Stability of nonautonomous differential equations in Hilbert spaces

Barreira, Luís; Valls, Clàudìa

doi:10.1016/j.jde.2005.05.008

Cited by 25 publications

(31 citation statements)

References 10 publications

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“…We note that Theorem 12 is a discrete time version of results in [2]. The following is another stability result.…”

Section: Stability For Nonuniform Contractionsmentioning

confidence: 78%

“…where C(n, 1) is defined in a similar manner to that in (2). Therefore, C(n, 1)e i , e j = U * n A(n, 1)e i , e j = A(n, 1)e i , U n e j = v i (n), u j (n) = 0 whenever i < j, in view of (27).…”

Section: Upper Triangular Reductionmentioning

confidence: 99%

“…The argument is inspired in the proof of Theorem 15 in [2]. Given an invertible k × k matrix C, let v 1 (n), .…”

Section: Nonuniform Exponential Contractionsmentioning

confidence: 99%

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Stability theory and Lyapunov regularity

Barreira

Valls

2007

Journal of Differential Equations

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We establish the stability under perturbations of the dynamics defined by a sequence of linear maps that may exhibit both nonuniform exponential contraction and expansion. This means that the constants determining the exponential behavior may increase exponentially as time approaches infinity. In particular, we establish the stability under perturbations of a nonuniform exponential contraction under appropriate conditions that are much more general than uniform asymptotic stability. The conditions are expressed in terms of the so-called regularity coefficient, which is an essential element of the theory of Lyapunov regularity developed by Lyapunov himself. We also obtain sharp lower and upper bounds for the regularity coefficient, thus allowing the application of our results to many concrete dynamics. It turns out that, using the theory of Lyapunov regularity, we can show that the nonuniform exponential behavior is ubiquitous, contrarily to what happens with the uniform exponential behavior that although robust is much less common. We also consider the case of infinite-dimensional systems.

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“…We note that Theorem 12 is a discrete time version of results in [2]. The following is another stability result.…”

Section: Stability For Nonuniform Contractionsmentioning

confidence: 78%

Section: Upper Triangular Reductionmentioning

confidence: 99%

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Stability theory and Lyapunov regularity

Barreira

Valls

2007

Journal of Differential Equations

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“…We emphasize that in a certain sense our assumptions are the weakest possible under which it is possible to establish the persistence of the stability. We refer to [3] for a detailed related discussion in the case of ordinary differential equations. We also refer to the book [9] for a detailed discussion of the geometric theory in the infinite-dimensional setting, with emphasis on delay differential equations.…”

Section: Introductionmentioning

confidence: 99%

“…Essentially, the existence of nonzero Lyapunov exponents leads to a nonuniform exponential behavior. We refer to the books [1,3] for detailed discussions. Here we consider only the case of negative Lyapunov exponents, and of delay r = 0 in equation (1.5), with Y = R p for some p ∈ N. In this case, each linear operator L m is a p × p matrix A m , and if lim sup…”

Section: Introductionmentioning

confidence: 99%

Stability Theory of General Contractions for Delay Equations

Barreira

Valls

2010

Integr. Equ. Oper. Theory

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We study the persistence of the asymptotic stability of delay equations both under linear and nonlinear perturbations. Namely, we consider nonautonomous linear delay equations v = L(t)vt with a nonuniform exponential contraction. Our main objective is to establish the persistence of the nonuniform exponential stability of the zero solution both under nonautonomous linear perturbations, i.e., for the equation v = (L(t) + M (t))vt, thus discussing the so-called robustness problem, and under a large class of nonlinear perturbations, namely for the equation v = L(t)vt + f (t, vt). In addition, we consider general contractions e −λρ(t) determined by an increasing function ρ that includes the usual exponential behavior with ρ(t) = t as a very special case. We also obtain corresponding results in the case of discrete time.Mathematics Subject Classification (2010). Primary 34K20.

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Stability of Nonautonomous Differential Equations

Barreira¹,

Valls²

2008

Lecture Notes in Mathematics

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163

158

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The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.Typesetting by the authors and SPi using a Springer L A T E X macro package Cover design: design & production GmbH, HeidelbergPrinted on acid-free paper SPIN: 12114993 41/SPi 5 4 3 2 1 0To our parents PrefaceThe main theme of this book is the stability of nonautonomous differential equations, with emphasis on the study of the existence and smoothness of invariant manifolds, and the Lyapunov stability of solutions. We always consider a nonuniform exponential behavior of the linear variational equations, given by the existence of a nonuniform exponential contraction or a nonuniform exponential dichotomy. Thus, the results hold for a much larger class of systems than in the "classical" theory of exponential dichotomies. The departure point of the book is our joint work on the construction of invariant manifolds for nonuniformly hyperbolic trajectories of nonautonomous differential equations in Banach spaces. We then consider several related developments, concerning the existence and regularity of topological conjugacies, the construction of center manifolds, the study of reversible and equivariant equations, and so on. The presentation is self-contained and intends to convey the full extent of our approach as well as its unified character. The book contributes towards a rigorous mathematical foundation for the theory in the infinite-dimensional setting, also with the hope that it may lead to further developments in the field. The exposition is directed to researchers as well as graduate students interested in differential equations and dynamical systems, particularly in stability theory.The first part of the book serves as an introduction to the other parts. After giving in Chapter 1 a detailed introduction to the main ideas and motivations behind the theory developed in the book, together with an overview of its contents, we introduce in Chapter 2 the concept of nonuniform exponential dichotomy, which is central in our approach, and we discuss some of its basic properties. Chapter 3 considers the problem of the robustness of nonuniform exponential dichotomies.In the second part of the book we discuss several consequences of local nature for a nonlinear system when the associated linear variational equation admits a nonuniform exponential dichotomy. In particular, we establish in Chapter 4 the existence of Lipschitz stable manifolds for nonautonomous equations in a Banach space. In Chapters 5 and 6 we establish the smooth-VIII Preface ness of the stable manifolds. We first consider the finite-dimensional case in Chapter 5, with the method of invariant families of cones. This approach uses in a decisive manner the compactness of the closed unit ball in the ambient space, and this is why we consider only finite-dimensional spaces in this chapter. Moreover, ...

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Stability of nonautonomous differential equations in Hilbert spaces

Cited by 25 publications

References 10 publications

Stability theory and Lyapunov regularity

Stability theory and Lyapunov regularity

Stability Theory of General Contractions for Delay Equations

Stability of Nonautonomous Differential Equations

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