Necessary Conditions for Stochastic Optimal Control Problems in Infinite Dimensions

Hélène Frankowska (1) , Xu Zhang (2)

1 IMJ - Institut de Mathématiques de Jussieu
2 MPIB - Max-Planck-Institut für Biochemie = Max Planck Institute of Biochemistry

The purpose of this paper is to establish the first and second order necessary conditions for stochastic optimal controls in infinite dimensions. The control system is governed by a stochastic evolution equation, in which both drift and diffusion terms may contain the control variable and the set of controls is allowed to be nonconvex. Only one adjoint equation is introduced to derive the first order necessary optimality condition either by means of the classical variational analysis approach or, under an additional assumption, by using differential calculus of set-valued maps. More importantly, in order to avoid the essential difficulty with the well-posedness of higher order adjoint equations, using again the classical variational analysis approach, only the first and the second order adjoint equations are needed to formulate the second order necessary optimality condition, in which the solutions to the second order adjoint equation are understood in the sense of the relaxed transposition. 2010 Mathematics Subject Classification: 93E20, 49J53, 60H15

Mots clés: Stochastic optimal control; First and second order necessary optimality conditions; Variational equation; Adjoint equation; Transposition solution; Maximum principle

Domaines: Mathématiques [math]; Optimisation et contrôle [math.OC]; Probabilités [math.PR]

Hélène Frankowska, Xu Zhang. Necessary Conditions for Stochastic Optimal Control Problems in Infinite Dimensions. Stochastic Processes and their Applications, 2020, 130 (7), pp.4081-4103. ⟨10.1016/j.spa.2019.11.010⟩. ⟨hal-02890403⟩