Geometric existence theory for the control-affine \(H_{\infty }\) problem. (English) Zbl 1124.93021

Mccaffrey, D.

J. Math. Anal. Appl. 324, No. 1, 682-695 (2006).

Summary: We consider the differential game formulation of the nonlinear state feedback \(H_{\infty }\) 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 \(\Lambda\). This manifold has a generating function \(S\) quadratic at infinity. A Ljusternik-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 \(H_{\infty }\) problem, and prove existence of a weak globally optimal set valued feedback solution in terms of \(\partial W\), the generalised gradient of \(W\). This feedback generalises, to a maximal region over which \(\Lambda\) is simply connected, the classical smooth feedback defined on the neighbourhood of the origin over which \(\Lambda\) has a well-defined projection onto state space.

MSC:

93B36	\(H^\infty\)-control
93B52	Feedback control
91A23	Differential games (aspects of game theory)

Keywords:

\(H_{\infty }\) control; Lagrangian manifold; Hamilton-Jacobi-Bellman equation; graph selector; Ljusternik-Schnirelmann minimax construction

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