OPTIMAL CONTROL IN THE MULTIPOINT BOUNDARY VALUE PROBLEM FOR 2B-PARABOLIC EQUATIONS

Abstract

The potential theory method was used to study the existence of a solution of a multi- point boundary value problem for a 2b-parabolic equation. Using the Green’s function of a homogeneous boundary value problem for a 2b-parabolic equation, the integral Fredholm equation of the second kind is placed in accordance with the multipoint boundary value problem. Taking into account the constraints on the coefficients of the nonlocal condition and using the sequential approximation method, an integrated image of the solution of the nonlocal problem at the initial moment of time and its estimation in the Holder spaces are found. Estimates of the solution of a nonlocal multipoint boundary value problem at fixed moments of time given in a nonlocal condition are found by means of estimates of the components of the Green’s function of the general boundary value problem for the 2b-parabolic equation. Taking into account the obtained estimates and constraints on coefficients in multipoint problem, estimates of the solution of the multipoint problem for the 2b-parabolic equations and its derivatives in Holder spaces are established. In addition, the uniqueness and integral image of the solution of the general multipoint problem for 2b-parabolic equations is justified. The obtained result is applied to the study of the optimal system control problem described by the general multipoint boundary value problem for 2b-parabolic equations. The case of simultaneous internal, initial and boundary value control of solutions to a multipoint parabolic boundary value problem is considered. The quality criterion is defined by the sum of volume and surface integrals. The necessary and sufficient conditions for the existence of an optimal solution of the system described by the general multipoint boundary value problem for 2b-parabolic equations with limited internal, initial and boundary value control are established.