Feedback Systems on a Reflexive Banach Space—Linearization

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 0 0 , A B , and prove that, if ( ) ( ) , , u v e f ; ( ) ( ) 0 0 , , u v e f , 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 a M . The results ob- tained 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 e H .