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State error estimates for the numerical approximation of sparse distributed control problems in the absence of Tikhonov regularization. (English) Zbl 1473.35242

Vietnam J. Math. 49, No. 3, 713-738 (2021); correction ibid. 51, No. 2, 565-566 (2023).
Summary: In this paper, we analyze optimal control problems of semilinear elliptic equations, where the controls are distributed. Box constraints for the controls are imposed and the cost functional does not involve the control itself, except possibly for a non-differentiable sparsity-promoting term. Under appropriate second order sufficient optimality conditions, first we estimate the difference between the discrete and continuous optimal states. Next, under an additional assumption on the optimal adjoint state, we prove error estimates for the controls and improve the estimates for the states.

MSC:

35J61 Semilinear elliptic equations
49K20 Optimality conditions for problems involving partial differential equations
49M25 Discrete approximations in optimal control
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