Relationships Between the Maximum Principle and Dynamic Programming for Infinite Dimensional Non-Markovian Stochastic Control Systems

TL;DR

This paper explores the deep connections between Pontryagin's maximum principle and dynamic programming in infinite-dimensional stochastic control systems, establishing new theoretical links in both smooth and non-smooth cases.

Contribution

It formulates the dynamic programming principle for infinite-dimensional stochastic systems and reveals explicit and sample-wise relationships between the two control principles.

Findings

Established the dynamic programming principle for infinite-dimensional stochastic systems.

Derived stochastic Hamilton-Jacobi-Bellman equations for the value function.

Uncovered new sample-wise relationships using nonsmooth analysis techniques.

Abstract

This paper investigates the relationship between Pontryagin's maximum principle and dynamic programming principle in the context of stochastic optimal control systems governed by stochastic evolution equations with random coefficients in separable Hilbert spaces. Our investigation proceeds through three contributions: (1). We first establish the formulation of the dynamic programming principle for this class of infinite-dimensional stochastic systems, subsequently deriving the associated stochastic Hamilton-Jacobi-Bellman equations that characterize the value function's evolution. (2). For systems with smooth value functions, we develop explicit correspondence relationships between Pontryagin's maximum principle and dynamic programming principle, elucidating their fundamental connections through precise mathematical characterizations. (3). In the more challenging non-smooth case, we employ tools in nonsmooth analysis and relaxed transposition solution techniques to uncover previously unknown sample-wise relationships between the two principles.