Stability rates for patchy vector fields

Fabio Ancona; Alberto Bressan

ESAIM: Control, Optimisation and Calculus of Variations (2004)

  • Volume: 10, Issue: 2, page 168-200
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Ancona, Fabio, and Bressan, Alberto. "Stability rates for patchy vector fields." ESAIM: Control, Optimisation and Calculus of Variations 10.2 (2004): 168-200. <http://eudml.org/doc/244945>.

@article{Ancona2004,
abstract = {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.},
author = {Ancona, Fabio, Bressan, Alberto},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Patchy vector field; impulsive perturbation; patchy vector field},
language = {eng},
number = {2},
pages = {168-200},
publisher = {EDP-Sciences},
title = {Stability rates for patchy vector fields},
url = {http://eudml.org/doc/244945},
volume = {10},
year = {2004},
}

TY - JOUR
AU - Ancona, Fabio
AU - Bressan, Alberto
TI - Stability rates for patchy vector fields
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2004
PB - EDP-Sciences
VL - 10
IS - 2
SP - 168
EP - 200
AB - 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.
LA - eng
KW - Patchy vector field; impulsive perturbation; patchy vector field
UR - http://eudml.org/doc/244945
ER -

References

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  1. [1] F. Ancona and A. Bressan, Patchy vector fields and asymptotic stabilization. ESAIM: COCV 4 (1999) 445-471. Zbl0924.34058MR1693900
  2. [2] F. Ancona and A. Bressan, Flow Stability of Patchy vector fields and Robust Feedback Stabilization. SIAM J. Control Optim. 41 (2003) 1455-1476. Zbl1055.34094MR1971958
  3. [3] A. Bressan, On differential systems with impulsive controls. Rend. Sem. Mat. Univ. Padova 78 (1987) 227-235. MR934514
  4. [4] F.H. Clarke, Yu.S. Ledyaev, L. Rifford and R.J. Stern, Feedback stabilization and Lyapunov functions. SIAM J. Control Optim. 39 (2000) 25-48. Zbl0961.93047MR1780907
  5. [5] F.H. Clarke, Yu.S. Ledyaev, E.D. Sontag and A.I. Subbotin, Asymptotic controllability implies feedback stabilization. IEEE Trans. Autom. Control 42 (1997) 1394-1407. Zbl0892.93053MR1472857
  6. [6] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory 178. Springer-Verlag, New York (1998). Zbl1047.49500MR1488695
  7. [7] L. Rifford, Existence of Lipschitz and semi-concave control-Lyapunov functions. SIAM J. Control Optim. 39 (2000) 1043-1064. Zbl0982.93068MR1814266
  8. [8] L. Rifford, Semi-concave control-Lyapunov functions and stabilizing feedbacks. SIAM J. Control Optim. 41 (2002) 659-681. Zbl1034.93053MR1939865
  9. [9] 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. Zbl0937.93034

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