The paper addresses the problems of $\mathcal{L}_2$-stability of a complex system containing local and remote subsystems, which interact by means of a communication channel that produces time-varying delays. Each subsystem is assumed to be QSR-dissipative and to satisfy the “liveness” condition according to J.C. Willems. The control procedure is then constructed under these assumptions on the subsystems. The basic elements of this procedure are the scattering transformations applied to the input-output parameters of the subsystems. As a result, the original input-output parameters of the subsystems transform into wave variables which are transmitted through the communication channel. To eliminate the destabilizing impact of time-varying delays during data transmission, wave variables are scaled by multiplying them by time-dependent factors that estimate the growth rates of time-varying delays. The combination of these two elements (scattering transformation and scaling) ensures $\mathcal{L}_2$-stability of the overall system with QSR-dissipative subsystems and time-varying communication channel delays. The proposed approach generalizes stabilization methods developed for remote interactions of passive systems.
Keywords: QSR-dissipative systems, scattering transformations, wave variables, $\mathcal{L}_2$-stability, communication time-varying
Received April 29, 2025
Revised May 12, 2025
Accepted May 19, 2025
Anastasiia Aleksandrovna Usova, Cand. Sci. (Phys.-Math.), PhD, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: ausova@imm.uran.ru
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Cite this article as: A.A. Usova. 2-stability of a remote interaction of QSR-dissipative systems with time-varying delays. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 2, pp. 262–279.
