# 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).

## References

- 1.Angeli D, Sontag ED (1999) Forward completeness, unboundedness observability, and their Lyapunov characterizations. Syst Control Lett 38(4–5):209–217MathSciNetCrossRefGoogle Scholar
- 2.Chueshov I (2015) Dynamics of quasi-stable dissipative systems. Springer, BerlinCrossRefGoogle Scholar
- 3.Curtain RF, Zwart H (1995) An introduction to infinite-dimensional linear systems theory. Springer, New YorkCrossRefGoogle Scholar
- 4.Dashkovskiy S, Mironchenko A (2013) Input-to-state stability of infinite-dimensional control systems. Math Control Signals Syst 25(1):1–35MathSciNetCrossRefGoogle Scholar
- 5.Datko R (1970) Extending a theorem of A. M. Liapunov to Hilbert space. J Math Anal Appl 32(3):610–616MathSciNetCrossRefGoogle Scholar
- 6.Hahn W (1967) Stability of motion. Springer, New YorkCrossRefGoogle Scholar
- 7.Hale JK, Verduyn Lunel SM (1993) Introduction to functional-differential equations, volume 99 of Applied mathematical sciences. Springer, New YorkCrossRefGoogle Scholar
- 8.Henry D (1981) Geometric theory of semilinear parabolic equations. Springer, BerlinCrossRefGoogle Scholar
- 9.Jacob B, Mironchenko A, Partington JR, Wirth F (2018) Remarks on input-to-state stability and non-coercive Lyapunov functions. In: Proceedings of the 57th IEEE Conference on Decision and Control (CDC 2018). Miami Beach, USA, pp 4803–4808Google Scholar
- 10.Karafyllis I, Jiang Z-P (2011) Stability and stabilization of nonlinear systems. Springer, LondonCrossRefGoogle Scholar
- 11.Kellett CM (2014) A compendium of comparison function results. Math Control Signals Syst 26(3):339–374MathSciNetCrossRefGoogle Scholar
- 12.Kellett CM (2015) Classical converse theorems in Lyapunov’s second method. Discrete Continuous Dyn Syst Ser B 20(8):2333–2360MathSciNetCrossRefGoogle Scholar
- 13.Kellett CM, Dower PM (2016) Input-to-state stability, integral input-to-state stability, and \(L_2\)-gain properties: qualitative equivalences and interconnected systems. IEEE Trans Automat Control 61(1):3–17MathSciNetCrossRefGoogle Scholar
- 14.Kurzweil J (1956) On the inversion of Lyapunov’s second theorem on stability of motion. Czechoslovak Math J 81:217–259zbMATHGoogle Scholar
- 15.Lin Y, Sontag ED, Wang Y (1996) A smooth converse Lyapunov theorem for robust stability. SIAM J Control Optim 34(1):124–160MathSciNetCrossRefGoogle Scholar
- 16.Littman W (1989) A generalization of a theorem of Datko and Pazy. In: Advances in computing and control, vol 130. Lecture notes in control and information sciences. Springer, Berlin, pp 318–323Google Scholar
- 17.Massera JL (1956) Contributions to stability theory. Ann Math 64(1):182–206MathSciNetCrossRefGoogle Scholar
- 18.Mironchenko A (2017) Uniform weak attractivity and criteria for practical global asymptotic stability. Syst Control Lett 105:92–99MathSciNetCrossRefGoogle Scholar
- 19.Mironchenko A, Wirth F (2017) A non-coercive Lyapunov framework for stability of distributed parameter systems. In: Proceedings of the 56th IEEE conference on decision and control. IEEE, pp 1900–1905Google Scholar
- 20.Mironchenko A, Wirth F (2017) Non-coercive Lyapunov functions for infinite-dimensional systems. J Differ Equ. https://doi.org/10.1016/j.jde.2018.11.026 CrossRefzbMATHGoogle Scholar
- 21.Mironchenko A, Wirth F (2018) Characterizations of input-to-state stability for infinite-dimensional systems. IEEE Trans Automat Control 63(6):1602–1617MathSciNetCrossRefGoogle Scholar
- 22.Pepe P, Jiang Z-P (2006) A Lyapunov–Krasovskii methodology for ISS and iISS of time-delay systems. Syst Control Lett 55(12):1006–1014MathSciNetCrossRefGoogle Scholar
- 23.Saks S (2005) Theory of the integral. Dover Publications, Mineola, NY (reprint of the 2nd rev. ed. (1937))Google Scholar
- 24.Schnaubelt R (2002) Feedbacks for nonautonomous regular linear systems. SIAM J Control Optim 41(4):1141–1165MathSciNetCrossRefGoogle Scholar
- 25.Sontag ED (1998) Comments on integral variants of ISS. Syst Control Lett 34(1–2):93–100MathSciNetCrossRefGoogle Scholar
- 26.Sontag ED, Wang Y (1996) New characterizations of input-to-state stability. IEEE Trans Automat Control 41(9):1283–1294MathSciNetCrossRefGoogle Scholar
- 27.Szarski J (1965) Differential inequalities. Polish Scientific Publishers PWN, WarszawazbMATHGoogle Scholar
- 28.Teel A, Panteley E, Loría A (2002) Integral characterizations of uniform asymptotic and exponential stability with applications. Math Control Signals Syst 15(3):177–201MathSciNetCrossRefGoogle Scholar
- 29.Yoshizawa T (1966) Stability Theory by Liapunov’s second method. Mathematical Society of JapanGoogle Scholar
- 30.Zames G (1966) On the input-output stability of time-varying nonlinear feedback systems Part one: Conditions derived using concepts of loop gain, conicity, and positivity. IEEE Trans Automat Control 11(2):228–238MathSciNetCrossRefGoogle Scholar