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Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems

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Abstract

Second-order sufficient optimality conditions are considered for a simplified class of semilinear parabolic equations with quadratic objective functional including distributed and terminal observation. Main emphasis is laid on problems where the objective functional does not include a Tikhonov regularization term. Here, standard second-order conditions cannot be expected to hold. For this case, new second-order conditions are established that are based on different types of critical cones. Depending on the choice of this cones, the second-order conditions are sufficient for local minima that are weak or strong in the sense of calculus of variations.

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Notes

  1. If N = 1, we assume throughout the paper that p ≥ 2.

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Acknowledgments

The first author was partially supported by Spanish Ministerio de Economía y Competitividad under projects MTM2011-22711 and MTM2014-57531-P. The second is supported by DFG in the framework of the Collaborative Research Center SFB 910, project B6.

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Correspondence to Fredi Tröltzsch.

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Dedicated to Prof. Dr. Eberhard Zeidler on the occasion of his 75th birthday.

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Casas, E., Tröltzsch, F. Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems. Vietnam J. Math. 44, 181–202 (2016). https://doi.org/10.1007/s10013-015-0175-6

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  • DOI: https://doi.org/10.1007/s10013-015-0175-6

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