0}=u_t\big|_{t=0}=0 && {\rm in}\,\,{\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.