Optimal control in the boundary value problem for elliptic equations with degeneration

Abstract

The problem of optimal control of the system described by the oblique derivative problem forthe elliptic equation of the second order is studied. Cases of internal and boundary managementare considered. The quality criterion is given by the sum of volume and surface integrals.The coefficients of the equation and the boundary condition allow power singularities of arbitraryorder 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 H\"{o}lder 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 bethe solution to the given problem.
 The necessary and sufficient conditions for the existence of the optimal solution of the systemdescribed by the boundary value problem for the elliptic equation with degeneracy have been established.