On optimal controls in coefficients for ill-posed non-linear elliptic Dirichlet boundary value problems. (English) Zbl 1398.49004
Summary: We consider an Optimal Control Problem (OCP) associated to a Dirichlet boundary value problem for nonlinear elliptic equations on a bounded domain \(\Omega\). We take the coefficient \(u(x)\in L^\infty(\Omega)\cap BV(\Omega)\) in the main part of the nonlinear differential operator as a control and in the linear part of differential operator we consider coefficients to be an unbounded skew-symmetric matrix \(A_{skew}\in L^q(\Omega;\mathbb{S}^N_{skew})\). 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.
MSC:
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 35D30 | Weak solutions to PDEs |
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