On the Use of Needle-Like Perturbations in Spatially Heterogeneous Control Systems

In this paper we consider a general control system involving a spatially heterogeneous dynamics. This means that the state space is partitioned into several disjoint regions and that each region has its own (smooth) control system. As a result, the dynamics discontinuously changes whenever the trajectory crosses an interface between two regions. In that spatially heterogeneous setting (and in contrast with the usual smooth case), a needle-like perturbation of the control may generate a perturbed trajectory that does not uniformly converge towards the nominal one, and may lead to the absence of a corresponding first-order variation vector. The first contribution of this paper is to illustrate this issue by means of a simple counterexample. Our second and main contribution is to provide a modified needle-like perturbation of the control (adapted to the spatially heterogeneous setting) which generates a perturbed trajectory that uniformly converges towards the nominal one, and leads to a corresponding first-order variation vector (which has the particularity of admitting a discontinuity jump at each interface crossing). This is made possible under several assumptions (including transverse crossing conditions), by introducing new tools such as auxiliary trajectories and auxiliary controls and by using a conic version of the implicit function theorem.