SLIDING MOTION ON TANGENTIAL SETS OF FILIPPOV SYSTEMS

The candidate’s proof reconstructs the paper’s Theorem 3: after φ-regularization, one can rewrite the dynamics as a singular perturbation problem possessing a slow manifold Stan diffeomorphic to M, and the reduced flow restricted to Stan is conjugate to the tangential sliding vector field Ztan. The paper proves this by straightening (h,η), rescaling x_n/ε, and constructing Stan = {(u,0,w*(u))} (with w*(u)=φ^{-1}(λ*(u,0,0))) and a direct conjugacy, whereas the candidate augments with a fast auxiliary variable λ to render the graph λ=φ(h/ε) invariant and then performs a slow-time reduction; both routes yield the same reduced dynamics under the same hypothesis 〈dηZ+,dηZ−〉=−‖dηZ+‖‖dηZ−‖≠0 on M. The statements and constructions match the paper’s definitions and Theorem 3 and its proof structure (Definition 1–2, Theorem 1, Definition 3, and Section 4) .