On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems

Abstract

We consider an optimal control problem associated to Dirichlet boundary value problem for non-linear elliptic equation on a bounded domain $Ω$ . We take the coefficient $u(x)∈ L^∞(Ω)\cap BV(Ω)$ in the main part of the non-linear differential operator as a control and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix $A_{skew}∈ L^q(Ω;\mathbb{S}^N_{skew})$ . We show that, in spite of unboundedness of the non-linear differential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-possed and solvable. At the same time, optimal solutions to such problem can inherit a singular character of the matrices $A^{skew}$ . We indicate two types of optimal solutions to the above problem and show that one of them can be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the two-parametric regularization of the initial OCP.