In this paper, optimal control problems with pointwise state constraints for linear and semilinear elliptic partial differential equations are studied. The problems are subject to perturbations in the problem data. Lipschitz stability with respect to perturbations of the optimal control and the state and adjoint variables is established initially for linear--quadratic problems. Both the distributed and Neumann boundary control cases are treated. Based on these results, and using an implicit function theorem for generalized equations, Lipschitz stability is also shown for an optimal control problem involving a semilinear elliptic equation.
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Roland Griesse, Lipschitz Stability of Solutions to Some State-Constrained Elliptic Optimal Control Problems. Z. Anal. Anwend. 25 (2006), no. 4, pp. 435–455DOI 10.4171/ZAA/1300