Structured stability radii and exponential stability tests for Volterra difference systems

Abstract

Uniform exponential (UE) stability of linear difference equations with infinite delay is studied using the notions of a stability radius and a phase space. The state space X is supposed to be an abstract Banach space. We work both with non-fading phase spaces c 0 (Z − , X ) and ℓ ∞ (Z − , X ) and with exponentially fading phase spaces of the ℓ p and c 0 types. For equations of the convolution type, several criteria of UE stability are obtained in terms of the Z-transform K(ζ) of the convolution kernel K(·), in terms of the input-state operator and of the resolvent (fundamental) matrix. These criteria do not impose additional positivity or compactness assumptions on coefficients K(j). Time-varying (non-convolution) difference equations are studied via structured UE stability radii r t of convolution equations. These radii correspond to a feedback scheme with delayed output and time-varying disturbances. We also consider stability radii r c associated with a time-invariant disturbance operator, unstructured stability radii, and stability radii corresponding to delayed feedback. For all these types of stability radii two-sided estimates are obtained. The estimates from above are given in terms of the Z-transform K(ζ), the estimate from below via the norm of the input-output operator. These estimates turn into explicit formulae if the state space X is Hilbert or if disturbances are time-invariant. The results on stability radii are applied to obtain various exponential stability tests for non-convolution equations. Several examples are provided.

"Admissibility and Polynomial Dichotomy of Discrete Nonautonomous Systems"

“…One of the most interesting topics in the asymptotic theory of dynamical systems is to explore whether an asymptotic behavior persists when the system is subjected to linear perturbations, i.e. to study its robustness (see [11,12,15,32,37,43,50,66] and also the recent works [3,4,6,10,20,49,58,63,69,70,72] and the references therein). There are various approaches in exploring robustness properties, some of the most representative being on the one hand those relying on direct estimates -that simply allow one to determine the explicit asymptotic behavior of the propagator of the perturbed system -and, on the other hand, those based on specific admissibility tools in which the asymptotic behavior of the perturbed system is implicitly deduced via admissibility criteria.…”

“…There are various approaches in exploring robustness properties, some of the most representative being on the one hand those relying on direct estimates -that simply allow one to determine the explicit asymptotic behavior of the propagator of the perturbed system -and, on the other hand, those based on specific admissibility tools in which the asymptotic behavior of the perturbed system is implicitly deduced via admissibility criteria. The second category of methods is often stronger (and requires combined arguments of functional analysis and control) as in some cases it provides not only the robustness property, but also a radius within which the "size" of the perturbation should fit (see [10,11,32,58,63,66] and the references therein). We note that, for the special case of nonuniform polynomial dichotomic behaviors, recent robustness results have been obtained in [18,19,65].…”

On exponential stability of nonlinear Volterra difference equations in phase spaces

“…In particular, problems of stability of Volterra difference equations have attracted much attention from researchers, during the last twenty years, see e.g. –, –, , , –, , and references therein. Very recently, E. Braverman and I. M. Karabash posed the following open problem:…”