Journal of Convex Analysis 09 (2002), No. 2, 625--648
Copyright Heldermann Verlag 2002
Nonconvex Duality and Viscosity Solutions of the Hamilton-Jacobi-Bellman Equation in Optimal Control
Nadia Raïssi
Laboratoire SIANO, Dép. des Mathématiques et d'Informatique, Faculté des Sciences, Université Ibn Tofail, Kénitra, Marocco
n.raissi@mailcity.com
Mustapha Serhani
Laboratoire SIANO, Dép. des Mathématiques et d'Informatique, Faculté des Sciences, Université Ibn Tofail, Kénitra, Marocco
mserhani@hotmail.com
We characterize the solutions of a nonconvex optimal control problem, using the Klötzler-Vinter nonconvex duality approach, in terms of generalized solutions of the Hamilton-Jacobi-Bellman equation (HJB). The dual problem is to find the supremum of the viscosity subsolutions of the HJB equation. We prove, without convexity assumptions, a weak duality between the primal and dual problems by using the technique of convolution and mollification. This weak duality provides necessary and sufficient conditions of optimality and leads to an error estimate. We also establish strong duality under an additional convexity hypothesis.
Keywords: Optimal control, Hamilton-Jacobi-Bellman equation, nonconvex duality, convolution, viscosity subsolution.
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