Technical stability with respect to measure of solutions of boundary value problems for distributed processes with a continuous control (Q1873549)

Let \(\varphi(x,t)\) satisfy a system of first-order partial differential vector equations \[ \partial\varphi/\partial t= A\varphi+ D\psi+ Gu_x,\qquad \partial\varphi/\partial x= C\varphi+ D\psi \] with \(\varphi(0,x)= \varphi_0(x)\). \(A\), \(D\), \(C\), \(G\), are matrices whose entries are continuous functions of \(x\) and \(t\), \(t\in I\), \(x\in D\). The author defines the technical stability of a controlled process with respect to the measure \(\rho= \rho(\varphi)= \int_D \varphi^*(t,x) \varphi(t,x) dx\), characterizing the deviation of \(\varphi\) from the zero function. The author derives sufficient conditions for technical stability both for finite and infinite time intervals using Lyapunov functionals and by following the comparison technique proposed originally by V. M. Matrosov. In proving his main theorem he follows the multipliers technique introduced in 1977 by Timur Sirazetdinov. The reviewer tried to visualize how Komornik's multiplier proposals, which are popular in the Western world, would work in the same situation.