Stability conditions for discontinuous vector fields

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 tha...

Controllability and stabilizability of bimodal systems with discontinuous vector fields

Collision Avoidance in Formation Control Using Discontinuous Vector Fields

This paper presents a novel collision avoidance approach in formation control for Multi-agent Robots. The control strategy consists in the mix of attractive vector fields and repulsive vector fields based on a scaled unstable focus centered at the position of another robot, instead of the vector fields obtained from the negative gradient of repulsive potential functions. The analysis of the clo...

The Two-Fold Singularity of Discontinuous Vector Fields

When a vector field in R is discontinuous on a smooth codimension one surface, it may be simultaneously tangent to both sides of the surface at generic isolated points (singularities). For a piecewise-smooth dynamical system governed by the vector field, we show that the local dynamics depends on a single quantity—the jump in direction of the vector field through the singularity. This quantity ...

On Algebraic Proofs of Stability for Homogeneous Vector Fields

We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sums of squares certificates and hence such a Lyapunov fun...