Stochastic maximum principle for optimal control problem with varying terminal time and non-convex control domain

TL;DR

This paper develops a stochastic maximum principle for control problems with a variable terminal time depending on the mean state, non-convex control domains, and control-dependent diffusion, using advanced adjoint equations.

Contribution

It introduces a novel stochastic maximum principle for problems with non-convex controls and variable terminal times influenced by the mean state.

Findings

Established first- and second-order adjoint equations for the new control system.

Proved the variational equation for the varying terminal time.

Validated results with two illustrative examples.

Abstract

In this paper, we consider a varying terminal time structure for the stochastic optimal control problem under state constraints, in which the terminal time varies with the mean value of the state. In this new stochastic optimal control system, the control domain does not need to be convex and the diffusion coefficient contains the control variable. To overcome the difficulty in the proof of the related Pontryagin's stochastic maximum principle, we develop asymptotic first- and second-order adjoint equations for the varying terminal time, and then establish its variational equation. In the end, two examples are given to verify the main results of this study.