First and second order necessary optimality conditions for controlled stochastic evolution equations with control and state constraints

• Published in: Journal of Differential Equations Vol. 268; no. 6; pp. 2949 - 3015
• Main Authors: Frankowska, Hélène, Lü, Qi
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 05.03.2020; Elsevier
• Subjects: Mathematics; Necessary optimality conditions; Optimization and Control; Set-valued analysis; Stochastic optimal control; secondary; Set-valued analysis; Necessary optimality conditions; Stochastic optimal control; primary; necessary optimality conditions; set-valued analysis
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2019.09.045

The purpose of this paper is to establish first and second order necessary optimality conditions for optimal control problems of stochastic evolution equations with control and state constraints. The control acts both in the drift and diffusion terms and the control region is a nonempty closed subset of a separable Hilbert space. We employ some classical set-valued analysis tools and theories of the transposition solution of vector-valued backward stochastic evolution equations and the relaxed-transposition solution of operator-valued backward stochastic evolution equations to derive these optimality conditions. The correction part of the second order adjoint equation, which does not appear in the first order optimality condition, plays a fundamental role in the second order optimality condition.