Abstract
Closed loop stabilization of impulsive control systems containing a measure in the dynamics is considered. It is proved that, as for regular affine systems, an almost everywhere continuous stabilizing impulsive feedback control law exists for such impulsive systems. An example illustrating the loop closing features is also presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Bressan Jr. and F. Rampazzo, Impulsive control systems with commutative vector fields. J. Optim. Theory Appl. 71 (1991), No. 1, 67–83.
A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls. Un. Mat. Ital. B (7) 2 (1988), No. 3, 641–656.
G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls. Differential Integral Equations 4 (1991), No. 4, 739–765.
D. Yu. Karamzin, Necessary conditions of the minimum in an impulse optimal control problem. J. Math. Sci. 139 (2006), No. 6, 7087–7150.
B. M. Miller and E. Ya. Rubinovich, Impulsive control in continuous and discrete-continuous systems. Kluwer Academic/Plenum Publishers, New York (2003).
F. L. Pereira and G. N. Silva, Stability for impulsive control systems. Dynam. Systems 17 (2002), No. 4, 421–434.
_____, Lyapunov stability for impulsive control systems. Differential Equations 40 (2004), No. 8, 1122–1130.
_____, Necessary conditions of optimality for vector-valued impulsive control problems. Systems Control Lett. 40 (2000), No. 3, 205–215.
L. Rifford, Existence of Lipschitz and semiconcave control-Lyapunov functions. SIAM J. Control Optim. 39 (2000), No. 4, 1043–1064.
_____, Semiconcave control-Lyapunov functions and stabilizing feedbacks. SIAM J. Control Optim. 41 (2002), No. 3, 659–681.
_____, Stratified semiconcave control-Lyapunov functions and the stabilization problem. Ann. Inst. H. Poincaré. Anal. Non Linéaire 22 (2005), No. 3, 343–384.
_____, On the existence of local smooth repulsive stabilizing feedbacks in dimension three. J. Differential Equations 226 (2006), No. 2, 429–500.
R. W. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures. J. Soc. Indust. Appl. Math. Ser. A 3 (1965), 191–205.
G. N. Silva and R. B. Vinter, Measure driven differential inclusions. J. Math. Anal. Appl. 202 (1996), No. 3, 727–746.
E. D. Sontag, A “universal” construction of Artstein’s theorem on nonlinear stabilization. Systems Control Lett. 13 (1989), No. 2, 117–123.
J. Warga, Optimal control of differential and functional equations. Academic Press, New York (1972).
P. R. Wolenski and S. Žabić, A sampling method and approximation results for impulsive systems. SIAM J. Control Optim. 46 (2007), No. 3, 983–998.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by CNPq-Brazil, grant No. 200875/06-0. This work was done while the second author was a visiting scholar on Sabbatical at the University of British Columbia, Department of Mathematics and Pacific Institute for the Mathematical Sciences (PIMS).
Rights and permissions
About this article
Cite this article
Code, W.J., Silva, G.N. Closed loop stability of measure-driven impulsive control systems. J Dyn Control Syst 16, 1–21 (2010). https://doi.org/10.1007/s10883-010-9085-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10883-010-9085-9