Necessary Condition for Near Optimal Control of Linear Forward-backward Stochastic Differential Equations

TL;DR

This paper establishes a necessary condition for near optimal control of linear forward-backward stochastic differential equations with control-dependent coefficients and non-convex control domains, extending previous results and providing new estimates.

Contribution

It introduces a necessary condition for near optimal control in complex FBSDE systems with non-convex control domains, advancing the theoretical framework.

Findings

Derived a necessary condition for near optimal control.

Extended the variational principle to non-convex control domains.

Provided new estimates to handle near optimality.

Abstract

This paper investigates the near optimal control for a kind of linear stochastic control systems governed by the forward backward stochastic differential equations, where both the drift and diffusion terms are allowed to depend on controls and the control domain is not assumed to be convex. In the previous work (Theorem 3.1) of the second and third authors [\textit{% Automatica} \textbf{46} (2010) 397-404], some problem of near optimal control with the control dependent diffusion is addressed and our current paper can be viewed as some direct response to it. The necessary condition of the near-optimality is established within the framework of optimality variational principle developed by Yong [\textit{SIAM J. Control Optim.} \textbf{48} (2010) 4119--4156] and obtained by the convergence technique to treat the optimal control of FBSDEs in unbounded control domains by Wu [% \textit{Automatica} \textbf{49} (2013) 1473--1480]. Some new estimates are given here to handle the near optimality. In addition, an illustrating example is discussed as well.