What are singular values of nonlinear operators?

Abstract

This paper is devoted to a characterization of singular values of nonlinear operators. Although eigenvalue and spectrum analysis for nonlinear operators has been studied by many researchers in mathematics literature, singular value analysis has not been investigated so much. In this paper, a framework of singular value analysis is proposed which is closely related to the operator gain. The proposed singular value analysis is based on the eigenvalue analysis of a special class of nonlinear operators called differentially self-adjoint. Some properties of those operators are clarified which are natural generalization of the linear case results. Furthermore, a sufficient condition for the existence of singular values is provided. The proposed analysis tools are expected to play an important role in nonlinear control systems theory as in the linear case.