On Neumann boundary control problem for ill-posed strongly nonlinear elliptic equation with p-Laplace operator and $$L^1$$L1-type of nonlinearity

Abstract

In this paper we study an optimal control problem for the mixed Dirichlet–Neumann boundary value problem for the strongly non-linear elliptic equation with p-Laplace operator and $$L^1$$-nonlinearity in their right-hand side. A density of surface traction u acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution $$y_d\in L^2(\varOmega )$$ and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any admissible control. After defining a suitable functional class in which we look for solutions and assuming that this problem admits at least one feasible solution, we prove the existence of optimal pairs. In order to handle the strong non-linearity in the right-hand side of elliptic equation, we involve a special two-parametric fictitious optimization problem. We derive existence of optimal solutions to the regularized optimization problems at each $$({\varepsilon },k)$$-level of approximation and discuss the asymptotic behaviour of the optimal solutions to regularized problems as the parameters $${\varepsilon }$$ and k tend to zero and infinity, respectively.