Linear Skew-product Semiflows Research Articles

Lyapunov Type Theorems for Exponential Stability of Linear Skew-Product Three-Parameter Semiflows with Discrete Time

For linear skew-product three-parameter semiflows with discrete time acting on an arbitrary Hilbert space, we obtain a complete characterization of exponential stability in terms of the existence of appropriate Lyapunov functions. As a nontrivial application of our work, we prove that the notion of an exponential stability persists under sufficiently small linear perturbations.

Individual exponential stability for linear skew-products semiflows over a semiflow

In this paper we consider individual stability for linear skew-product semiflows over semiflows by using the standard Perron’s method. To our knowledge, in the study of this problem, the constant from the boundedness theorem has been so far always assumed to be independent of the variable from the metric space. The novelty of our approach consists in being able to choose this constant for each value of the metric space variable in a separate manner.

Discrete-time theorems for global and pointwise dichotomies of cocycles over semiflows

In this paper, we consider linear skew-product semiflows on bundles of Banach fibers over a locally compact metric space. Our aim is to give discrete-time theorems for the existence of global and pointwise continuous-time dichotomies with no invariant unstable manifolds. We involve here a concept of exponential dichotomy for skew-product semiflows weaker than the concept used by Sacker and Sell (J Differ Equ 15:429–458, 1974; 22:478–522, 1976) and Magalhaes (in: Chow, Hale (eds) Dynamics of infinite dimensional systems, NATO Advanced Science Institutes Series F: Computer and Systems Sciences, Springer, New York, vol 37, pp 161–168, 1987); our definition (of no past exponential dichotomy) follows roughly the definition given by Chow and Leiva (Proc Am Math Soc 124:1071–1081, 1996) in the sense that we allow the unstable subspace to have infinite dimension. The main improvement is that we go even more general and we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable subspace is invariant under the cocycle). Roughly speaking, we prove that if the solution of the corresponding inhomogeneous variational difference equation belongs to any sequence space (on which the right shift is an isometry) for every inhomogeneity from the same class of sequence spaces, then the continuous-time solutions of the variational homogeneous differential equation will exhibit a (no past) exponential dichotomic behavior. This approach has many advantages among which we emphasize on the facts that the above condition is very general (since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical p-summable spaces, sequence Orlicz spaces, etc.). Since we use a discrete-time technique we are not forced to require any continuity or measurability hypotheses on the trajectories of the exponentially bounded cocycle. Also, it is worth to mention that from discrete-time conditions we get informations about the continuous-time behavior of the solutions of differential variational equations.

The Selgrade Decomposition for Linear Semiflows on Banach Spaces

We extend Selgrade's Theorem, Morse spectrum, and related concepts to the setting of linear skew product semiflows on a separable Banach bundle. We recover a characterization, well-known in the finite-dimensional setting, of exponentially separated subbundles as attractor-repeller pairs for the associated semiflow on the projective bundle.

Nonuniform exponential dichotomy for linear skew-product semiflows over semiflows

The aim of this paper is to obtain necessary and sufficient conditions for the existence of a nonuniform exponential dichotomy over a general class of linear skew-product semiflows (over semiflows) on a Banach space. We extend Datko’s classical result to the case of the exponential nonuniform dichotomy of linear skew-product semiflows over semiflows on a Banach space, by using Lyapunov norms.

Extending Some Results of L. Barreira and C. Valls to the Case of Linear Skew-Product Semiflows

In a paper from 2010, Barreira and Valls (J Differ Equ 249:2889–2904, 2010) use Lyapunov norms to set up admissibility conditions for nonuniform exponential contractions. In 2011, the same authors extend their analysis from Barreira and Valls (J Differ Equ 249:2889–2904, 2010) to the case of nonuniform exponential dichotomy for linear evolution families with (non)uniform exponential growth (see Barreira and Valls in Discret Contin Dyn Syst 30(1):39–53, 2011). Following their approach, we are able to choose appropriate “test-functions” to establish admissibility-type conditions for the existence of a (non)uniform exponential dichotomy of a strongly continuous cocycle with (non)uniform exponential growth.

Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

We determine sufficient conditions for uniform and strict persistence in the case of skew-product semiflows generated by solutions of non-autonomous families of cooperative systems of ODEs or delay FDEs in terms of the principal spectrums of some associated linear skew-product semiflows which admit a continuous separation. Our conditions are also necessary in the linear case. We apply our results to a noncooperative almost periodic Nicholson system with a patch structure, whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution.

An extension of a theorem of R. Datko to the case of (non)uniform exponential stability of linear skew-product semiflows

In 1970 R. Datko proved that the trajectories of a C0-semigroup {T(t)}t≥0 on a Hilbert space X, exhibit an exponential decay if and only if they stay in L2(R+,X). Datko's theorem was crucial in extending the Lyapunov operator equation to the case of autonomous systems x˙=Ax with unbounded A. Extensions have been done to the case of uniform exponential stability of evolution families and more recently of LSPS (linear skew-product semiflow). The aim of the present paper is to extend Datko's result to the general case of (non)uniform exponential stability of the LSPS that does not necessarily possess a uniform exponential growth.

