Feedback Systems on a Reflexive Banach Space—Linearization

Abstract

The aim of our work is to formulate and demonstrate the results of the normality, the Lipschitz continuity, of a nonlinear feedback system described by the monotone maximal operators and hemicontinuous, defined on real reflexive Banach spaces, as well as the approximation in a neighborhood of zero, of solutions of a feedback system [A,B] assumed to be non-linear, by solutions of another linear, This approximation allows us to obtain appropriate estimates of the solutions. These estimates have a significant effect on the study of the robust stability and sensitivity of such a system see [1] [2] [3]. We then consider a linear FS , and prove that, if ; , with the respective solutions of FS’s [A,B] and corresponding to the given (u,v) in . There exists,, positive real constants such that, . These results are the subject of theorems 3.1, ... , 3.3. The proofs of these theorems are based on our lemmas 3.2, ... , 3.5, devoted according to the hypotheses on A and B, to the existence of the inverse of the operator I+BA and . The results obtained and demonstrated along this document, present an extension in general Banach space of those in [4] on a Hilbert space H and those in [5] on a extended Hilbert space .