Second-order sufficient conditions for strong solutions to optimal control problems

Joseph Frederic Bonnans 1, 2 Xavier Dupuis 1, 2 Laurent Pfeiffer 1, 2
2 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées
Abstract : In this report, given a reference feasible trajectory of an optimal control problem, we say that the quadratic growth property for bounded strong solutions holds if the cost function of the problem has a quadratic growth over the set of feasible trajectories with a bounded control and with a state variable sufficiently close to the reference state variable. Our sufficient second-order optimality conditions in Pontryagin form ensure this property and ensure a fortiori that the reference trajectory is a bounded strong solution. Our proof relies on a decomposition principle, which is a particular second-order expansion of the Lagrangian of the problem.
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Joseph Frederic Bonnans, Xavier Dupuis, Laurent Pfeiffer. Second-order sufficient conditions for strong solutions to optimal control problems. ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 20 (03), pp.704-724. ⟨10.1051/cocv/2013080⟩. ⟨hal-00825260⟩

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