Geometric existence theory for the control-affine H∞ problem

We consider the differential game formulation of the nonlinear state feedback H∞ 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 Λ. 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 H∞ problem, and prove existence of a weak globally optimal set valued feedback solution in terms of ∂W, the generalised gradient of W. This feedback generalises, to a maximal region over which Λ is simply connected, the classical smooth feedback defined on the neighbourhood of the origin over which Λ has a well-defined projection onto state space.