Optimal impulsive control of delay systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 767-779.

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.

DOI : https://doi.org/10.1051/cocv:2008009
Classification : 49K15,  49K25
Mots clés : optimal control, impulse control, switched systems, delay systems, calculus of variation
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author = {Delmotte, Florent and Verriest, Erik I. and Egerstedt, Magnus},
title = {Optimal impulsive control of delay systems},
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Delmotte, Florent; Verriest, Erik I.; Egerstedt, Magnus. Optimal impulsive control of delay systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 767-779. doi : 10.1051/cocv:2008009. http://www.numdam.org/articles/10.1051/cocv:2008009/

[1] R.M. Anderson and R.M. May, Directly transmitted infectious diseases: Control by vaccination. Science 215 (1982) 1053-1060. | MR 646224

[2] D.D. Bainov and P.S. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications. Ellis Horwood Limited, Chichester, West Sussex (1989). | MR 1010418 | Zbl 0683.34032

[3] 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). | MR 1266625 | Zbl 0815.34001

[4] 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). | MR 1331144 | Zbl 0828.34002

[5] M.S. Branicky, V.S. Borkar and S.K. Mitter, A unified framework for hybrid control: Model and optimal control theory. IEEE Trans. Automatic Control 43 (1998) 31-45. | MR 1604250 | Zbl 0951.93002

[6] A.E. Bryson and Y.C. Ho, Applied Optimal Control. Routledge (1975).

[7] 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.

[8] K.L. Cooke and P. Van Den Driessche, Analysis of an seirs epidemic model with two delays. J. Math. Biology 35 (1996) 240-260. | MR 1478050 | Zbl 0865.92019

[9] 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.

[10] 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. | MR 292559

[11] H.E. Gollwitzer, Applications of the method of steepest descent to optimal control problems. Master's thesis, University of Minnesota, USA (1965).

[12] 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.

[13] R. Rishel, Application of an extended Pontryagin principle. IEEE Trans. Automatic Control 11 (1966) 167-170. | MR 204189

[14] 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.

[15] 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.

[16] 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).

[17] 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).

[18] 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.

[19] E.I. Verriest, F. Delmotte and M. Egerstedt, Control strategies for epidemics by vaccination. Automatica (submitted).

[20] W. Wendi and M. Zhien, Global dynamics of an epidemic model with time delay. Nonlinear Analysis: Real World Applications archive 3 (2002) 365-373. | MR 1905456 | Zbl 0998.92038

[21] 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.

[22] T. Yang, Impulsive control. IEEE Trans. Automatic Control 44 (1999) 1081-1083. | MR 1690562 | Zbl 0954.49022

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