Geometric existence theory for the control-affine H∞ problem

[摘要] We consider the differential game formulation of the nonlinear state feedback H-infinity control problem, in which the control term enters linearly in the dynamics and quadratically in the cost. Under well-known conditions on the linearisation of this problem around the equilibrium point at the origin, there exists a stable Lagrangian manifold A. This manifold has a generating function S quadratic at infinity. A Lusternick-Schnirelman minimax construction produces from S a Lipschitz function W over state space. We show that, for problems in general position, -W is the lower value function for the Hoc problem, and prove existence of a weak globally optimal set valued feedback solution in terms of a W, the generalised gradient of W. This feedback generalises, to a maximal region over which A is simply connected, the classical smooth feedback defined on the neighbourhood of the origin over which A has a well-defined projection onto state space. (c) 2005 Elsevier Inc. All rights reserved.

[发布日期] 2006-12-01

[关键词] H-infinity control;Lagrangian manifold;Hamilton-Jacobi-Bellman equation;graph selector