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)
