Algorithm for Numerical Method of Solution of the Optimal Control Problem for Semilinear Sobolev Type Models on Basis of Decomposition Method

Abstract

The question of finding the numerical solutions of optimal control problems in mathematical models based on the degenerate semilinear equations of mathematical physics is one of the most important. It is so because of a large number of applications of such problems. In the case of non-linear state equation the search for the numerical solution of optimal control problem becomes more difficult. A lot of initial-boundary value problems for the equations and the systems of equations which are not resolved with respect to time derivative are considered in the framework of abstract Sobolev type equations that make up the vast field of non-classical equations of mathematical physics. We are interested in the optimal control problem to solutions of the Showalter -- Sidorov problem for the semilinear Sobolev type equation. The article proposes to use a decomposition method, allowing to linearize a non-linear equation, and the penalty method, which allows to find an approximate solution of the problem. Our numerical method is based on the Galerkin method, and the method of decomposition. Using it we find approximate solutions of optimal control problem for the Hoff mathematical model and the generalized mathematical model of deformation of the structure of I-beams.