When do Trajectories have Bounded Sensitivity to Cumulative Perturbations?

Abstract: We investigate sensitivity to cumulative perturbations for a few dynamical system classes of practical interest. A system is said to have bounded sensitivity to cumulative perturbations (bounded sensitivity, for short) if an additive disturbance leads to a change in the state trajectory that is bounded by a constant multiple of the size of the cumulative disturbance. As our main result, we show that there exist dynamical systems in the form of (negative) gradient field of a convex function that have unbounded sensitivity. We show that the result holds even when the convex potential function is piecewise linear. This resolves a question raised in [1], wherein it was shown that the (negative) (sub)gradient field of a piecewise linear and convex function has bounded sensitivity if the number of linear pieces is finite. Our results establish that the finiteness assumption is indeed necessary. Among our other results, we provide a necessary and sufficient condition for a linear dynamical system to have bounded sensitivity to cumulative perturbations. We also establish that the bounded sensitivity property is preserved, when a dynamical system with bounded sensitivity undergoes certain transformations. These transformations include convolution, time discretization, and spreading of a system (a transformation that captures approximate solutions of a system).