Uniform Stability for Time-Varying Infinite-Dimensional Discrete Linear Systems

Stability for time-varying discrete linear systems in a Banachspace is investigated. On the one hand is established a fairlycomplete collection of necessary and sufficient conditions foruniform asymptotic equistability for input-free systems. Thisincludes uniform and strong power equistability, and uniformand strong l_p-equistability, among other technical conditionswhich also play an essential role in stability theory. On theother hand, it is shown that uniform asymptotic equistabilityfor input-free systems is equivalent to each of the followingconcepts of uniform stability for forced systems: l_p-input l_p-state,e_o-input e_o-state, bounded-input bounded-state, l_p-input bounded-state(with p>1), e_o-input bounded-state, and convergent-inputbounded-state; these are also equivalent to their nonuniformcounterparts. For time-varying convergent systems, the aboveis also equivalent to convergent-input convergent-state stability.The proofs presented here are all ‘lementary’ inthe sense that they are based essentially only on the Banach–Steinhaustheorem.