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.

Keywords: optimal control, impulse control, switched systems, delay systems, calculus of variation

@article{COCV_2008__14_4_767_0, author = {Delmotte, Florent and Verriest, Erik I. and Egerstedt, Magnus}, title = {Optimal impulsive control of delay systems}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {767--779}, publisher = {EDP-Sciences}, volume = {14}, number = {4}, year = {2008}, doi = {10.1051/cocv:2008009}, mrnumber = {2451795}, zbl = {1148.49017}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2008009/} }

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 PY - 2008 SP - 767 EP - 779 VL - 14 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008009/ DO - 10.1051/cocv:2008009 LA - en ID - COCV_2008__14_4_767_0 ER -

%0 Journal Article %A Delmotte, Florent %A Verriest, Erik I. %A Egerstedt, Magnus %T Optimal impulsive control of delay systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 767-779 %V 14 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008009/ %R 10.1051/cocv:2008009 %G en %F COCV_2008__14_4_767_0

Delmotte, Florent; Verriest, Erik I.; Egerstedt, Magnus. Optimal impulsive control of delay systems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 4, pp. 767-779. doi : 10.1051/cocv:2008009. http://archive.numdam.org/articles/10.1051/cocv:2008009/

[1] Directly transmitted infectious diseases: Control by vaccination. Science 215 (1982) 1053-1060. | MR

and ,[2] Systems with Impulse Effect: Stability, Theory and Applications. Ellis Horwood Limited, Chichester, West Sussex (1989). | MR | Zbl

and ,[3] Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics 66. Longman Scientific, Harlow (1993). | MR | Zbl

and ,[4] Impulsive Differential Equations: Asymptotic Properties of the Solutions, Series on Advances in Mathematics for Applied Sciences 28. World Scientific (1995). | MR | Zbl

and ,[5] A unified framework for hybrid control: Model and optimal control theory. IEEE Trans. Automatic Control 43 (1998) 31-45. | MR | Zbl

, and ,[6] Applied Optimal Control. Routledge (1975).

and ,[7] 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.

and ,[8] Analysis of an seirs epidemic model with two delays. J. Math. Biology 35 (1996) 240-260. | MR | Zbl

and ,[9] 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.

, and ,[10] 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

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

,[12] Time-optimal control of the swing using impulse control actions, in Proceedings of the 1998 American Control Conference 1 (1998) V200-204.

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

,[14] Optimal impulsive control problems with state constraints, in Proceedings of the 32nd IEEE Conference on Decision and Control 4 (1993) 3811-3812.

and ,[15] A maximum principle for hybrid optimal control problems, in Proceedings of the 38th IEEE Conference on Decision and Control 1 (1999) 425-430.

,[16] Regularization method for optimally switched and impulse systems with biomedical applications, in Proceedings of the 42nd IEEE Conference on Decision and Control (2003).

,[17] Optimal impulsive control for point delay systems with refractory period, in IFAC Workshop on Time-Delay Systems, Leuven, Belgium (2004).

, and ,[18] Control of epidemics by vaccination, in Proceedings of the 2005 American Control Conference 2 (2005) 985-990.

, and ,[19] Control strategies for epidemics by vaccination. Automatica (submitted).

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

and ,[21] Optimal control of switched autonomous systems, in Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV (2002) 4401-4406.

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

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