Stability

Abstract

The analysis of stability of mechanical systems is a classic topic in mechanics and dynamics that has affected a number of mathematical and engineering disciplines. In his classic work, Lagrange [1788] investigated the stability of mechanical systems at local minima of the potential function using energy arguments. In a work widely recognized to be one of the first on control theory, Maxwell [1868] analyzed the stability of certain mechanical governors using linearization. At about the same time, Thomson and Tait [1867] studied the asymptotic stability of mechanical systems subject to dissipative forces also via linear methods. Finally, Lyapunov [1892] developed the key elements of a stability notion and of stability criteria applicable to a broad class of nonlinear systems; this work laid the foundations for modern stability theory. We present the Lyapunov Stability Critera in Theorem 6.14. So-called invariance principles were later developed to establish stability properties of dynamical systems on the basis of weaker requirements than those required by Lyapunov’s original criteria. Early work on invariance principles in stability is due to Barbashin and Krasovskiĭ [1952]; LaSalle presented his Invariance Principle in [LaSalle 1968]. Recent influential works on stability include [LaSalle and Lefschetz 1962], [Hahn 1963, 1967] and [Chetaev 1955]. Nowadays, stability theory is a cornerstone of dynamical systems and control theory; examples of modern treatments in nonlinear control monographs include [Khalil 2001, Sastry 1999, Sontag 1998], and dynamical systems references include [Arnol’d 1992, Guckenheimer and Holmes 1990, Hirsch and Smale 1974, Merkin 1997].