6. On Tikhonov Regularization of Optimal Distributed Control Problem for an Ill-Posed Elliptic Equation with p-Laplace Operator and \(L^1\)-type of Non-linearity

We discuss the existence of solutions to an optimal control problem for the Dirichlet boundary value problem for strongly non-linear p-Laplace equations with \(L_1\)-type of nonlinearity and \(p\ge 2\). The control variable u is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution \(y_d\in L^p(\varOmega )\) and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this we make use of a variant of the classical Tikhonov regularization. We eliminate the differential constraints between control and state and allow such pairs run freely in their respective sets of feasibility by introducing some additional variable which plays the role of “defect”. We show that this special residual function can be determined in a unique way. We introduce a special family of regularized optimization problems and show that each of these problems is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and propose the way for their approximation.