On an evolutionary dynamical system of the first order with boundary control

TL;DR

This paper investigates the controllability of a first-order evolutionary dynamic system with boundary control, linked to symmetric operators, establishing a condition based on the absence of maximal symmetric parts.

Contribution

It provides a new criterion for controllability of boundary-controlled systems associated with symmetric operators, advancing the theoretical understanding of such dynamical systems.

Findings

Controllability depends on the symmetric operator having no maximal symmetric parts.

Established a necessary and sufficient condition for controllability.

Connected the controllability of the system to properties of the symmetric operator.

Abstract

The work is carried out as part of the program to construct a new functio\-nal (so-called {\it wave}) model of symmetric operators. It is shown that an abstract evolutionary dynamic system of the first order (with respect to time) with boundary control, which is determined by a symmetric operator $L_0:{\mathscr H}\to{\mathscr H}$, is controllable if and only if $L_0$ has no maximal symmetric parts in~${\mathscr H}$.