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Second order analysis for the optimal control of parabolic equations under control and final state constraints

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Abstract

We consider the optimal control of a semilinear parabolic equation with pointwise bound constraints on the control and finitely many integral constraints on the final state. Using the standard Robinson’s constraint qualification, we provide a second order necessary condition over a set of strictly critical directions. The main feature of this result is that the qualification condition needed for the second order analysis is the same as for classical finite-dimensional problems and does not imply the uniqueness of the Lagrange multiplier. We establish also a second order sufficient optimality condition which implies, for problems with a quadratic Hamiltonian, the equivalence between solutions satisfying the quadratic growth property in the L 1 and \(L^{\infty }\) topologies.

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  1. XLIM - DMI UMR CNRS 7252 Faculté des Sciences et Techniques, Université de Limoges, Limoges, France

    F. J. Silva

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  1. F. J. Silva
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Correspondence to F. J. Silva.

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Dedicated to Professor Lionel Thibault for his pioneering contributions in variational analysis.

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Silva, F.J. Second order analysis for the optimal control of parabolic equations under control and final state constraints. Set-Valued Var. Anal 24, 57–81 (2016). https://doi.org/10.1007/s11228-015-0337-4

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