A New Set of Stability Criteria Extending Lyapunov's Direct Method
Abstract
A dynamical system is a mathematical model described by a high
dimensional ordinary differential equation for a wide variety of real
world phenomena, which can be as simple as a clock pendulum or as
complex as a chaotic Lorenz system. Stability is an important topic in
the studies of the dynamical system. A major challenge is that the
analytical solution of a time-varying nonlinear dynamical system is in
general not known. Lyapunov's direct method is a classical approach used
for many decades to study stability without explicitly solving the
dynamical system, and has been successfully employed in numerous
applications ranging from aerospace guidance systems, chaos theory, to
traffic assignment. Roughly speaking, an equilibrium is stable if an
energy function monotonically decreases along the trajectory of the
dynamical system. This paper extends Lyapunov's direct method by
allowing the energy function to follow a rich set of dynamics. More
precisely, the paper proves two theorems, one on globally uniformly
asymptotic stability and the other on stability in the sense of
Lyapunov, where stability is guaranteed provided that the evolution of
the energy function satisfies an inequality of a non-negative Hurwitz
polynomial differential operator, which uses not only the first-order
but also high-order time derivatives of the energy function. The
classical Lyapunov theorems are special cases of the extended theorems.
the paper provides an example in which the new theorem successfully
determines stability while the classical Lyapunov's direct method fails.