#### Determining stability

Suppose a velocity field is given along with an equilibrium point . Let denote a candidate Lyapunov function, which will be used as an auxiliary device for establishing the stability of . An appropriate must be determined for the particular vector field . 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 succeeds in establishing stability, then it is promoted to being called a Lyapunov function for .

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

 (15.3)

This results in a new function , which indicates for each the change in along the direction of .

Several concepts are needed to determine stability. Let a function be called a hill if it is continuous, strictly increasing, and . This can be considered as a one-dimensional navigation function, which has a single local minimum at the goal, 0. A function is called locally positive definite if there exists some open set and a hill function such that and for all . If can be chosen as , and if is bounded, then is called globally positive definite or just positive definite. In some spaces this may not be possible due to the topology of ; such issues arose when constructing navigation functions in Section 8.4.4. If is unbounded, then must additionally approach infinity as approaches infinity to yield a positive definite [846]. For , a quadratic form , for which 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 is locally positive definite at . If there exists an open set for which , and on all , then is Lyapunov stable. If is also locally positive definite on , then is asymptotically stable. If and are both globally positive definite, then is globally asymptotically stable.

Example 15..1 (Establishing Stability via Lyapunov Functions)   Let . Let , and we will attempt to show that is stable. Let the candidate Lyapunov function be . The Lie derivative (15.3) produces . It is clear that and are both globally positive definite; hence, 0 is a global, asymptotically stable equilibrium point of .

Steven M LaValle 2020-08-14