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

Published online by Cambridge University Press:  14 March 2014

J. Frédéric Bonnans ,
Xavier Dupuis  and
Laurent Pfeiffer
J. Frédéric Bonnans
Affiliation:
Inria Saclay and CMAP, Ecole Polytechnique. Route de Saclay, 91128 Palaiseau Cedex, France. frederic.bonnans@inria.fr; xavier.dupuis@cmap.polytechnique.fr; laurent.pfeiffer@polytechnique.edu
Xavier Dupuis
Affiliation:
Inria Saclay and CMAP, Ecole Polytechnique. Route de Saclay, 91128 Palaiseau Cedex, France. frederic.bonnans@inria.fr; xavier.dupuis@cmap.polytechnique.fr; laurent.pfeiffer@polytechnique.edu
Laurent Pfeiffer
Affiliation:
Inria Saclay and CMAP, Ecole Polytechnique. Route de Saclay, 91128 Palaiseau Cedex, France. frederic.bonnans@inria.fr; xavier.dupuis@cmap.polytechnique.fr; laurent.pfeiffer@polytechnique.edu
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Abstract

In this article, 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.

Keywords

Optimal controlsecond-order sufficient conditionsquadratic growthbounded strong solutionsPontryagin multiplierspure state and mixed control-state constraintsdecomposition principle
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 3 , July 2014 , pp. 704 - 724
Copyright
© EDP Sciences, SMAI, 2014

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