Linearization techniques for $\mathbb{L}^{\infty}$-control problems and dynamic programming principles in classical and $\mathbb{L}^{\infty}$-control problems

Dan Goreac (1) , Oana Silvia Serea (2)

1 PS
2 LAMPS - LAboratoire de Modélisation Pluridisciplinaire et Simulations

The aim of the paper is to provide a linearization approach to the $\mathbb{L}^{\infty}$-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the $\mathbb{L}^{p}$ approach and the associated linear formulations. This seems to be the most appropriate tool for treating $\mathbb{L}^{\infty}$ problems in continuous and lower semicontinuous setting.

Mots clés: essential supremum; HJ equations; occupational measures; $\mathbb{L}^{p}$ approximations; Dynamic programming principle

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Dan Goreac, Oana Silvia Serea. Linearization techniques for $\mathbb{L}^{\infty}$-control problems and dynamic programming principles in classical and $\mathbb{L}^{\infty}$-control problems. ESAIM: Control, Optimisation and Calculus of Variations, 2012, 18 (3), pp.836-855. ⟨10.1051/cocv/2011183⟩. ⟨hal-00759482⟩