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Changing supply functions in input/state stable systems. (English) Zbl 0832.93047

The authors consider the system \(\dot x= f(x, u)\), \(f: \mathbb{R}^n\times \mathbb{R}^m\to \mathbb{R}^n\), locally Lipschitz with \(f(0, 0)= 0\). The system is ISS if there exists a positive definite \(V(x)\), \(V(0)= 0\), radially bounded \(\mathbb{R}^n\to \mathbb{R}\) (a storage function) and two functions (in \(K_\infty\)) \(\alpha\) and \(\gamma\) such that \(\dot V= (\nabla V\cdot f)\leq \gamma(|u|)- \alpha(|x|)\). A combination of the \(\gamma\) and \(\alpha\) functions characterizes the “input to gain” for the system. The authors prove some relations for positive definiteness of the storage function and the existence of \(K_\infty\) functions assuring that dissipation estimates for some composite systems are available.
Reviewer: V.Komkov (Roswell)

MSC:

93D10 Popov-type stability of feedback systems
93C10 Nonlinear systems in control theory
93A99 General systems theory
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