Structured stability radii and exponential stability tests for Volterra difference systems

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 XX is an abstract Banach space. We work with non-fading phase spaces c0(Z−,X)c0(Z−,X) and ℓ∞(Z−,X)ℓ∞(Z−,X) and with exponentially fading phase spaces of the ℓpℓp and c0c0 types. For equations of the convolution type, several criteria of UE stability are obtained in terms of the Z-transform K̂(ζ) of the convolution kernel K(⋅)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)K(j). Time-varying (non-convolution) difference equations are studied via structured UE stability radii rt of convolution equations. These radii correspond to a feedback scheme with delayed output and time-varying disturbances. We also consider stability radii rc 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 K̂(ζ), the estimate from below via the norm of the input–output operator. These estimates turn into explicit formulas if the state space XX 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.