Impulsive delay systems with general end-times

In this paper we study impulsive control systems with state time delays in the dynamics and scalar inputs, and derive related necessary optimality conditions. We adopt a definition of state trajectories, motivated by applications in the aerospace field, in which impulse controls are interpreted as idealizations of high-intensity control actions of short duration. In previous work, the authors have shown that, for such systems, the limiting state trajectories due to different distribution sense approximations of a given measure control by strict sense, i.e. absolutely continuous, controls may not be the same. (This is a surprising phenomenon since, for delay-free impulse control systems with scalar input, the limiting state trajectory is unique.) Under the hypothesis that the end-time is an integer multiple of the delay time (the ‘commensurate end-time’ assumption), it has also been shown however that, if measure controls are supplemented by ‘attached controls’ then, together, they unambiguously define a state trajectory. For controls in this broader sense, we can prove closure properties of reachable sets for state trajectories, provide conditions for existence of minimizers for impulsive optimal control problems with time delay and derive necessary conditions of optimality. The key idea is to use re-parameterization techniques to reformulate the impulse control system as a classical control system; the desired properties of the impulse control system are then demonstrated by establishing the analogous properties of the reformulated control system. In applications we can expect that the end-time and the time delay will be independently specified. The commensurate end-time hypothesis is therefore highly artificial. The purpose of this paper is to show that closure properties of reachable sets are retained and necessary conditions of optimality can be derived, even when the commensurate end-time hypothesis is discarded. These goals are achieved by means of a new kind of parameterization, which reduces the impulse control system with time delay to a coupled family of control systems, known as ‘multiprocess’ systems.