Second Order Conditions for L2 Local Optimality in PDE Control

Abstract : In the second order analysis of infinite dimension optimization problems, we have to deal with the so-called two-norm discrepancy. As a consequence of this fact, the second order optimality conditions usually imply local optimality in the L ∞  sense. However, we have observed that the L2 local optimality can be proved for many control problems of partial differential equations. This can be deduced from the standard second order conditions. To this end, we make some quite realistic assumptions on the second derivative of the cost functional. These assumptions do not hold if the control does not appear explicitly in the cost functional. In this case, the optimal control is usually of bang-bang type. For this type of problems we also formulate some new second order optimality conditions that lead to the strict L2 local optimality of the bang-bang controls.
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Dietmar Hömberg; Fredi Tröltzsch. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. Springer, IFIP Advances in Information and Communication Technology, AICT-391, pp.1-12, 2013, System Modeling and Optimization. 〈10.1007/978-3-642-36062-6_1〉
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Eduardo Casas. Second Order Conditions for L2 Local Optimality in PDE Control. Dietmar Hömberg; Fredi Tröltzsch. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. Springer, IFIP Advances in Information and Communication Technology, AICT-391, pp.1-12, 2013, System Modeling and Optimization. 〈10.1007/978-3-642-36062-6_1〉. 〈hal-01347511〉

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