Stability Analysis of Optimal Control Problems with a Second-order State Constraint

Audrey Hermant 1, 2
2 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
CNRS - Centre National de la Recherche Scientifique : UMR7641, X - École polytechnique, UMA - Unité de Mathématiques Appliquées, Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
Abstract : This paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second-order. The strengthened Legendre-Clebsch condition is assumed to hold, and no assumption on the structure of the contact set is made. Under a weak second-order sufficient condition (taking into account the active constraints), we show that the solutions are Lipschitz continuous w.r.t. the perturbation parameter in the $L^2$ norm, and Hölder continuous in the $L^\infty$ norm. We use a generalized implicit function theorem in metric spaces by Dontchev and Hager [SIAM J. Control Optim., 1998]. The difficulty is that multipliers associated with second-order state constraints have a low regularity (they are only bounded measures). We obtain Lipschitz stability of a ``primitive'' of the state constraint multiplier.
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SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2009, 20 (1), pp.104-129. 〈10.1137/070707993〉
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Audrey Hermant. Stability Analysis of Optimal Control Problems with a Second-order State Constraint. SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2009, 20 (1), pp.104-129. 〈10.1137/070707993〉. 〈inria-00186968v2〉

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