4.9 Lyapunov's Indirect Method

In this section we are concerned with the problem of investigating stability properties of an equilibrium state of a nonlinear system based on its linearization about the given equilibrium. We devise a method that allows one to determine whether the equilibrium of the nonlinear system is asymptotically stable or unstable based on the dynamics of the linearized system about this equilibrium. The method is sometimes referred to as the Lyapunov's first method or Lyapunov's indirect method. For simplicity of notation, we assume that the equilibrium point to be tested for stability is the origin. We consider a nonlinear system model,

where is a continuously differentiable function from a domain into . We assume that the origin x = 0 is in the interior of and is an equilibrium state of (4.57), that is, f( 0) = 0. Let

be the Jacobian matrix of f evaluated at x = 0. Then, applying Taylor's theorem to the right-hand side of (4.57) yields

where the function g represents higher-order terms and has the property

The above suggests that in a "small" neighborhood of the origin, we can approximate the nonlinear system (4.57) with its linearized system about the origin, ? = Ax. The following theorems give conditions under which we can infer about stability properties of the origin of the nonlinear system based on stability properties of the linearized system.