Abstract
The celebrated Lyapunov function method (or the direct Lyapunov method) introduced in the Ph.D. thesis of A. M. Lyapunov in 1892 is a simple effective tool for stability analysis of differential equations. The main advantage of this method lies in the fact that a decision on stability or instability can be made by means of a certain investigation of the right-hand side of a differential equation without finding its solutions. Initially, the Lyapunov function method was limited by a regular class of ODE with continuous right-hand sides. The later evolution of the ODE theory and its applications had required extensions of this method to differential equations with discontinuous right-hand sides, functional differential equations, PDEs, and evolution systems.
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References
Kurzweil J (1956) On the inversion of Ljapunov’s second theorem on stability of motion. Czechoslov Math J 6(2):217–259 [in Russian]
Zubov VI (1957) Methods of A. M. Lyapunov and their applications. In: Noordhoff L (1964) (Translated from Russian: V. I. Zubov, Metody Lyapunova i ih primenenie, Leningrad, LGU)
Lyapunov AM (1992) The general problem of the stability of motion. Taylor & Francis
Bacciotti A, Rosier L (2001) Lyapunov functions and stability in control theory. Springer
Roxin E (1966) On finite stability in control systems. Rend Circ Mat Palermo 15:273–283
Polyakov A, Poznyak A (2009) Reaching time estimation for “super-twisting” second order sliding mode controller via Lyapunov function designing. IEEE Trans Autom Control 54(8):1951–1955
Polyakov A, Poznyak A (2009) Lyapunov function design for finite-time convergence analysis: “twisting” controller for second order sliding mode realization. Automatica 45:444–448
Moreno J, Osorio M (2012) Strict Lyapunov functions for the super-twisting algorithm. IEEE Trans Autom Control 57:1035–1040
Natanson IP (1955) Theory of functions of a real variable. Frederick Ungar Publishing Co., New York
Filippov AF (1988) Differential equations with discontinuous right-hand sides. Kluwer Academic Publishers
Clarke F, Ledyaev YS, Stern R, Wolenski P (1995) Qualitative properties of trajectories of control systems: a survey. J Dyn Control Syst 1(1):1–48
Aubin JP, Ekeland I (1984) Applied nonlinear analysis. Wiley, New York
Clarke F (1990) Optimization and nonsmooth analysis. SIAM, Philadelphia
Clarke FH, Ledyaev YuS, Stern RJ (1998) Asymptotic stability and smooth Lyapunov functions. J Differ Equ 149:69–114
Mironchenko A, Wirth F (2019) Non-coercive Lyapunov functions for infinite-dimensional systems. J Differ Equ
Levant A (1993) Sliding order and sliding accuracy in sliding mode control. Int J Control 58(6):1247–1263
Lopez-Ramirez F, Efimov D, Polyakov A, Perruquetti W (2018) On necessary and sufficient conditions for fixed-time stability of continuous autonomous systems. In: 2018 European control conference (ECC), pp 197–200
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Polyakov, A. (2020). Method of Lyapunov Functions. In: Generalized Homogeneity in Systems and Control. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-38449-4_5
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DOI: https://doi.org/10.1007/978-3-030-38449-4_5
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