Effects of Impulsive Control in Systems with Delay

Abstract

We investigate the potential use of impulsive inputs for linear and bilinear systems with delays. To set the ideas, we first consider ODE systems. Of interest is the extension to time-variant linear systems, where besides the jump effect on the states we also make concrete investigations of the microscopic singular behavior of the state. We then discuss impulsive bilinear systems. Here it is found that one cannot proceed in a distributional sense. New generalized functions, introduced by Colombeau are discussed. In particular, we settle some issues about the ‘right’ interpretation for extending Schwartz's theory of distributions and applications to differential equations. Armed with this new perspective we investigate linear time-invariant (LTI) delay systems and classes of bilinear delay systems with impulsive inputs, and define new relevant reachability matrices along the way.