Differential geometric aspects of impulsive control theory

A general question one has to face in the study of an optimal control problem is the representation of the closure of the set of trajectories. Loosely speaking, two main problems lie at the basis of the non triviality of this question: the lack of sufficient convexity assumptions (which is at the origin of the introduction of the so-called chattering controls) and the occurrence of unbounded controls. Here, we disregard the former question (e.g. by assuming sufficient convexity hypotheses) and focus on the latter. An obvious question is the following: how to interpret the dynamic equations when a sequence of controls tends, say, to a distribution? It happens that, while there is no difficulty to give a robust notion of solution to x = g0(x) + Gξ x(t) = x tϵ[t, T] (1) when ξ is a first order distribution - i.e., ξ = u, u ϵ L¹_loc- a distributional approach is not adequate for a nonlinear system of the form x = g0(x) + G(x)ξ = g0(x) + g1(x)ξ1 + ... + g_m(x)ξ_m x(t) = x (2). The reason why this drawback occurs is in fact a differential geometric one, namely the non-commutativity of Lie brackets [g_i, g_j]. We give a brief account of various consequences of this fact and of some arguments which can be utilized to frame the problem in an appropriate setting.