Stability and Stabilizability for Linear Systems of Difference Equations

On Stabilizability and Detectability of Variational Control Systems

“…A remarkable class of Banach function spaces is represented by the translations invariant spaces. These spaces have a special role in the study of the asymptotic properties of the dynamical systems using control type techniques (see Sasu (2008), Sasu & Sasu (2004) …”

“…A special application of our main results will be the study of the connections between the exponential stability and the stabilizability and detectability of nonautonomous control systems in infinite dimensional spaces. The nonautonomous case treated in this chapter will include as consequences many interesting situations among which we mention the results obtained by Clark, Latushkin, Montgomery-Smith and Randolph (see (Clark et al, 2000)) and the authors (see (Sasu & Sasu, 2004)) concerning the connections between stabilizability, detectability and exponential stability.…”

“…Two fundamental concepts related to the asymptotic behavior of the associated perturbed systems (see (Clark et al, 2000), (Curtain & Zwart, 1995), (Sasu & Sasu, 2004)) are described by stabilizability and detectability as follows: Definition 1.5. The system (π, B, C) is said to be: (i) stabilizable if there exists a mapping F ∈C s (Θ, B(X, U)) such that the system (S π BF ) is uniformly exponentially stable; (ii) detectable if there exists a mapping K ∈C s (Θ, B(Y, X)) such that the system (S π KC ) is uniformly exponentially stable.…”

Stability of Difference Equations and Applications to Robustness Problems

“…In the study of the asymptotic behavior of discrete-time systems, there is an increasing interest in finding methods arising from control theory see 5-8, 21-27, 29-32 . This is motivated by the fact that besides their large applicability area, the control-type techniques can be also applied to the analysis of the robustness of diverse properties in the presence of perturbations see 5,25,26,30,32,33 . In this context it is natural to extend the study to the variational case.…”

“…The input-output methods or the so-called "theorems of Perron type" have proved to be important tools in the study of the asymptotic behavior of difference equations like stability see 27,29,30 , expansiveness see 6 , dichotomy see 1,5,8,15,17,19,20,32 , and trichotomy as well see 7, 18, 20 . A distinct method for the study of exponential stability relies on the convergence of some associated series and this was used in 28 for difference equations and in 4 for variational difference equations. The exponential stability of difference equations with several variable delays and variable coefficients was studied in 16 , where the authors obtained interesting conditions for global exponential stability using new computational formulas with respect to the coefficients.…”