Short Communication
Differential Equations and Mathematical Physics

On determination of gradient in optimal control problems for frictionless mechanical oscillatory systems

A. S. Zinchenko^a, A. A. Nekhaev^b, A. M. Romanenkov<sup>ab

a</sup> Moscow Aviation Institute (National Research University), Moscow, 125993, Russian Federation
^b Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, Moscow, 119333, Russian Federation

Abstract: This paper investigates the problem of gradient computation for an optimal control algorithm applied to a distributed system. The mathematical model of the system is described by an initial-boundary value problem for a linear high-order hyperbolic partial differential equation. The study considers an oscillatory process without energy dissipation. The proposed model covers a wide class of applied problems, including vibrations of strings, beams, rods, and other one-dimensional elastic mechanical systems, as well as systems reducible to these cases. By using the method of integral estimates, we prove a uniqueness theorem for the solution and derive an explicit expression for the gradient of the minimized quadratic functional.

Keywords: optimal control, hyperbolic equations, oscillatory systems, gradient method, initial-boundary value problems

UDC: 517.977.56

MSC: 49K20, 35Lxx

Received: November 13, 2024
Revised: May 23, 2025
Accepted: June 2, 2025
First online: July 3, 2025

DOI: 10.14498/vsgtu2133

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