Bounded real Lemma of nonlinear systems and its application to H ∞ state feedback control

This paper is concerned with H_∞ control of nonlinear systems. First, some necessary and sufficient conditions for nonlinear systems to be internally stable and to have the L₂ gain less than a specified number γ are given. These conditions are reduced to the well known bounded real lemma when one restricts the systems to be linear ones. The relations between internal stability of nonlinear systems and stabilizing solutions of the Hamilton-Jacobi equation are also clarified, which are peculiar to nonlinear systems. Second, based on the above result on the L₂ gain, a sufficient condition is given for the existence of nonlinear H_∞ state feedback control, which completely corresponds to the linear case