Approximate Solution for a Class of Optimal Control Problems in Distributed Parameter Systems

Abstract

An approximate solution is developed using the direct method for a specific class of optimal control problems in systems governed by nonlinear boundary value problems of parabolic type. These problems are particularly significant in the modeling and optimization of dynamic processes distributed over space and time. The methodology is based on constructing finite-dimensional approximations of the original infinite-dimensional problem, allowing for a practical computational approach. By applying a priori estimate for the solutions of systems of linear ordinary differential equations, the convergence of the proposed direct method is rigorously proven. This convergence guarantees that the approximate solutions approach the exact solution of the original control problem in terms of minimizing the given functional. Furthermore, a constructive scheme for generating a minimizing sequence of controls is introduced, which depends on the chosen class of admissible controls. This scheme provides a systematic way to approach optimality in practical applications. As a practical illustration, the study presents an example related to determining the optimal technological regime for the operation of gas wells, which demonstrates the applicability of the proposed method to real-world engineering problems. The developed approach can serve as a valuable computational tool for solving similar optimal control problems in distributed parameter systems.