Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T01:31:15.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability rates for patchy vector fields

Published online by Cambridge University Press:  15 March 2004

Fabio Ancona  and
Alberto Bressan
Fabio Ancona
Affiliation:
Dipartimento di Matematica and C.I.R.A.M., Università di Bologna, Piazza Porta S. Donato 5, Bologna 40127, Italy; [email protected].
Alberto Bressan
Affiliation:
S.I.S.S.A., Via Beirut 4, Trieste 34014, Italy; [email protected].
Get access

Abstract

This paper is concerned with the stability of the set of trajectoriesof a patchy vector field, in the presence of impulsiveperturbations. Patchy vector fields arediscontinuous, piecewise smooth vector fieldsthat were introduced in Ancona and Bressan (1999) to study feedback stabilization problems.For patchy vector fields in the plane, with polygonalpatches 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.

Keywords

Patchy vector fieldimpulsive perturbation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 2 , April 2004 , pp. 168 - 200
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ancona, F. and Bressan, A., Patchy vector fields and asymptotic stabilization. ESAIM: COCV 4 (1999) 445-471. CrossRef
Ancona, F. and Bressan, A., Flow Stability of Patchy vector fields and Robust Feedback Stabilization. SIAM J. Control Optim. 41 (2003) 1455-1476. CrossRef
Bressan, A., On differential systems with impulsive controls. Rend. Sem. Mat. Univ. Padova 78 (1987) 227-235.
Clarke, F.H., Ledyaev, Yu.S., Rifford, L. and Stern, R.J., Feedback stabilization and Lyapunov functions. SIAM J. Control Optim. 39 (2000) 25-48. CrossRef
Clarke, F.H., Ledyaev, Yu.S., Sontag, E.D. and Subbotin, A.I., Asymptotic controllability implies feedback stabilization. IEEE Trans. Autom. Control 42 (1997) 1394-1407. CrossRef
F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory 178. Springer-Verlag, New York (1998).
Rifford, L., Existence of Lipschitz and semi-concave control-Lyapunov functions. SIAM J. Control Optim. 39 (2000) 1043-1064. CrossRef
Rifford, L., Semi-concave control-Lyapunov functions and stabilizing feedbacks. SIAM J. Control Optim. 41 (2002) 659-681. CrossRef
E.D. Sontag, Stability and stabilization: discontinuities and the effect of disturbances, in Proc. NATO Advanced Study Institute – Nonlinear Analysis, Differential Equations, and Control, Montreal, Jul/Aug 1998, F.H. Clarke and R.J. Stern Eds., Kluwer (1999) 551-598.