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 control, second-order sufficient conditions, quadratic growth, bounded strong solutions, Pontryagin multipliers, pure state and mixed control-state constraints, decomposition principle

@article{COCV_2014__20_3_704_0, author = {Fr\'ed\'eric Bonnans, J. and Dupuis, Xavier and Pfeiffer, Laurent}, title = {Second-order sufficient conditions for strong solutions to optimal control problems}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {704--724}, publisher = {EDP-Sciences}, volume = {20}, number = {3}, year = {2014}, doi = {10.1051/cocv/2013080}, mrnumber = {3264220}, zbl = {1293.49039}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2013080/} }

TY - JOUR AU - Frédéric Bonnans, J. AU - Dupuis, Xavier AU - Pfeiffer, Laurent TI - Second-order sufficient conditions for strong solutions to optimal control problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 704 EP - 724 VL - 20 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2013080/ DO - 10.1051/cocv/2013080 LA - en ID - COCV_2014__20_3_704_0 ER -

%0 Journal Article %A Frédéric Bonnans, J. %A Dupuis, Xavier %A Pfeiffer, Laurent %T Second-order sufficient conditions for strong solutions to optimal control problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 704-724 %V 20 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2013080/ %R 10.1051/cocv/2013080 %G en %F COCV_2014__20_3_704_0

Frédéric Bonnans, J.; Dupuis, Xavier; Pfeiffer, Laurent. Second-order sufficient conditions for strong solutions to optimal control problems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, pp. 704-724. doi : 10.1051/cocv/2013080. http://archive.numdam.org/articles/10.1051/cocv/2013080/

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