Bogdan Sasu

On nonuniform exponential dichotomy of evolution operators in Banach spaces

The problem of nonuniform exponential dichotomy of evolution operators in Banach spaces is considered. Connections between this concept and admissibility of the pair (C0,C0) are established. Generalizations to the nonuniform case of some results of Van Min, Räbiger and Schnaubelt ([MRS]) are obtained. It is shown that an implication from the uniform case is not true for nonuniform exponential dichotomy. The results are applicable for general time-varying linear equations with unbounded coefficients in Banach spaces.

Stability and Stabilizability for Linear Systems of Difference Equations

The aim of this paper is to characterize the uniform exponential stability of difference equations. We obtain very general input–output conditions for stability of difference equations using diverse vector valued sequence spaces. As an application, we obtain an estimation for the lower bound of the stability radius of a linear control system of difference equations. Finally, we characterize the stability of systems of difference equations in terms of stabilizability and detectability, obtaining discrete-time versions for a result due to Clark, Latushkin, Montgomery-Smith and Randolph.

Exponential trichotomy for variational difference equations

The aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of variational difference equations in terms of the solvability of an associated discrete-time system. First, we obtain the structure of the trichotomy projections. Then, we associate with a linear system of variational difference equations (A) an input–output system (S A ). The linear space of all sequences with finite support contained in Z + is denoted by ℱ(Z, X). We show that the uniform admissibility of the pair (ℓ∞(Z, X), ℱ(Z, X)) for the system (S A ) is a sufficient condition for the existence of uniform exponential trichotomy of the system (A). Next, we prove that the system (A) is uniformly exponentially trichotomic if and only if the pair (ℓ∞(Z, X), ℱ(Z, X)) is uniformly admissible for the associated input–output system (S A ). We also prove that the uniform exponential trichotomy of a linear skew-product flow is equivalent with the uniform exponential trichotomy of the variational difference equation associated with it. We apply our results to the study of the uniform exponential trichotomy of general linear skew-product flows in infinite-dimensional spaces.

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.

Input–output control systems and dichotomy of variational difference equations

We propose a new and unified approach for the study of dichotomy of variational difference equations, establishing a link between control methods and basic techniques from interpolation theory. We obtain necessary and sufficient conditions for the existence of uniform dichotomy and, respectively, for uniform exponential dichotomy of variational difference equations in terms of the admissibility of general pairs of sequence spaces. We provide a classification of the main classes of sequence spaces where the input spaces and the output spaces may belong to, for each dichotomy property and prove that the hypotheses on the underlying sequence spaces cannot be removed. The obtained results extend the framework to the study of dichotomy of variational difference equations, hold without any requirement on the coefficients and are applicable to all systems of variational difference equations.

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.

A Zabczyk type method for the study of the exponential trichotomy of discrete dynamical systems

The aim of this paper is to present new discrete-time characterizations for uniform exponential trichotomy of dynamical systems. We obtain necessary and sufficient conditions for the existence of uniform exponential trichotomy of discrete dynamical systems, using a set of computational conditions of Zabczyk type. The main results are applied to deduce new criteria for the detection of the uniform exponential trichotomy of dynamical systems modeled by skew-product flows.

Integral Equations and Exponential Trichotomy of Skew-Product Flows

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.

Translation Invariant Spaces and Asymptotic Properties of Variational Equations

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.

Integral equations in the study of the asymptotic behavior of skew-product flows

The aim of this paper is to present an inedit perspective concerning the study of the asymptotic behavior of variational systems, using distinct techniques compared with those in the existing literature. We give new and complete answers concerning the study of exponential dichotomy of skew-product flows in terms of the solvability of integral equations between L p -spaces, motivating the techniques by illustrative examples. We present a comparative analysis between the integral admissibility on the real line and on the half-line, pointing out the main differences as well as the advantages of each case.