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,
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 . 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.
Steven M LaValle 2020-08-14