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.
Highlights
IntroductionIn the last decades an increasing interest was focused on the asymptotic properties of the most general class of evolution equations—the variational systems and a number of open questions were answered, increasing the applicability area to partial differential equations and to systems arising from the linearization of nonlinear equations see 1–10 and the references therein
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
In the last decades an increasing interest was focused on the asymptotic properties of the most general class of evolution equations—the variational systems and a number of open questions were answered, increasing the applicability area to partial differential equations and to systems arising from the linearization of nonlinear equations see 1–10 and the references therein
Summary
In the last decades an increasing interest was focused on the asymptotic properties of the most general class of evolution equations—the variational systems and a number of open questions were answered, increasing the applicability area to partial differential equations and to systems arising from the linearization of nonlinear equations see 1–10 and the references therein. 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 applied to the analysis of the robustness of diverse properties in the presence of perturbations see 5, 25, 26, 30, 32, 33. Our target is to obtain a lower bound for the stability radius of variational systems of difference equations as well as to determine the largest class of Banach sequence spaces within the robustness properties hold. With this purpose we associate with the system. We note that the study is done without any restriction or assumption on the coefficients, the obtained results being applicable for any system of variational difference equations
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