First and second order necessary conditions for stochastic optimal control problems

Joseph Frédéric Bonnans (1, 2) , Francisco J. Silva (3)

1 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 Dipartimento di Matematica "Guido Castelnuovo", Università Roma La Sapienza

In this work we consider a stochastic optimal control problem with either convex control constraints or finitely many equality and inequality constraints over the final state. Using the variational approach, we are able to obtain first and second order expansions for the state and cost function, around a local minimum. This fact allows us to prove general first order necessary condition and, under a geometrical assumption over the constraint set, second order necessary conditions are also established. We end by giving second order optimality conditions for problems with constraints on expectations of the final state.

Domaines: Optimisation et contrôle [math.OC]

Joseph Frédéric Bonnans, Francisco J. Silva. First and second order necessary conditions for stochastic optimal control problems. [Research Report] RR-7454, 2010. ⟨inria-00537227v1⟩