Piecewise Smooth Vector Fields Research Articles

Stability rates for patchy vector fields

This paper is concerned with the stability of the set of trajectories of a patchy vector field, in the presence of impulsive perturbations. Patchy vector fields are discontinuous, piecewise smooth vector fields that were introduced in Ancona and Bressan (1999) to study feedback stabilization problems. For patchy vector fields in the plane, with polygonal patches in generic position, we show that the distance between a perturbed trajectory and an unperturbed one is of the same order of magnitude as the impulsive forcing term.

Discontinuity mappings for vector fields with higher order continuity

This paper concerns the derivation of discontinuity mappings in dynamical systems defined by piecewise smooth vector fields. The cases of trajectories that either cross or are quadratically tangent to a co-dimension 1 surface in state space are treated. The vector field is assumed to have C n−1 continuity across the surface, and it is shown that the discontinuity mappings are equal to the identity up to order n, and the structure of the mappings and the lowest order term is derived. An example where the mapping can be derived in another way is used to illustrate the theory.

Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization

The paper is concerned with patchy vector fields, a class of discontinuous, piecewise smooth vector fields that were introduced by the authors to study feedback stabilization problems. We prove the stability of the corresponding solution set w.r. t. a wide class of impulsive perturbations. These results yield the robustness of patchy feedback controls in the presence of measurement errors and external disturbances.