Team-optimal closed-loop Stackelberg strategies in hierarchical control problems

This paper is concerned with the derivation of closed-loop Stackelberg (CLS) solutions of a class of continuous-time two-player nonzero-sum differential games characterized by linear state dynamics and quadratic cost functionals. Explicit conditions are obtained for both the finite and infinite horizon problems under which the CLS solution is a representation of the optimal feedback solution of a related team problem which is defined as the joint minimization of the leader's cost function. First, a specific class of representations is considered which depend linearly on the current and initial values of the state, and then the results are extended to encompass a more general class of linear strategies that also incorporate the whole past trajec-tory. The conditions obtained all involve solutions of linear matrix equations and are amenable to computational analysis for explicit determination of CLS strategies. 1. Introduction AN APPROPRIATE solution concept for hierarchical multicriteria decision problems is the Stackelberg solution concept which was first introduced in economics within the context of static economic competition (yon Stackelberg, 1934). Its dynamic version later entered the control literature through the works of Chen and Cruz (1972) and Simaan and Cruz (1973a, b), and found applications in nonzero-sum differential games where one player has enough ability or power to enforce his strategy on the other player(s). Within the context of two-player differential games, the more powerful player is called the leader and the other one is called the follower. An extension is, of course, possible to 'one leader'-'many followers' and even to "many leaders' 'many followers' situations. The initial application of the Stackelberg solution concept to differential games has been within the framework of the open-loop information structure, and the extension to closed-loop information structures has remained as a challenge for a. long time. The main difficulty arises from the fact that under closed-loop information, the reaction set of the follower cannot be determined explicitly, even in the case of linear quadratic (LQ) problems, thus resulting in a nonclassical control problem faced by the leader. One way to circumvent this difficulty is to assume specific parametric structures for the strategy of the leader, in which case the follower's reaction can be explicitly determined, and then the problem faced by the leader is to optimize on those parameters subject to the constraint imposed by the reaction set of the follower. Such an approach has, in fact, been adopted by Medanic (1977),