Stabilizability for nonlinear systems of difference equations

International Journal of Mathematics and Mathematical Sciences

2014

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The aim of this paper is to characterize the exponential stability of linear systems of difference equations with slowly varying coefficients. Our approach is based on the generalization of the freezing method for difference equations combined with new estimates for the norm of bounded linear operators. The main novelty of this work is that we use estimates for the absolute values of entries of a matrix-valued function, instead of bounds on its eigenvalues. By this method, new explicit stability criteria for linear nonautonomous systems are derived.

“…provided the series in (14) converges. Important contributions to the theory of matrix-valued functions can be found in Verde-Star [21].…”

Section: Preliminaries and Problem Statementmentioning

confidence: 96%

“…Recently, Gil and Medina [11,12] and Medina [13][14][15] begun the study of stability and stabilizability theory for discrete-time systems by means of new estimates for the powers of matrix-valued and operator-valued functions.…”

Section: Introductionmentioning

confidence: 99%

Exponential Stability Criteria for Nonautonomous Difference Systems

International Journal of Mathematics and Mathematical Sciences

2014

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“…Such an approach has already been successfully applied in the field of discrete‐time systems. See, for example, Desoer , Gil'&Medina , Solo , and Medina .…”

Section: Introductionmentioning

confidence: 99%

Exponential stability of slowly varying discrete systems with multiple state delays

Intl J Robust & Nonlinear

2012

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Exponential stability criteria for discrete time‐delay systems

Intl J Robust & Nonlinear

Martı́nez

2013

SUMMARYIn this paper, we study the exponential stability of linear discrete time-delay systems with slowly varying coefficients and nonlinear perturbations. We establish the robustness of the exponential stability in Hilbert spaces, in the sense that the exponential stability for a given linear equation persists under sufficiently small perturbations. As an application of the main results, we discuss the exponential stability of a general nonlinear system. The main novelty of this work is that we always consider the exponential behavior of solutions with respect to an specific ball.