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Optimal impulsive control of delay systems

Published online by Cambridge University Press:  30 January 2008

Florent Delmotte ,
Erik I. Verriest  and
Magnus Egerstedt
Florent Delmotte
Affiliation:
School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30302, USA; [email protected]; [email protected]; [email protected]
Erik I. Verriest
Affiliation:
School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30302, USA; [email protected]; [email protected]; [email protected]
Magnus Egerstedt
Affiliation:
School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30302, USA; [email protected]; [email protected]; [email protected]
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Abstract

In this paper, we solve an optimal control problem using thecalculus of variation. The system under consideration is aswitched autonomous delay system that undergoes jumps at theswitching times. The control variables are the instants when theswitches occur, and a set of scalars which determine the jumpamplitudes. Optimality conditions involving analytic expressionsfor the partial derivatives of a given cost function with respectto the control variables are derived using the calculus ofvariation. A locally optimal impulsive control strategy can thenbe found using a numerical gradient descent algorithm.

Keywords

Optimal controlimpulsecontrolswitched systemsdelay systemscalculus of variation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008 , pp. 767 - 779
Copyright
© EDP Sciences, SMAI, 2008

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References

Anderson, R.M. and Directly, R.M. May transmitted infectious diseases: Control by vaccination. Science 215 (1982) 1053–1060. CrossRef
D.D. Bainov and P.S. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications. Ellis Horwood Limited, Chichester, West Sussex (1989).
D.D. Bainov and P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics 66. 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 Sciences 28. World Scientific (1995).
Branicky, M.S., Borkar, V.S. and Mitter, S.K., A unified framework for hybrid control: Model and optimal control theory. IEEE Trans. Automatic Control 43 (1998) 31–45. CrossRef
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 Control 4, Orlando, FL (2001) 3569–3574.
Cooke, K.L. and van den Driessche, P., Analysis of an seirs epidemic model with two delays. J. Math. Biology 35 (1996) 240–260. CrossRef
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 Control AC-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 Conference 1 (1998) V200–204.
Rishel, R., Application of an extended Pontryagin principle. IEEE Trans. Automatic Control 11 (1966) 167–170. CrossRef
G.N. Silva and R.B. Vinter, Optimal impulsive control problems with state constraints, in Proceedings of the 32nd IEEE Conference on Decision and Control 4 (1993) 3811–3812. CrossRef
H.J. Sussmann, A maximum principle for hybrid optimal control problems, in Proceedings of the 38th IEEE Conference on Decision and Control 1 (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).
E.I. Verriest, F. Delmotte and M. Egerstedt, Control of epidemics by vaccination, in Proceedings of the 2005 American Control Conference 2 (2005) 985–990. CrossRef
E.I. Verriest, F. Delmotte and M. Egerstedt, Control strategies for epidemics by vaccination. Automatica (submitted).
Wendi, W. and Zhien, M., Global dynamics of an epidemic model with time delay. Nonlinear Analysis: Real World Applications archive 3 (2002) 365–373. CrossRef
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.
Yang, T., Impulsive control. IEEE Trans. Automatic Control 44 (1999) 1081–1083. CrossRef