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Article Contents

# Sparse optimal control for the heat equation with mixed control-state constraints

• * Corresponding author: Fredi Tröltzsch

Dedicated to Prof. Dr. Fréderic Bonnans on the occasion of his 60th birthday

The first author was partially supported by Spanish Ministerio de Economía, Industria y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P. The second author was supported by the Collaborative Research Center SFB 910, TU Berlin, project B6

• A problem of sparse optimal control for the heat equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic tracking type functional is to be minimized that includes a Tikhonov regularization term and the $L^1$-norm of the control accounting for the sparsity. Special emphasis is laid on existence and regularity of Lagrange multipliers for the mixed control-state constraints. To this aim, a duality theorem for linear programming problems in Hilbert spaces is proved and applied to the given optimal control problem.

Mathematics Subject Classification: Primary: 49K20, 49N10; Secondary: 90C05, 90C46.

 Citation:

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