H∞ control, stabilization, and input-output stability of nonlinear systems with homogeneous properties

In this paper, we discuss the problems about H(infinity) control, stabilization and input-output stability of nonlinear systems, which may not satisfy known regularity conditions related to smoothness. To this end, a class of homogeneous systems and systems that can be approximated by homogeneous systems, are concentrated on. New relationships are established between H(infinity) control and stabilization, and between L(p) stability and Lyapunov stability. At first, with Hamilton-Jacobi-Isaacs inequality, the nonlinear H(infinity) control problem of the systems with homogeneous properties is discussed. The results show that their stabilizability via homogeneous feedback, in the case without exogenous input signals, implies the solvability of their H(infinity) control problem. Then, simply formulated results on input-output stability are obtained, based on the relations among the homogeneity degrees concerned with the considered systems. The conclusions hold globally for homogeneous systems and locally for those that can be homogeneously approximated. (C) 2001 Elsevier Science Ltd. All rights reserved.