Controllability of systems described by linear discrete Volterra equations (Q1883832)

An algebraic approach of studying the structural properties of linear Volterra equations is developed. It uses discrete transformations of functions and operators in the ring of power series in order to apply purely algebraic methods. The main structural characteristics of a linear Volterra equation (stability, controllability, boundedness of solutions,\dots) depend on the Volterra operator it generates. Therefore, necessary and sufficient conditions for a discrete Volterra operator to be surjective and injective are given, the spectrum of such operators is studied. Then conditions for the stability and complete controllability of the Volterra equation are given, and conditions for the existence, boundedness and stability of their solutions are proved. The results are useful tools for investigating digital pattern processing, visualization, and modelling technological processes.