Dynamics at a Switching Intersection: Hierarchy, Isonomy, and Multiple Sliding

Abstract

If a set of ordinary differential equations is discontinuous along some threshold, solutions can be found that are continuous, if sometimes multivalued. We show the extent to which unique solutions can be found in general cases when the threshold takes the form of finitely many intersecting manifolds. If the intersections are transversal, finitely many solutions can be found that slide along the threshold. They are obtained by a hierarchical application of convex combinations to form a differential inclusion. The system chooses between these solutions by means of an instantaneous dummy system. No assumptions on attractivity are required, and all switches are treated equally, so the standard “Filippov” method is extended to intersections of discontinuity manifolds in the most natural way possible. The corresponding result in the setting of equivalent control is also given, allowing systems more general than typical linear control forms to be solved.