Approximation of optimal control problems in the coefficient for the p-Laplace equation. I. Convergence result - Casas, Eduardo; Kogut, Peter I.; Leugering, Günter (SIAM Journal on Control and Optimization, 2016)
We study a Dirichlet optimal control problem for a quasi-linear monotone elliptic equation, the so-called weighted p-Laplacian problem. The coefficient of the -Laplacian, the weight u, we take as a control in BV (Ω) ∩ L∞(Ω). In this article, we use box-type constraints for the control such that there is a strictly positive lower and some upper bound. In order to handle the inherent degeneracy of the -Laplacian, we use a regularization, sometimes referred to as the -Laplacian. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted -Laplacian, where we approximate the nonlinearity by a bounded monotone function, parametrized by κ. Further, we discuss the asymptotic behavior of the solutions to the regularized problem on each (ϵ, κ)-level as the parameters tend to zero and infinity, respectively.
Keywords: Control in coefficients; Nonlinear Dirichlet problem; Optimal control