ON OPTIMAL CONTROL PROBLEM FOR AN ILL-POSED STRONGLY NONLINEAR ELLIPTIC EQUATION WITH p-LAPLACE OPERATOR AND L~1-TYPE OF NONLINEARITY

摘要

We study an optimal control problem for one class of non-linear elliptic equations with p-Laplace operator and L-1-nonlinearity. 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, we reformulate the original problem and prove the existence of optimal pairs. In order to ensure the validity of such reformulation, we provide its substantiation using a special family of fictitious optimal control problems. The idea to involve the fictitious optimization problems was mainly inspired by the brilliant book of V.S. Mel'nik and V.I. Ivanenko "Variational Methods in Control Problems for the Systems with Distributed Parameters", Kyiv, 1998.