On Approximation of an Optimal Control Problem for Ill-Posed Strongly Nonlinear Elliptic Equation with p-Laplace Operator

Abstract

We study an optimal control problem for one class of non-linear elliptic equations with p-Laplace operator and L ¹-nonlinearity in their right-hand side. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any given control. After defining a suitable functional class in which we look for solutions and a special cost functional, we prove the existence of optimal pairs. In order to handle the inherent degeneracy of the p-Laplacian and strong non-linearity in the right-hand side of elliptic equation, we use a two-parametric (ε, k)-regularization of p-Laplace operator, where we approximate it by a bounded monotone operator, and involve a special fictitious optimization problem. We derive existence of optimal solutions to the parametrized optimization problems at each (ε, k)-level of approximation. We also deduce the differentiability of the state for approximating problem with respect to the controls and obtain an optimality system based on the Lagrange principle. Further we discuss the asymptotic behaviour of the optimal solutions to regularized problems as the parameters ε and k tend to zero and infinity, respectively.