It will be important to allow vector fields that are smooth only over
a finite number of patches. At a *switching boundary* between two
patches, a discontinuous jump may occur. For example, suppose that an
-dimensional switching boundary,
, is defined
as

(8.17) |

in which is a function . If has dimension and is not singular, then has dimension . Define

(8.18) |

and

(8.19) |

The definitions are similar to the construction of implicit models using geometric primitives in Section 3.1.2. Suppose that is smooth over and but experiences a discontinuous jump at . Such differential equations model

Let denote an open ball of radius centered at . Let denote the set

for which | (8.20) |

Let denote any subset of that has measure zero (i.e., ). Let denote the convex hull of a set, , of points in . A path is called a

in which the intersections are taken over all possible and sets, , of measure zero. The expression (8.21) is actually called a

Many bizarre vector fields can yield solutions in the sense of
Filipov. The switching boundary model is relatively simple among
those permitted by Filipov's condition. Figure 8.7
shows various cases that can occur at the switching boundary .
For the case of consistent flow, solutions occur as you may
intuitively expect. Filipov's condition, (8.21),
requires that at the velocity vector of points between
vectors before and after crossing (for example, it can point down,
which is the average of the two directions). The magnitude must also
be between the two magnitudes. For the inward flow case, the integral
curve moves along , assuming the vectors inside of point in the
same direction (within the convex hull) as the vectors on either side
of the boundary. In applications that involve physical systems, this
may lead to oscillations around . This can be alleviated by
regularization, which thickens the boundary [846] (the subject
of *sliding-mode control* addresses this issue [303]).
The outward flow case can lead to nonuniqueness if the initial state
lies in . However, trajectories that start outside of will not
cross , and there will be no such troubles. If the flow is tangent
on both sides of a boundary, then other forms of nonuniqueness may
occur. The tangent-flow case will be avoided in this chapter.

Steven M LaValle 2012-04-20