The goal of this paper is to prove the first and second order optimality conditions for some control problems governed by semilinear elliptic equations with pointwise control constraints and finitely many equality and inequality pointwise state constraints. To carry out the analysis we formulate a regularity assumption which is equivalent to the first order optimality conditions. Though the presence of pointwise state constraints leads to a discontinuous adjoint state, we prove that the optimal control is Lipschitz in the whole domain. Necessary and sufficient second order conditions are proved with a minimal gap between them.

Classification: 49K20, 35J25

Keywords: elliptic control problems, pointwise state constraints, Pontryagin's principle, second order optimality conditions

@article{COCV_2008__14_3_575_0, author = {Casas, Eduardo}, title = {Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {14}, number = {3}, year = {2008}, pages = {575-589}, doi = {10.1051/cocv:2007063}, zbl = {pre05309732}, mrnumber = {2434067}, language = {en}, url = {http://www.numdam.org/item/COCV_2008__14_3_575_0} }

Casas, Eduardo. Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 3, pp. 575-589. doi : 10.1051/cocv:2007063. http://www.numdam.org/item/COCV_2008__14_3_575_0/

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