Closed-loop solvability of delayed control problems: A stochastic Volterra system approach
Weijun Meng, Tianxiao Wang, Ji-Feng Zhang
TL;DR
This work addresses closed-loop solvability for a general stochastic LQ control problem with time-varying coefficients and simultaneous state and control delays in both the dynamics and the cost. It introduces a transformative approach that recasts the delayed system as a finite-dimensional SVIE without delay, enabling a Riccati–Volterra framework and a Type-II extended backward SVIE to characterize optimal feedback controls. The authors construct an explicit optimal closed-loop strategy $(K_1^*,K_2^*,K_3^*,K_4^*,v^*)$ and derive a corresponding value function, proving solvability under mild, integrability-based assumptions and showing consistency with known special cases. The framework unifies and extends previous results, bypasses infinite-dimensional lifting, and applies to stochastic integro-differential systems with memory, broadening the applicability of closed-loop LQ control in delayed stochastic environments.
Abstract
A general and new stochastic linear quadratic optimal control problem is studied, where the coefficients are allowed to be time-varying, and both state delay and control delay can appear simultaneously in the state equation and the cost functional. The closed-loop outcome control of this delayed problem is given by a new Riccati system whose solvability is carefully established. To this end, a novel method is introduced to transform the delayed problem into a control problem driven by a stochastic Volterra integral system without delay. This method offers several advantages: it bypasses the difficulty of decoupling the forward delayed state equation and the backward anticipated adjoint equation, avoids the introduction of infinite-dimensional spaces and unbounded control operators, and ensures that the closed-loop outcome control depends only on past state and control, without relying on future state or complex conditional expectation calculations. Finally, several particular important stochastic systems are discussed. It is found that the model can cover a class of stochastic integro-differential systems, whose closed-loop solvability has not been available before.