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

  • Andrii MironchenkoEmail author
  • Fabian Wirth
Original Article


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.


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



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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© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Computer Science and MathematicsUniversity of PassauPassauGermany

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