A standard framework for studying `impulse’ control systems, in which the control input is a measure, is to interpret state trajectories as limits of classical state trajectories for a conventional control system, with affine control dependence and for which the control is unbounded. Discontinuities in the state trajectory can occur at the atoms of the measure control. If the measure control is scalar valued, then all limiting state trajectories are the same and discontinuities in the state trajectories at any particular time are determined by the solution to a differential equation, which depends on the magnitude of the atom and the state-dependent input gain function. By means of reparameterization techniques, which accord time the role of a state variable and which replace the original impulse control system by a classical control system, we can establish important properties of impulse control systems, such as the representation of the reachable set as the closure of the reachable set for classical control inputs and existence of minimizers for optimal impulse control problems, by simply demonstrating analogous properties of the reparameterized control system. This approach breaks down when the time dependence of the drift term in the impulse control system is not Lipschitz continuous because the right side of the reparameterized dynamic equations are not Lipschitz continuous w.r.t. the state variables and therefore is not amenable to standard analysis. In this paper, it is shown that density and closure properties of impulse control systems are retained, for impulse control systems in which the drift term is merely measurable regarding its time dependence and when the input gain function is state dependent. Difficulties arising from the irregular state dependence of the reparameterized control system can be simply overcome by switching between the original and reparameterized control system descriptions in the convergence analysis.
Marchini et al. (Thu,) studied this question.