OPTIMAL CONTROL IN THE DIRICHLET PROBLEM FOR ELLIPTIC EQUATIONS WITH DEGENERATION

Abstract

The theory of optimal control of systems, which is described by partial differential equations, is rich in results and is actively developing nowadays. The popularity of this kind of research is connected with its active use in solving problems of natural science, in particular hydro and gas dynamics, heat physics, diffusion, and the theory of biological populations. The problem of optimal control of the system described by the Dirichlet problem for the elliptic equation of the second order is studied. Cases of internal control are considered. The quality criterion is given by the volumetric integral. The coefficients of the equation admit power singularities of arbitrary order in any variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special Hölder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions to auxiliary problems, the limiting value of which will be the solution to the given problem. The necessary and sufficient conditions for the existence of the optimal solution of the system described by the Dirichlet problem for the elliptic equation with degeneracy have been established.