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

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.