## 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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