By C.J. Harris and J.M.E. Valenca (Eds.)

ISBN-10: 0123276802

ISBN-13: 9780123276803

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**Extra resources for The Stability of Input-Output Dynamical Systems**

**Example text**

Let p: C,+ represent the map which assigns f~ C, to (f,) E when fT = PTf, since fT = the element (fT) lies in k. 2 T rt t c 1. MATHEMATICAL PRELIMINARIES 19 The coherence of this definition follows from the equivalence f s ( t ) = f T ( f ) for all S > T. Finally @ is one-to-one since @f = 0 means that PTf = 0 for all T>O and therefore f ( t ) - 0 . As vector spaces we can then conclude are isomorphic. Let us now compare topologies. l. , r } which can be rewritten as U= f? {fECe:IIPT,fll<&) I=' n C, n p g 1 ) ~ , , ( t I ) n = I=I where B,,(&) is the open ball in C ,of radius 6,.

Consider then f l = f h with h an absolutely continuous homeomorphism of Zl into Z. ) is also absolutely continuous and its variation in Zl coincides with the variation of f in Z. Given a path f : Z X it is always possible to define a homeomorphism h :[0, 13- X such that f and f 0 h are equivalent. Hence it is always possible to think of a path in X as being defined by a function whose domain is [0, 11. In this work, unless otherwise stated, it is assumed that a path is defined by a function f(-)whose domain is [0, 13.

8 Every positive measurable function f :R , 4R , can be expressed as the limit everywhere of an increasing sequence of positive simple functions. 8 is fundamental in the following definition of an integral. Let p be a Borel measure; we can define in S ( B ) a positive linear functional fi : S(B)+ R through for fi(4)= CIP(EI)+ CZP(EZ)+. + c,p(En) * 4 = c 1 x E I + . *+cnxE, Clearly if 4 2 0 , then fi(4)rO;therefore fi is a positive functional. , czxE,+. + cnxE, be such that C$ 2 0 and Conversely, let 4 = c l x E + f i ( $ ) = O .

### The Stability of Input-Output Dynamical Systems by C.J. Harris and J.M.E. Valenca (Eds.)

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