New PDF release: An Introduction to Infinite Dimensional Dynamical Systems —

By Jack K. Hale, Luis T. Magalhães, Waldyr M. Oliva (auth.)

ISBN-10: 0387909311

ISBN-13: 9780387909318

ISBN-10: 1475744935

ISBN-13: 9781475744934

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Extra info for An Introduction to Infinite Dimensional Dynamical Systems — Geometric Theory

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J for some e lR} by as a Ck g ejV. 1-4)). 4, taking has finite codimension in CO. be a basis for a linear complement of range(A - I). Let By the argument of the preceding paragraph, one can get the value of the solution i y t* of yet) = f'(xt)Yt + gi(xt ), yo = 0, arbitrarily close to choosing the i gl'· •. ,gJ appropriately in ~ can be made so close that Yt* J in the definition of f' (xt)Y t + g(x t ). as ro Thus, for oi ro i = Yti *, i form a basis for a r = 0nx*t* (tn* .... •• 'Yt* linear complement of range(A - I).

From Lemma 1/1 E,/. 1, , / is nonempty, compact, connected and invariant. To prove that ,/ is the maximal compact invariant set, suppose is any compact invariant set. it follows that Hc Since K attracts H Hand H is invariant, (i'. nr K and, therefore, H c <7'. ~ It remains to prove that ,/ is independent of the choice of the o compact set , / = ,/(K) K which attracts all compact sets of C and ,/(K l ) = nn>O ~nrKl - 0 tracts all compact sets of C where Both /(K) compact, and they are attracted by both /(K I ) c K and /(K) c Kl ~nrKI,,/(KI) is a compact set which atand ,/(K I ) K and c ~nrK For this, denote KI .

O t ~ T. Since. 4. A(G) = nn>O ~nr(C ). ~tC it follows that A(G) c U. The preceding argument requires the use of converse theorems on asymptotic stability. establishing the existence of "Liapunov functions". An alternative proof can be given as follows. A(F) is compact and attracts CO. 4. the set U denote an arbitrarily small neighborhood of A(F). say consisting of all points at a distance from A(F) smaller than a certain one can show that £ > 0. Based on Gronwall's inequality ~~(~) and ~~(~) can be made as close as -52- desired, uniformly in ~ € CO G € V c~l, by choosing V to be a and sufficiently small neighborhood of F in ~l.

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An Introduction to Infinite Dimensional Dynamical Systems — Geometric Theory by Jack K. Hale, Luis T. Magalhães, Waldyr M. Oliva (auth.)


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