# Existence of non-coercive Lyapunov functions is equivalent to integral uniform global asymptotic stability

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## Abstract

In this paper, a class of abstract dynamical systems is considered which encompasses a wide range of nonlinear finite- and infinite-dimensional systems. We show that the existence of a non-coercive Lyapunov function without any further requirements on the flow of the forward complete system ensures an integral version of uniform global asymptotic stability. We prove that also the converse statement holds without any further requirements on regularity of the system. Furthermore, we give a characterization of uniform global asymptotic stability in terms of the integral stability properties and analyze which stability properties can be ensured by the existence of a non-coercive Lyapunov function, provided either the flow has a kind of uniform continuity near the equilibrium or the system is robustly forward complete.

## Keywords

Nonlinear control systems Infinite-dimensional systems Lyapunov methods Global asymptotic stability## Notes

### Acknowledgements

This research has been supported by the German Research Foundation (DFG) within the project “Input-to-state stability and stabilization of distributed parameter systems” (Grant Wi 1458/13-1).

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