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 $c_0 (\ZZ^-, \X)$ and $\ll^\infty (\ZZ^-, \X)$ and with exponentially fading phase spaces of the $\ll^p$ and $c_0$ types. For equations of the convolution type, several criteria of UE stability are obtained in terms of the Z-transform $\wh K (ζ)$ of the convolution kernel $K (\cdot)$, 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 $\wh K (ζ)$, 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.