The Solution of an Optimal Control Problem for a Third-Order Equation in the Presence of Distributed and Starting Controls

Abstract

There are a large number of scientific articles and monographs devoted to various problems of control theory in systems described by partial differential equations, which are quite well stated in monographs [1-6] and in many scientific articles. However, in all these works, problems are considered that are related mainly to equations of classical types studied in courses of mathematical physics. Recently, papers that solve various problems obtained from the practice and associated with fourth-order partial differential equations have appeared. Along with it, there are numerous applied problems described by third-order equations or the so called equations of variable type. Despite the fact that the initial-boundary value problems for such equations are studied well, there are only a few works [12-18] in which control problems in such systems are investigated. In this paper, the problem of the quadratic functional minimum for the initial-boundary value problem for a third-order linear equation is studied. First, using the Fourier method, a representation of the solution of the initial-boundary value problem is obtained, and then the optimal control problem is reduced to a conditional extremum problem and its solution is obtained in the form of a series.