8.1 Introduction

In many practical situations, it is desirable for the system under consideration to remain near its equilibrium state. In such cases, we say that the system is stable at the equilibrium position. If a system tends to move away from equilibrium, but it is possible to apply a control to keep it near the equilibrium, this process is called stabilization of the system. In this chapter, we present the stability properties of linear and nonlinear dynamical systems at equilibrium points, as well as the stabilizability of control systems.

In Section 8.2, definitions of equilibrium point, stability, asymptotic stability, and instability of equilibrium points are given. In Section 8.3, stability of the trivial solutions of a linear time-invariant system is studied using the eigenvalues of the coefficient matrix. Some stability results for nonautonomous linear systems are also presented. For some nonlinear systems, stability is analyzed through linearization in Section 8.4. In Section 8.5, Lyapunov stability theorems for time-invariant systems are presented with examples. Stability analysis using matrix equations is also given in this section. Lyapunov theorems for time-varying systems are presented in Section 8.6. The gradient method and the Krasovskii formula for constructing Lyapunov functions are also presented.