Lyapunov Stability Theory

Abstract

The study of the stability of dynamical systems has a very rich history. Many famous mathematicians, physicists, and astronomers worked on axiomatizing the concepts of stability. A problem, which attracted a great deal of early interest was the problem of stability of the solar system, generalized under the title “the N-body stability problem.” One of the first to state formally what he called the principle of “least total energy” was Torricelli (1608–1647), who said that a system of bodies was at a stable equilibrium point if it was a point of (locally) minimal total energy. In the middle of the eighteenth century, Laplace and Lagrange took the Torricelli principle one step further: They showed that if the system is conservative (that is, it conserves total energy—kinetic plus potential), then a state corresponding to zero kinetic energy and minimum potential energy is a stable equilibrium point. In turn, several others showed that Torricelli’s principle also holds when the systems are dissipative, i.e., total energy decreases along trajectories of the system. However, the abstract definition of stability for a dynamical system not necessarily derived for a conservative or dissipative system and a characterization of stability were not made till 1892 by a Russian mathematician/engineer, Lyapunov, in response to certain open problems in determining stable configurations of rotating bodies of fluids posed by Poincaré. The original paper of Lyapunov of 1892, was translated into French very shortly there after, but its English translation appeared only recently in [193].