Stochastic irregular linear-quadratic optimal control with dual controllers and two-layer asymmetric partial observations

Abstract

Linear-quadratic (LQ) optimal control problems have been widely studied under symmetric information and regularity assumptions. However, in many practical systems, multiple controllers operate with different information sets, and the standard regularity condition may fail due to singular weighting matrices. This paper investigates a stochastic irregular LQ optimal control problem with dual controllers and two-layer asymmetric partial observations, in which one controller has strictly less information than the other. The main contribution is the derivation of both the solvability conditions and the explicit feedback-form solutions for this problem. To this end, we reformulate the optimization problem as a system of forward-backward stochastic differential equations (FBSDEs) that naturally captures the asymmetric information structure. By solving these FBSDEs via four interconnected Riccati equations, we obtain explicit optimal controllers for both the regular and irregular cases. In the irregular case, where the standard Riccati equation is not directly solvable, we introduce auxiliary Riccati equations and derive additional existence conditions involving singular matrix decompositions. A detailed technical comparison with existing results confirms that our solution is not a straightforward extension, as prior works either addressed irregular LQ control under full information or asymmetric information under regular conditions. This work provides a unified framework for optimal control problems where information asymmetry and singularity arise simultaneously.