Download Stochastic Systems: Estimation, Identification, and Adaptive by P.R. Kumar, Pravin Varaiya PDF

By P.R. Kumar, Pravin Varaiya

ISBN-10: 013846684X

ISBN-13: 9780138466848

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Extra resources for Stochastic Systems: Estimation, Identification, and Adaptive Control

Example text

L N in//3 to be "weakly convergent" to x E ~ , if ~ ' ( x j - x) j ~ oo ~0 V9v E 2 B * and write x j ~ x . Obviously the weak limit is unique. The uniform boundedness principle (BanachSteinhaus theorem), cf. Yosida [72], implies that each bounded sequence (xj)jenv has a weakly convergent subsequence (Xjk)ke~V such that xjk ~ x. In turn, each weakly convergent sequence is bounded. Let A : B 1 ~ J~2 be a linear operator. Its dual A' : IB~ ~ ~ is defined by (A'~)(x) = ~(Ax) Vx ~ B1 If A is bounded on B 1 , then A' is a bounded linear operator too.

Another common approach, see Taylor [81] or Gilkey [84], is based on the Fourier transformation of fast falling distribution on ~'~, and a process of gluing sections by appropriate transition functions. The latter, however, cannot be adopted to the case of 0-manifolds. 33 By construction it is clear that the spaces W : , P F ( F ) are Banach, and the spaces H : F ( F ) are Hilbert spaces. e. ~1 + ~1 = 1, and p > 1. By elementary analysis cd < pc p + Td r for all c, d 9 ~ . 3) follows by estimating (IOl+o(F))q I+,o> l < I~l+o(F) IOljo(F) < x (l~ljo(~)~ ~ 1 II~IIL~ II011L~ -- II~IIL~ II011L~ -- P \ ~ ] +q \ and integration over M.

B1) may depend on the weight parameter a, cf. Owen [81]. This becomes comprehensible by observing that the weighted Sobolev spaces W~'VF(F) correspond to the classical Sobolev spaces W~,PF(/F a) of sections in the induced vector bundle F a over an appropriate compactification of G C ~x~,~, cf. Choquet-Bruhat and DeWitt-Morette [89]. Obviously, different weights are related to different compact~cations. Consequently, the topologies of the corresponding induced bundles s will, in general, differ.

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Stochastic Systems: Estimation, Identification, and Adaptive Control by P.R. Kumar, Pravin Varaiya


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