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New characterizations of input-to-state stability. (English) Zbl 0862.93051
This paper studies some stability properties of the state response with respect to inputs of smooth nonlinear systems of the form \[ \dot x=f(x,u). \] The starting point is the so-called ISS property. This means that the norm of the solution is majorized at each future time \(t\) by an expression \(\beta(|\xi|,t)+\gamma(|u|)\), where \(\xi\) is the initial state, \(|u|\) is the \(L_\infty\) norm of the input and \(\beta\), \(\gamma\) are suitable comparison function.
The authors introduce a list of (apparently different) properties which are in fact proved to be equivalent to ISS. This variety of possible formulations turns out to be useful in several applications.

93D25 Input-output approaches in control theory
93C10 Nonlinear systems in control theory
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