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In this paper, we study a linear-quadratic partially observed Stackelbergstochastic differential game problem in which a single leader and multiplefollowers are involved. We consider more practical formulation for partialinformation that none of them can observed the complete information and thefollowers know more than the leader. Some completely different methodsincluding orthogonal decomposition are applied to overcome the difficultiescaused by partially observability which improves the tools and relaxes theconstraint condition imposed on admissible control in the existing literature.More precisely, the followers encounter the standard linear-quadratic partiallyobserved optimal control problems, however, a kind of forward-backwardindefinite linear-quadratic partially observed optimal control problem isconsidered by the leader. Instead of maximum principle of forward-backwardcontrol systems, inspired by the existing work related to definite case andclassical forward control system, some distinct forward-backwardlinear-quadratic decoupling techniques including the method of completion ofsquares are applied to solve the leader's problem. More interestingly, wedevelop the deterministic formation control in multi-agent system with aframework of Stackelberg differential game and extend it to the stochasticcase. The optimal strategies are obtained by our theoretical result suitably.

A linear-quadratic partially observed Stackelberg stochastic  differential game with multiple followers and its application to multi-agent  formation control