Abstract:
We consider an integral variational control system on a Banach space and we study the connections between its uniform exponential stability and the ((？

Abstract:
We give very general characterizations for uniform exponential dichotomy of variational difference equations. We propose a new method in the study of exponential dichotomy based on the convergence of some associated series of nonlinear trajectories. The obtained results are applied to difference equations and also to linear skew-product flows.

Abstract:
We give new and very general characterizations for uniform exponential dichotomy of variational difference equations in terms of the admissibility of pairs of sequence spaces over ？ with respect to an associated control system. We establish in the variational case the connections between the admissibility of certain pairs of sequence spaces over ？ and the admissibility of the corresponding pairs of sequence spaces over ？. We apply our results to the study of the existence of exponential dichotomy of linear skew-product flows.

Abstract:
We present a new approach for the theorems of Perron type for exponential expansiveness of one-parameter semigroups in terms of l^p(N, X) spaces. We prove that an exponentially bounded semigroup is exponentially expansive if and only if the pair (l^p (N, X), l^q(N, X)) is completely admissible relative to a discrete equation associated to the semigroup, where p, q ∈ [1, ∞), p ≥ q. We apply our results in order to obtain very general characterizations for exponential expansiveness of C_0-semigroups in terms of the complete admissibility of the pair (L^ p (R_+ , X), L^ q (R_+ , X)) and for exponential dichotomy, respectively, in terms of the admissibility of the pair (L^p(R_+,X), L^q(R_+,X)).

Abstract:
The aim of this paper is to obtain new necessary and sufficient conditions for the uniform exponential stability of variational difference equations with applications to robustness problems. We prove characterizations for exponential stability of variational difference equations using translation invariant sequence spaces and emphasize the importance of each hypothesis. We introduce a new concept of stability radius for a variational system of difference equations with respect to a perturbation structure and deduce a very general estimate for the lower bound of . All the results are obtained without any restriction concerning the coefficients, being applicable for any system of variational difference equations.

Abstract:
The aim of this paper is to obtain new necessary and sufficient conditions for the uniform exponential stability of variational difference equations with applications to robustness problems. We prove characterizations for exponential stability of variational difference equations using translation invariant sequence spaces and emphasize the importance of each hypothesis. We introduce a new concept of stability radius rstab(A;B,C) for a variational system of difference equations (A) with respect to a perturbation structure (B,C) and deduce a very general estimate for the lower bound of rstab(A;B,C). All the results are obtained without any restriction concerning the coefficients, being applicable for any system of variational difference equations.

Abstract:
We are interested in an open problem concerning the integral characterizations of the uniform exponential trichotomy of skew-product flows. We introduce a new admissibility concept which relies on a double solvability of an associated integral equation and prove that this provides several interesting asymptotic properties. The main results will establish the connections between this new admissibility concept and the existence of the most general case of exponential trichotomy. We obtain for the first time necessary and sufficient characterizations for the uniform exponential trichotomy of skew-product flows in infinite-dimensional spaces, using integral equations. Our techniques also provide a nice link between the asymptotic methods in the theory of difference equations, the qualitative theory of dynamical systems in continuous time, and certain related control problems.

Abstract:
We present a new perspective concerning the study of the asymptotic behavior of variational equations by employing function spaces techniques. We give a complete description of the dichotomous behaviors of the most general case of skew-product flows, without any assumption concerning the flow, the cocycle or the splitting of the state space, our study being based only on the solvability of some associated control systems between certain function spaces. The main results do not only point out new necessary and sufficient conditions for the existence of uniform and exponential dichotomy of skew-product flows, but also provide a clear chart of the connections between the classes of translation invariant function spaces that play the role of the input or output classes with respect to certain control systems. Finally, we emphasize the significance of each underlying hypothesis by illustrative examples and present several interesting applications.

Abstract:
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