Nonrobustness of asymptotic stability of impulsive systems with inputs

TL;DR

This paper demonstrates that stability properties of impulsive systems are not inherently robust to inputs, providing counterexamples and discussing conditions under which some robustness can be achieved.

Contribution

It shows that key stability properties like 0-GUAS do not imply CICS or iISS in impulsive systems, highlighting the limits of robustness without constraints.

Findings

Zero-input 0-GUAS does not imply CICS.

0-GUAS and UBEBS do not imply iISS.

Constraints on impulse timing can restore some robustness properties.

Abstract

Suitable continuity and boundedness assumptions on the function f defining the dynamics of a time-varying nonimpulsive system with inputs are known to make the system inherit stability properties from the zero-input system. Whether this type of robustness holds or not for impulsive systems was still an open question. By means of suitable (counter)examples, we show that such stability robustness with respect to the inclusion of inputs cannot hold in general, not even for impulsive systems with time-invariant flow and jump maps. In particular, we show that zero-input global uniform asymptotic stability (0-GUAS) does not imply converging input converging state (CICS), and that 0-GUAS and uniform bounded-energy input bounded state (UBEBS) do not imply integral input-to-state stability (iISS). We also comment on available existing results that, however, show that suitable constraints on the allowed impulse-time sequences indeed make some of these robustness properties possible.