15.1.1 Stability

The subject of stability addresses properties of a vector field with respect to a given point. Let $X$ denote a smooth manifold on which the vector field is defined; $X$ may be a C-space or a phase space. The given point is denoted as ${x_{G}}$ and can be interpreted in motion planning applications as the goal state. Stability characterizes how ${x_{G}}$ is approached from other states in $X$ by integrating the vector field.

The given vector field $f$ is considered as a velocity field, which is represented as

$${\dot x}= f(x) .$$ (15.1)

This looks like a state transition equation that is missing actions. If a system of the form ${\dot x}= f(x,u)$ is given, then $u$ can be fixed by designing a feedback plan $\pi : X \rightarrow U$. This yields ${\dot x}= f(x,\pi (x))$, which is a vector field on $X$ without any further dependency on actions. The dynamic programming approach in Section 14.5 computed such a solution. The process of designing a stable feedback plan is referred to in control literature as feedback stabilization.