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
Let denote an open ball of radius centered at . Let denote the set
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  (the subject of sliding-mode control addresses this issue ). 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 2020-08-14