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
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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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
Keywords
- Optimal control
- Parabolic equation
- Semilinear equation
- Second-order optimality conditions
- Weak local minimum
- Strong local minimum