Point control of linear hyperbolic integro-differential systems

Authors

DOI:

https://doi.org/10.17721/1812-5409.2024/2.4

Keywords:

integro-differential equations, linear hyperbolic equation, Volterra operator, viscoelastic media, optimal control, point control

Abstract

Background. The work focuses on the optimal control of distributed systems described by linear hyperbolic integro-differential equations with partial derivatives and Volterra-type integral components. Such integro-differential equations are a standard subject in applied mathematics and frequently arise in studies of processes in viscoelastic media (such as amorphous polymers, semi-crystalline polymers, biopolymers, metals at very high temperatures, bituminous materials, and more). The primary goal is to prove the existence of optimal control for distributed systems modeled by these equations.

Methods. We apply methods of functional analysis and the theory of distributions. The study is conducted in specially defined Hilbert spaces, where the control operator in the system's right-hand side includes generalized Dirac delta functions. We establish the main result on the existence of optimal control based on the theory of a priori estimates in negative norms, building on the foundational work of Yu. M. Berezansky and further developed by V. P. Didenko, S. I. Lyashko, and their colleagues.

Results. We formulate an optimal control problem, where control of the system is governed by a control operator appearing in the right-hand side of the initial-boundary value problem. The control operator acts into spaces of generalized functions, modeling pointwise control of the system. Further, we propose appropriate Hilbert spaces for the problem's operator and the space of admissible controls. Moreover, we provide a priori estimates in negative norms, define generalized solutions, and prove the well-posedness of the initial-boundary value problem. Finally, a general theorem on the existence of optimal control is establish and the well-definedness and weak continuity of the control operator are proved. Based on these statements, we formulate a theorem on the existence of optimal control for the problem, imposing restrictions on the admissible control set and the quality criterion to be minimized.

Сonclusions. We prove a theorem providing sufficient conditions for the existence of optimal control for the considered system. In particular, we demonstrate that the control operator corresponding to pointwise control is well-defined and weakly continuous from the control space to the space of right-hand sides.

Pages of the article in the issue: 20 - 28

Language of the article: Ukrainian English

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Published

2025-01-29

How to Cite

Anikushyn, A., & Hranishak, K. (2025). Point control of linear hyperbolic integro-differential systems. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, 79(2), 20–28. https://doi.org/10.17721/1812-5409.2024/2.4

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Section

Differential equations, mathematical physics and mechanics