Volterra systems with impulses and delays
by
Staros A. Belbas
Math. Dept., University of Alabama, Tuscaloosa, AL 35487-0350, USA.

It is well known that wide classes of nolinear systems can be modelled by input-output relations in the form of multi-linear Volterra systems involving multiple integrals. A more general form, with general nonlinear (not necessarily multi-linear) terms, arises, for example, when one wants to represent the solution of controlled analytic ordinary differential equations via series of integral terms. In order to model also systems with impulses and delays, we extend the concept of nonlinear Volterra series representations of input-output relations, to include concentrated delays (as distinct from the distributed memory of ordinary Volterra-series systems) and impulsive terms. These Volterra operators can represent, among other things, the solutions of delay-differential equations with impulses. We analyze the properties of delayed-impulsive nonlinear Volterra series (DINVS), the solvability of integral equations involving DINVS, and the variational calculus concerning optimal control of systems described by DINVS. These have application to feedback design for systems described by Volterra input-output relations, and to design of realizable control laws for systems with partial observations of the state.

Date received: March 24, 2007