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,

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

Steven M LaValle 2012-04-20