A linear-quadratic partially observed Stackelberg stochastic differential game with multiple followers and its application to multi-agent formation control
Yichun Li, Yaozhong Hu, Jingtao Shi, Yueyang Zheng
TL;DR
The paper tackles a $LQ$ partially observed Stackelberg stochastic differential game with one leader and multiple followers under asymmetrical information. It develops an orthogonal-decomposition approach to separate costs and enable a completely observed reformulation, then solves the followers' problems via filtering, Riccati equations, and BSDEs to obtain state-filtered feedback controls. For the leader, it treats a forward–backward stochastic system with conditional mean-field terms, employing dimension enlargement and completion-of-squares to derive a linear feedback control expressed through filtered states and adjoint processes, again governed by coupled Riccati equations and BSDEs. The methodology is then applied to stochastic multi-agent formation control on graphs, yielding explicit, distributed feedback strategies for followers and a centralized leader, thereby providing a tractable framework for emergent formation behavior under partial information with potential extensions to full-information or privacy-preserving settings.
Abstract
In this paper, we study a linear-quadratic partially observed Stackelberg stochastic differential game problem in which a single leader and multiple followers are involved. We consider more practical formulation for partial information that none of them can observed the complete information and the followers know more than the leader. Some completely different methods including orthogonal decomposition are applied to overcome the difficulties caused by partially observability which improves the tools and relaxes the constraint condition imposed on admissible control in the existing literature. More precisely, the followers encounter the standard linear-quadratic partially observed optimal control problems, however, a kind of forward-backward indefinite linear-quadratic partially observed optimal control problem is considered by the leader. Instead of maximum principle of forward-backward control systems, inspired by the existing work related to definite case and classical forward control system, some distinct forward-backward linear-quadratic decoupling techniques including the method of completion of squares are applied to solve the leader's problem. More interestingly, we develop the deterministic formation control in multi-agent system with a framework of Stackelberg differential game and extend it to the stochastic case. The optimal strategies are obtained by our theoretical result suitably.