Upper semicontinuity of the solution map to a parametric elliptic optimal control problem
One investigates a parametric optimal control problem of the form: find a couple of control functions \((u,v)\), such that if \(y\) is the solution of a controlled Robin boundary value problem to an elliptic partial differential equation, an integral objective functional \(J (y, u, v, \mu)\) gets minimized. The coefficients in the elliptic model are assumed to satisfy some auxiliary constraints and depend on parameter \(\lambda\). The control variables act in the domain \(\Omega\) as well as on the boundary \(\partial \Omega\) = \(\Gamma\). The second section of the paper is devoted to basic definitions and results on \((s,\omega)\)-upper semicontinuity and \((s,s)\)-upper semicontinuity. In the third section one shows some sufficient conditions for the existence and uniqueness of a solution to the Robin boundary problem. Under some restrictive assumptions on coefficients, parameters and controls one proves necessary optimality conditions for the existence of optimal solutions. One writes the adjoint equation and shows the existence and uniqueness of an admissible solution to the stated control problem. The proofs can be found in the fourth section. In the fifth section some illustrative examples are presented.
scientific article; zbMATH DE number 7381142
scientific article