Asymptotic stability equals exponential stability, and ISS equals finite energy gain---if you twist your eyes
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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.
14 pages, several references added, remarks section added, clarified construction
14 pages, several references added, remarks section added, clarified construction