Sliding vector fields for non-smooth dynamical systems having intersecting switching manifolds

摘要

We consider a differential equation ˙p = X(p), p ϵ R3 with discontinuous right-hand side and discontinuities occurring on an algebraic variety ∑. We discuss the dynamics of the sliding mode which occurs when for any initial condition near p ϵ ∑ the corresponding solution trajectories are attracted to ∑. First we suppose that ∑ = H-1(0) where H is a polynomial function and 0 ϵ R is a regular value. In this case ∑ is locally di↵eomorphic to the set F = {(x, y, z) ϵ R3; z = 0} (Filippov). Second we suppose that ∑ is the inverse image of a non–regular value. We focus our attention to the equations defined around singularities as described in [8]. More precisely, we restrict the degeneracy of the singularity so as to admit only those which appear when the regularity conditions in the definition of smooth surfaces of R3 in terms of implicit functions and immersions are broken in a stable manner. In this case ∑ is locally diffeomorphic to one of the following sets D = {(x, y, z) ϵ R3; xy = 0} (double crossing); T = {(x, y, z) ϵ R3; xyz = 0} (triple crossing); C = {(x, y, z) ϵ R3; z2-x2-y2 = 0}(cone) or W = {(x, y, z) ϵ R3; zx2-y2 = 0} (Whitney’s umbrella).