Open-Loop Impulse Control

Abstract

This chapter describes how to find optimal open-loop impulse controls [1,2,3,4,5, 7,8,9]. We begin by defining an impulse control system and proving the existence and uniqueness of its trajectories. Then we set up the basic problem of open-loop impulse control. This is how to transfer the system from a given initial state to a given target state within given time under a control of minimum variation. A key point in solving the open-loop impulse control problem is the construction of reachability sets for the system. Here we indicate how to construct such sets and study their properties. After that we present some simple model examples. The solution to the optimal impulse control problem problem is given by the Maximum Rule for Impulse Controls, an analogue of Pontryagin’s Maximum Principle for ordinary controls [7, 10]. The linearity of the considered system implies that the Maximum Rule indicates not only the necessary, but also some sufficient conditions of optimality. We further describe and prove an important feature of the problem which is that there exists an optimal control as a combination of a finite number of impulses, whose number is not greater than the system dimension [8]. Then we discuss some extensions of the basic impulse control problem. These include the control of a subset of coordinates, problems with set-valued boundary conditions, and the problem of time-optimal impulse control. Finally we treat a problem of the Mayer–Bolza type (a Stiltjes integral-terminal functional), that is further used in solving the problem of closed-loop impulse control.