15.1.2 Lyapunov Functions

Suppose a velocity field ${\dot x}= f(x)$ is given along with an equilibrium point, ${x_{G}}$. Can the various forms of stability be easily determined? One of the most powerful methods to prove stability is to construct a Lyapunov function. This will be introduced shortly, but first some alternatives are briefly mentioned.

If $f(x)$ is linear, which means that $f(x) = Ax$ for some constant $n \times n$ matrix $A$ and $X = {\mathbb{R}}^n$, then stability questions with respect to the origin, ${x_{G}}= 0$, are answered by finding the eigenvalues of $A$ [192]. The state $x = 0$ is asymptotically stable if and only if all eigenvalues of $A$ have negative real parts. Consider the scalar case, ${\dot x}= a x$, for which $X = {\mathbb{R}}$ and $a$ is a constant. The solution to this differential equation is $x(t) = x(0) \,e^{a t}$, which converges to 0 only if $a < 0$. This can be easily extended to the case in which $X = {\mathbb{R}}^n$ and $A$ is an $n \times n$ diagonal matrix for which each diagonal entry (or eigenvalue) is negative. For a general matrix, real or complex eigenvalues determine the stability (complex eigenvalues cause oscillations). Conditions also exist for Lyapunov stability. Every equilibrium state of ${\dot x}= A x$ is Lyapunov stable if the eigenvalues of $A$ all have nonpositive real parts, and the eigenvalues with zero real parts are distinct roots of the characteristic polynomial of $A$.

If $f(x)$ is nonlinear, then stability can sometimes be inferred by linearizing $f(x)$ about ${x_{G}}$ and performing linear stability analysis. In many cases, however, this procedure is inconclusive (see Chapter 6 of [156]). Proving the stability of a vector field is a challenging task for most nonlinear systems. One approach is based on LaSalle's invariance principle [39,156,585] and is particularly useful for showing convergence to any of multiple goal states (see Section 5.4 of [846]). The other major approach is to construct a Lyapunov function, which is used as an intermediate tool to indirectly establish stability. If this method fails, then it still may be possible to show stability using other means. Therefore, it is a sufficient condition for stability, but not a necessary one.