Closed-loop Stackelberg strategies with applications in the optimal control of multilevel systems

Abstract

This paper develops a new approach to obtain the closed-loop Stackelberg (CLS) solution of an important class of two-person nonzero-sum dynamic games characterized by linear state dynamics and quadratic cost functionals. The new technique makes use of an important property of nonunique representations of a closed-loop strategy, and it relates the CLS solution to a particular representation of the optimal solution of a team problem. It is shown that, under certain conditions, the CLS strategies for the leader are linear and of the one-step memory type, while those of the follower can be realized in linear feedback form. Exact expressions are given for the optimal coefficient matrices involved, which can be determined recursively. These results are then extended to multilevel discrete-time control of linear-quadratic systems which are characterized by one central controller and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> second-level controllers. Conditions are obtained under which a one-step memory strategy of the central controller forces the other controllers to a team-optimal solution, while each one of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> second-level controllers is in fact minimizing its own cost function.