Lars Grüne, Eduardo D Sontag, Fabian Wirth
On the equivalence between asymptotic and exponential stability, and between ISS and finite $H_\infty$ gain
The paper is published: Proceedings of the 38th IEEE Conference on Decision and Control, USA, 1200 - 1205, 1999
MSC:
34D05 Asymptotic properties
93B36 ${H}^ \infty$-control
Abstract:
In this paper we show that uniformly global asymptotic stability for a family of ordinary differential
equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of
variables. The same is shown for input-to-state stability and input-to-state exponential stability, and for
input-to-state exponential stability and a nonlinear $H_\infty$ estimate.

Keywords: asymptotic stability, exponential stability, input-to-state stability, state transformation