Some results on the qualitative theory of semiflows

We point out some results on the qualitative theory of three-parameter semiflows by analyzing the stability of the associated linear skew-product semiflow. Extensions of the well-known theorems due to Datko, Pazy, Rolewicz are obtained.

THE RELATION BETWEEN THE UNIFORM EXPONENTIAL DICHOTOMY AND THE UNIFORM ADMISSIBILITY OF THE PAIR (lp, lq) on ⊝

The goal of this paper is to present a characterization of the uniform exponential dichotomy property for linear skew-product semiflows using a discrete-time approach.

Discretized characterizations of exponential dichotomy of linear skew-product semiflows over semiflows

Using the method of discretization, we investigate the necessary and sufficient conditions for the existence of exponential dichotomy of linear skew-product semiflows over semiflows through the existence of discrete exponential dichotomy of the discretized linear-skew product semiflows. We then apply the obtained results to consider the roughness of exponential dichotomy of linear-skew product semiflows.

Exponential dichotomy and admissibility of linearized skew-product semiflows defined on a compact positively invariant subset of semiflows

We study exponential dichotomy of linear skew-product semiflows which come from linearizing skew-product semiflows on a compact positively invariant subset M of semiflows and construct the relationship between continuous separation and exponential dichotomy under assumptions that skew-product semiflows are eventually strongly monotone. In addition, we deduce that the exponential dichotomy is trivial when M is hyperbolically stable, and the hyperbolic instability of M is the necessary condition of the state space admitting a trivial separation in another forms. Simultaneously, we list some conditions for hyperbolic stability and instability of M . At last, we construct a sufficient and necessary condition for exponential dichotomy of linear skew-product semiflows in terms of the admissibility of the pair ( B ( R + , X ) , B 0 ( R + , X ) ) .

Functionals on normed function spaces and exponential instability of linear skew-product semiflows

The aim of this paper is to give necessary and sufficient conditions for uniform exponential instability of linear skew-product semiflows in terms of functionals on normed function spaces. We obtain the versions of some results due to Datko, Pazy, Neerven and Rolewicz for the case of instability of linear skew-product semiflows.

Existence and robustness of exponential dichotomy of linear skew-product semiflows over semiflows

In this paper we investigate the exponential dichotomy of linear skew-product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on Banach bundles. We characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy.

On the uniform exponential stability of linear skew‐product semiflows

The problem of uniform exponential stability of linear skew‐product semiflows on locally compact metric space with Banach fibers, is discussed. It is established a connection between the uniform exponential stability of linear skewproduct semiflows and some admissibility‐type condition. This approach is based on the method of “test functions”, using a very large class of function spaces, the so‐called Orlicz spaces.

Schäffer spaces and uniform exponential stability of linear skew-product semiflows

We study the exponential stability of linear skew-product semiflows on locally compact metric space with Banach fibers. Our main tool is the admissibility of a pair of the so-called Schäffer spaces. This characterization is a very general one, it includes as particular cases many interesting situations among them we can mention some results due to Clark, Datko, Latushkin, van Minh, Montgomery–Smith, Randolph, Räbiger, Schnaubelt.

A lower bound for the stability radius of time-varying systems

We introduce and characterize the stability radius of systems whose state evolution is described by linear skew-product semiflows. We obtain a lower bound for the stability radius in terms of the Perron operators associated to the linear skew-product semiflow. We generalize a result due to Hinrichsen and Pritchard.

Exponential instability of linear skew-product semiflows in terms of Banach function spaces

In this paper we obtain necessary and sufficient conditions for uniform exponential instability of linear skew-product semiflows in terms of Banach sequence spaces and Banach function spaces, respectively. We deduce the versions of some theorems due to Datko, Neerven, Przyluski, Rolewicz and Zabczyk, for the case of instability of linear skew-product semiflows.

Integral characterizations for stability of linear skew-product semiflows

We give necessary and sufficient conditions for uniform exponential stability of linear skew-product semiflows. Using the theory of Banach function spaces we obtain integral characterizations for this concept. We extend a stability theorem obtained by Rolewicz for evolution families, at the general case of linear skew-product semiflows. Mathematics subject classification (2000): 34D05, 46E30, 93D20.

Theorems of Perron type for uniform exponential dichotomy of linear skew-product semiflows.

We study the connections between the uniform exponential dichotomy of a discrete linear skew-product semiflow and the uniform admissibility of the pair $(c_{0}(\n,X),c_{00}(\n, X))$. We give necessary and sufficient conditions for uniform exponential dichotomy of linear skew-product semiflows in terms of the uniform admissibility of the pairs $(c_{0}(\n, X),c_{00}(\n, X))$ and $(C_0(\r, X),$ $C_{00}(\r, X))$, respectively. We generalize a dichotomy theorem due to Van Minh, Rabiger and Schnaubelt for the case of linear skew-product semiflows.