# Optimal impulsive control of delay systems

Florent Delmotte; Erik I. Verriest; Magnus Egerstedt

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 14, Issue: 4, page 767-779
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topDelmotte, Florent, Verriest, Erik I., and Egerstedt, Magnus. "Optimal impulsive control of delay systems." ESAIM: Control, Optimisation and Calculus of Variations 14.4 (2008): 767-779. <http://eudml.org/doc/250312>.

@article{Delmotte2008,

abstract = {
In this paper, we solve an optimal control problem using the
calculus of variation. The system under consideration is a
switched autonomous delay system that undergoes jumps at the
switching times. The control variables are the instants when the
switches occur, and a set of scalars which determine the jump
amplitudes. Optimality conditions involving analytic expressions
for the partial derivatives of a given cost function with respect
to the control variables are derived using the calculus of
variation. A locally optimal impulsive control strategy can then
be found using a numerical gradient descent algorithm.
},

author = {Delmotte, Florent, Verriest, Erik I., Egerstedt, Magnus},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Optimal control; impulse
control; switched systems; delay systems; calculus of variation.; optimal control; impulse control; calculus of variation},

language = {eng},

month = {1},

number = {4},

pages = {767-779},

publisher = {EDP Sciences},

title = {Optimal impulsive control of delay systems},

url = {http://eudml.org/doc/250312},

volume = {14},

year = {2008},

}

TY - JOUR

AU - Delmotte, Florent

AU - Verriest, Erik I.

AU - Egerstedt, Magnus

TI - Optimal impulsive control of delay systems

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2008/1//

PB - EDP Sciences

VL - 14

IS - 4

SP - 767

EP - 779

AB -
In this paper, we solve an optimal control problem using the
calculus of variation. The system under consideration is a
switched autonomous delay system that undergoes jumps at the
switching times. The control variables are the instants when the
switches occur, and a set of scalars which determine the jump
amplitudes. Optimality conditions involving analytic expressions
for the partial derivatives of a given cost function with respect
to the control variables are derived using the calculus of
variation. A locally optimal impulsive control strategy can then
be found using a numerical gradient descent algorithm.

LA - eng

KW - Optimal control; impulse
control; switched systems; delay systems; calculus of variation.; optimal control; impulse control; calculus of variation

UR - http://eudml.org/doc/250312

ER -

## References

top- R.M. Anderson and R.M. May, Directly transmitted infectious diseases: Control by vaccination. Science215 (1982) 1053–1060. Zbl1225.37099
- D.D. Bainov and P.S. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications. Ellis Horwood Limited, Chichester, West Sussex (1989). Zbl0676.34035
- D.D. Bainov and P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics66. Longman Scientific, Harlow (1993).
- D.D. Bainov and P.S. Simeonov, Impulsive Differential Equations: Asymptotic Properties of the Solutions, Series on Advances in Mathematics for Applied Sciences28. World Scientific (1995). Zbl0828.34002
- M.S. Branicky, V.S. Borkar and S.K. Mitter, A unified framework for hybrid control: Model and optimal control theory. IEEE Trans. Automatic Control43 (1998) 31–45. Zbl0951.93002
- A.E. Bryson and Y.C. Ho, Applied Optimal Control. Routledge (1975).
- J. Chudoung and C. Beck, The minimum principle for deterministic impulsive control systems, in Proceedings of the 40th IEEE Conference on Decision and Control4, Orlando, FL (2001) 3569–3574.
- K.L. Cooke and P. van den Driessche, Analysis of an seirs epidemic model with two delays. J. Math. Biology35 (1996) 240–260. Zbl0865.92019
- M. Egerstedt, Y. Wardi and F. Delmotte, Optimal control of switching times in switched dynamical systems, in Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii (2003) 2138–2143.
- E.G. Gilbert and G.A. Harasty, A class of fixed-time fuel-optimal impulsive control problems and an efficient algorithm for their solution. IEEE Trans. Automatic ControlAC-16 (1971) 1–11.
- H.E. Gollwitzer, Applications of the method of steepest descent to optimal control problems. Master's thesis, University of Minnesota, USA (1965).
- J.C. Luo and E.B. Lee, Time-optimal control of the swing using impulse control actions, in Proceedings of the 1998 American Control Conference1 (1998) V200–204.
- R. Rishel, Application of an extended Pontryagin principle. IEEE Trans. Automatic Control11 (1966) 167–170.
- G.N. Silva and R.B. Vinter, Optimal impulsive control problems with state constraints, in Proceedings of the 32nd IEEE Conference on Decision and Control4 (1993) 3811–3812.
- H.J. Sussmann, A maximum principle for hybrid optimal control problems, in Proceedings of the 38th IEEE Conference on Decision and Control1 (1999) 425–430.
- E.I. Verriest, Regularization method for optimally switched and impulse systems with biomedical applications, in Proceedings of the 42nd IEEE Conference on Decision and Control (2003).
- E.I. Verriest, F. Delmotte and M. Egerstedt, Optimal impulsive control for point delay systems with refractory period, in IFAC Workshop on Time-Delay Systems, Leuven, Belgium (2004). Zbl1148.49017
- E.I. Verriest, F. Delmotte and M. Egerstedt, Control of epidemics by vaccination, in Proceedings of the 2005 American Control Conference2 (2005) 985–990.
- E.I. Verriest, F. Delmotte and M. Egerstedt, Control strategies for epidemics by vaccination. Automatica (submitted). Zbl1148.49017
- W. Wendi and M. Zhien, Global dynamics of an epidemic model with time delay. Nonlinear Analysis: Real World Applications archive3 (2002) 365–373. Zbl0998.92038
- X. Xu and P. Antsaklis, Optimal control of switched autonomous systems, in Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV (2002) 4401–4406.
- T. Yang, Impulsive control. IEEE Trans. Automatic Control44 (1999) 1081–1083. Zbl0954.49022

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.