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},
year = {2008},
publisher = {EDP Sciences},
volume = {14},
number = {4},
doi = {10.1051/cocv:2008009},
mrnumber = {2451795},
zbl = {1148.49017},
language = {en},
url = {https://www.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 - https://www.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 https://www.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, Tome 14 (2008) no. 4, pp. 767-779. doi: 10.1051/cocv:2008009
[1] and , Directly transmitted infectious diseases: Control by vaccination. Science 215 (1982) 1053-1060. | MR
[2] and , Systems with Impulse Effect: Stability, Theory and Applications. Ellis Horwood Limited, Chichester, West Sussex (1989). | Zbl | MR
[3] and , Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics 66. Longman Scientific, Harlow (1993). | Zbl | MR
[4] and , Impulsive Differential Equations: Asymptotic Properties of the Solutions, Series on Advances in Mathematics for Applied Sciences 28. World Scientific (1995). | Zbl | MR
[5] , and , A unified framework for hybrid control: Model and optimal control theory. IEEE Trans. Automatic Control 43 (1998) 31-45. | Zbl | MR
[6] and , Applied Optimal Control. Routledge (1975).
[7] and , 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] and , Analysis of an seirs epidemic model with two delays. J. Math. Biology 35 (1996) 240-260. | Zbl | MR
[9] , and , 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] and , 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
[11] , Applications of the method of steepest descent to optimal control problems. Master's thesis, University of Minnesota, USA (1965).
[12] and , Time-optimal control of the swing using impulse control actions, in Proceedings of the 1998 American Control Conference 1 (1998) V200-204.
[13] , Application of an extended Pontryagin principle. IEEE Trans. Automatic Control 11 (1966) 167-170. | MR
[14] and , Optimal impulsive control problems with state constraints, in Proceedings of the 32nd IEEE Conference on Decision and Control 4 (1993) 3811-3812.
[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] , and , Optimal impulsive control for point delay systems with refractory period, in IFAC Workshop on Time-Delay Systems, Leuven, Belgium (2004).
[18] , and , Control of epidemics by vaccination, in Proceedings of the 2005 American Control Conference 2 (2005) 985-990.
[19] , and , Control strategies for epidemics by vaccination. Automatica (submitted).
[20] and , Global dynamics of an epidemic model with time delay. Nonlinear Analysis: Real World Applications archive 3 (2002) 365-373. | Zbl | MR
[21] and , Optimal control of switched autonomous systems, in Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV (2002) 4401-4406.
[22] , Impulsive control. IEEE Trans. Automatic Control 44 (1999) 1081-1083. | Zbl | MR
Cité par Sources :
