## 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.

### MSC:

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

### Keywords:

input/output behaviour; Lyapunov methods; nonlinear systems
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