Optimal control of the systems governed by linear hyperbolic integro-differential equations

Andrii Anikushyn; Viktor Lyashko; Oleksandr Samosonok

doi:10.17721/1812-5409.2024/1.22

Authors

Andrii Anikushyn Taras Shevchenko National University of Kyiv https://orcid.org/0000-0003-1109-6290
Viktor Lyashko V.M. Glushkov Institute of Cybernetics, Kyiv https://orcid.org/0009-0002-7787-7241
Oleksandr Samosonok V.M. Glushkov Institute of Cybernetics, Kyiv

DOI:

https://doi.org/10.17721/1812-5409.2024/1.22

Keywords:

a priory inequalities, linear hyperbolic equation, viscoelasticity, generalized solution, optimal control, integro-differential equation

Abstract

The work is dedicated to the study of hyperbolic integro-differential equations with partial derivatives. Integro-differential equations of this type have long been a standard object of study in applied mathematics and often arise in the investigation of processes in viscoelastic media (amorphous polymers, semicrystalline polymers, biopolymers, metals at very high temperatures, bituminous materials, etc.).

The main goal is to prove the existence and uniqueness of generalized solutions to the corresponding initial-boundary value problem, as well as to investigate the existence of optimal control for systems described by these models.

The main results regarding the well-posedness of the initial-boundary value problem and the existence of optimal control are obtained using the method of a priori estimates in negative norms. In particular, estimates in negative norms for the integro-differential operator in certain special spaces are obtained. By utilizing the results of S.I. Lyashko and building upon the proven a priori estimates, various definitions of generalized solutions are formulated in the work, and a result regarding their equivalence is provided. A theorem concerning the well-posedness, i.e., the existence, uniqueness, and continuous dependence of the generalized solution on the right-hand side of the equation, is presented.

The work begins with a review of relevant results with similar formulations, referencing works with physical justification of the model. References to the application of the a priori estimates methodology for differential and integro-differential equations are provided. The problem statement, constraints on equation parameters, and functional spaces necessary for the investigation are described. Section 3 provides proven a priori estimates, which are a central part of the work. Section 4 contains definitions of generalized solutions and theorems describing their properties, including the well-posedness of the initial-boundary value problem. Section 5 is dedicated to the investigation of the existence of optimal control. Control is exerted through the right-hand side of the equation using a certain special operator. The work examines some examples of such control operators (illustrating various control mechanisms) and the corresponding function spaces. By utilizing proven inequalities and relying on general theorems from the theory of a priori estimates in negative norms, conditions for the existence of optimal control are established. In particular, restrictions are imposed on the admissible set of controls and the cost criterion in different scenarios.

Pages of the article in the issue: 111 - 118

Language of the article: English

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