Determining stability

Suppose a velocity field ${\dot x}= f(x)$ is given along with an equilibrium point ${x_{G}}$. Let $\phi$ denote a candidate Lyapunov function, which will be used as an auxiliary device for establishing the stability of $f$. An appropriate $\phi$ must be determined for the particular vector field $f$. This may be quite challenging in itself, and the details are not covered here. In a sense, the procedure can be characterized as ``guess and verify,'' which is the way that many solution techniques for differential equations are described. If $\phi$ succeeds in establishing stability, then it is promoted to being called a Lyapunov function for $f$.

It will be important to characterize how $\phi$ varies in the direction of flow induced by $f$. This is measured by the Lie derivative,

$${\dot \phi}(x) = \sum_{i=1}^n \frac{\partial\phi}{\partial x_i} f_i(x) .$$ (15.3)

This results in a new function ${\dot \phi}(x)$, which indicates for each $x$ the change in $\phi$ along the direction of ${\dot x}= f(x)$.

Several concepts are needed to determine stability. Let a function $h: [0,\infty) \rightarrow [0,\infty)$ be called a hill if it is continuous, strictly increasing, and $h(0) = 0$. This can be considered as a one-dimensional navigation function, which has a single local minimum at the goal, 0. A function $\phi : X \rightarrow [0,\infty)$ is called locally positive definite if there exists some open set $O \subseteq X$ and a hill function $h$ such that $\phi({x_{G}}) = 0$ and $\phi(x) \geq h(\Vert x\Vert)$ for all $x \in O$. If $O$ can be chosen as $O = X$, and if $X$ is bounded, then $\phi$ is called globally positive definite or just positive definite. In some spaces this may not be possible due to the topology of $X$; such issues arose when constructing navigation functions in Section 8.4.4. If $X$ is unbounded, then $h$ must additionally approach infinity as $\Vert x\Vert$ approaches infinity to yield a positive definite $\phi$ [846]. For $X = {\mathbb{R}}^n$, a quadratic form $x^T M x$, for which $M$ is a positive definite matrix, is a globally positive definite function. This motivates the use of quadratic forms in Lyapunov stability analysis.

The Lyapunov theorems can now be stated [156,846]. Suppose that $\phi$ is locally positive definite at ${x_{G}}$. If there exists an open set $O$ for which ${x_{G}}\in O$, and ${\dot \phi}(x) \leq 0$ on all $x \in O$, then $f$ is Lyapunov stable. If $-{\dot \phi}(x)$ is also locally positive definite on $O$, then $f$ is asymptotically stable. If $\phi$ and $-{\dot \phi}$ are both globally positive definite, then $f$ is globally asymptotically stable.

Example 15..1 (Establishing Stability via Lyapunov Functions)   Let $X = {\mathbb{R}}$. Let ${\dot x}= f(x) = -x^5$, and we will attempt to show that $x = 0$ is stable. Let the candidate Lyapunov function be $\phi(x) = \frac{1}{2} x^2$. The Lie derivative (15.3) produces ${\dot \phi}(x) = -x^6$. It is clear that $\phi$ and $-{\dot \phi}$ are both globally positive definite; hence, 0 is a global, asymptotically stable equilibrium point of $f$. $\blacksquare$