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

Abstract

We consider an optimal control problem associated to Dirichlet boundary value problem for linear elliptic equations on a bounded domain Ω. We take the matrix-valued coefficients $\mathit{A}={\mathit{A}}^{\mathrm{sym}}+{\mathit{A}}^{\mathrm{skew}}$ of such system as a control in ${\mathit{L}}^1\mathrm{(}\mathrm{\Omega };{\mathbb{S}}_{\mathrm{sym}}^{\mathit{N}}\mathrm{)}\oplus {\mathit{L}}^{2\mathit{p}}\mathrm{(}\mathrm{\Omega };{\mathbb{S}}_{\mathrm{skew}}^{\mathit{N}}\mathrm{)}$ . One of the important features of the class of admissible controls is the fact that the matrices $\mathit{A}\mathrm{(}\mathit{x}\mathrm{)}$ are unbounded on Ω and eigenvalues of the symmetric parts ${\mathit{A}}^{\mathrm{sym}}$ may vanish in Ω. In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP is well-possed and admits at least one solution. At the same time, optimal solutions to such problem can inherit a singular character of the matrices ${\mathit{A}}^{\mathrm{skew}}$ . We indicate two types of optimal solutions to the above problem and show that one of them can not be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the Steklov smoothing of matrix-valued controls A.