Lyapunov Stability of Motion

Abstract

The main intent of this chapter is to introduce the essential mathematical tools for stability analysis of continuous finite-dimensional dynamical systems. We begin with an overview of sufficient conditions to guarantee existence and uniqueness of the system solutions, followed by a collection of Lyapunov-based methods for studying stability of the system equilibriums and trajectories. The beginning of what is known today as the Lyapunov stability theory can be traced back to the original publication of Alexander Mikhailovich Lyapunov’s doctoral thesis on “the general problem of the stability of motion,” which he defended at the University of Moscow in 1892. Our interest and emphasis on the Lyapunov’s stability methods stem from the fact that these methodologies lay out the much needed theoretical framework and the foundation for performing design and analysis of adaptive controllers. In this chapter, we review selected (but not inclusive) methods due to Lyapunov. This selection is primarily driven by our interest in justifying the design of stable model reference adaptive controllers, with predictable and guaranteed closed-loop performance, for a wide class of nonlinear nonautonomous multi-input multi-output dynamical systems.

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-1-4471-4396-3_15