Stability rates for patchy vector fields
ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 2, pp. 168-200.

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.

DOI : 10.1051/cocv:2004003
Classification : 34A37, 34D, 93D09, 93D15
Mots clés : Patchy vector field, impulsive perturbation
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     title = {Stability rates for patchy vector fields},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
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     publisher = {EDP-Sciences},
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Ancona, Fabio; Bressan, Alberto. Stability rates for patchy vector fields. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 2, pp. 168-200. doi : 10.1051/cocv:2004003. http://archive.numdam.org/articles/10.1051/cocv:2004003/

[1] F. Ancona and A. Bressan, Patchy vector fields and asymptotic stabilization. ESAIM: COCV 4 (1999) 445-471. | Numdam | MR | Zbl

[2] F. Ancona and A. Bressan, Flow Stability of Patchy vector fields and Robust Feedback Stabilization. SIAM J. Control Optim. 41 (2003) 1455-1476. | MR | Zbl

[3] A. Bressan, On differential systems with impulsive controls. Rend. Sem. Mat. Univ. Padova 78 (1987) 227-235. | Numdam | MR

[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. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[7] L. Rifford, Existence of Lipschitz and semi-concave control-Lyapunov functions. SIAM J. Control Optim. 39 (2000) 1043-1064. | MR | Zbl

[8] L. Rifford, Semi-concave control-Lyapunov functions and stabilizing feedbacks. SIAM J. Control Optim. 41 (2002) 659-681. | MR | Zbl

[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. | Zbl

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