Abstract dynamical system with boundary control. IV

M. I. Belishev^a, S. A. Simonov<sup>abc

a</sup> St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
^b Alferov Federal State Budgetary Institution of Higher Education and Science Saint Petersburg National Research Academic University of the Russian Academy of Sciences, St. Petersburg
^c ITMO University

Abstract: The paper continues the study of the general properties of evolutionary dynamical systems of the type
\begin{align*} & u''(t)+L_0^*u(t) = 0 && \text{ in } {{\mathscr H}}, t\in(0,T), & u(0)=u'(0)=0 && \text{ in } {{\mathscr H}}, &\Gamma_1 u(t) = f(t) && \text{ in } {{\mathscr K}}, t\in[0,T], \end{align*}
where $\mathscr H$ is a Hilbert space, $L_0$ is a positive-definite symmetric operator in $\mathscr H$, $\Gamma_1:{\mathrm {Dom}}\,L_0^*\to \mathscr K:={\mathrm {Ker}}\,L_0^*$ is one of the boundary operators from the Green formula $({L_0^*} u,v)-(u,{L_0^*}v)=(\Gamma_1u,\Gamma_2v)-(\Gamma_2u,\Gamma_1v)$, $f=f(t)$ is a $\mathscr K$-valued function of time (boundary control), $u=u^f(t)$ is the solution (trajectory), which is an $\mathscr H$-valued function of time. Compared to previous works on this topic, the novelty lies in the properties of the response operator $R^T: f\mapsto \Gamma_2 u^f(\cdot)$, which acts in $\mathscr F^T:=L_2([0,T];\mathscr K)$ on a relevant ${\mathrm{Dom}}\,R^T$.

Key words and phrases: triangular factorization of operators, nest theory, functional models.

UDC: 517.98

Received: 05.10.2025