## 15.1.1 Stability

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

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

 (15.1)

This looks like a state transition equation that is missing actions. If a system of the form is given, then can be fixed by designing a feedback plan . This yields , which is a vector field on 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.

Subsections
Steven M LaValle 2020-08-14