Elliptic and Parabolic Equations - Abstract
Schiela, Anton
We consider the derivation of first order optimality conditions in state constrained optimal control, subject to an elliptic partial differential equation. To be able to show existence of Lagrangian multipliers, one has to chose as a topological framework spaces of continuous functions. Hence, the differential operator involved, has to be considered as a closed, unbounded operator [ A : C(overline Omega) supset D_q to (W_Gamma^1,q'(Omega))^*, ] where $q>d$ and $q'$ is its dual exponent. By the new regularity results of sc Haller-Dintelmann, Rehberg, et. al. citeHaReMeSc2009 this is possible under very mild assumptions on the coefficients, the boundary conditions and the domain of the PDE. In order to derive an adjoint differential equation, one has to study subsequently the adjoint operator of $A$: [ A^* : W_Gamma^1,q'(Omega) supset D_q^* to C(overline Omega)^* cong M(overline Omega) ] In the case, where $D_q$ is a subspace of some $W^1,q_Gamma(Omega)$ with $q>d$, this is straightforward. However, if no such inclusion holds, then the interpretation of $A^*$ is a subtle task. In particular, uniqueness issues are very delicate. bibliographystyleplain beginthebibliography1 bibitemHaReMeSc2009 R. Haller-Dintelmann, J. Rehberg, C. Meyer, A. Schiela. newblock Hölder continuity and optimal control for nonsmooth elliptic problems. newblock Appl. Math. Optim., 60 (2009) pp. 397--428. endthebibliography