• André Preumont
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 246)


This short chapter examines the main results of Lyapunov’s theory of stability which constitutes a formidable tool for analyzing the stability of nonlinear mechanical system. The chapter begins with recalling the various concepts of stability (in the sense of Lyapunov, asymptotic stability, etc.) and the Routh–Hurwitz criterion for linear invariant systems. Next, Lyapunov direct method is introduced, the concept of Lyapunov function candidate, the stability theorem, the extension of Lasalle’s theorem, the instability theorem, and the particular case of a linear system. Lyapunov indirect method is then applied to the local stability of a nonlinear system about the equilibrium point. Finally, the concept of energy absorbing collocated control is introduced for collocated actuator/sensor pairs. The chapter concludes with a short list of references and a set of problems.


Routh–Hurwitz criterion Lyapunov direct method Lyapunov function Asymptotic stability Lasalle theorem Lyapunov equation Lyapunov indirect method Energy absorbing collocated control 


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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Active Structures LaboratoryUniversité Libre de BruxellesBrusselsBelgium

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