This paper provides a convergent numerical approximation of the Pareto optimal set for finite-horizon multiobjective optimal control problems in which the objective space is not necessarily convex. Our approach is based on Viability Theory. We first introduce a set-valued return function V and show that the epigraph of V equals the viability kernel of a certain related augmented dynamical system. We then introduce an approximate set-valued return function with finite set-values as the solution of a multiobjective dynamic programming equation. The epigraph of this approximate set-valued return function equals to the finite discrete viability kernel resulting from the convergent numerical approximation of the viability kernel proposed in [P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre. Birkhauser, Boston (1999) 177-247. P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Set-Valued Analysis 8 (2000) 111-126]. As a result, the epigraph of the approximate set-valued return function converges to the epigraph of V. The approximate set-valued return function finally provides the proposed numerical approximation of the Pareto optimal set for every initial time and state. Several numerical examples illustrate our approach.
Mots clés : multiobjective optimal control, Pareto optimality, viability theory, convergent numerical approximation, dynamic programming
@article{COCV_2014__20_1_95_0,
author = {Guigue, Alexis},
title = {Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {95--115},
publisher = {EDP-Sciences},
volume = {20},
number = {1},
year = {2014},
doi = {10.1051/cocv/2013056},
mrnumber = {3182692},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2013056/}
}
TY - JOUR AU - Guigue, Alexis TI - Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 95 EP - 115 VL - 20 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2013056/ DO - 10.1051/cocv/2013056 LA - en ID - COCV_2014__20_1_95_0 ER -
%0 Journal Article %A Guigue, Alexis %T Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 95-115 %V 20 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2013056/ %R 10.1051/cocv/2013056 %G en %F COCV_2014__20_1_95_0
Guigue, Alexis. Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 95-115. doi : 10.1051/cocv/2013056. http://www.numdam.org/articles/10.1051/cocv/2013056/
[1] , Viability theory. Birkhauser, Boston (1991). | MR | Zbl
[2] and , Set-Valued Analysis. Birkhauser, Boston (1990). | MR | Zbl
[3] and , Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Boston (1997). | MR | Zbl
[4] , and , Set-valued numerical analysis for optimal control and differential games. Stochastic and differential games: Theory and numerical methods. Annals of the international Society of Dynamic Games, edited by M. Bardi, T.E.S. Raghavan, T. Parthasarathy. Birkhauser, Boston (1999) 177-247. | MR | Zbl
[5] , and , Numerical schemes for discontinuous value functions of optimal control. Set-Valued Anal. 8 (2000) 111-126. | MR | Zbl
[6] , and , Differential games through viability theory: Old and recent results. Advances in Dynamic Game Theory. Annals of the international Society of Dynamic Games, edited by S. Jorgensen, M. Quincampoix, T.L. Vincent, T. Basar. Birkhauser, Boston (2007) 3-35. | MR | Zbl
[7] , and , Optimal multi-objective low-thrust spacecraft trajectories. Comput. Methods Appl. Mech. Engrg. 186 (2000) 387-402. | Zbl
[8] and , A multi-criteria non-linear optimization model for the control and management of a tropical fishery. Mar. Resour. Econ. 7 (1992) 23-40.
[9] , Multi-objective optimization using evolutionary algorithms. John Wiley and Sons, Chichister (2001). | MR | Zbl
[10] and , Scale of viability and minimal time of crisis. Set-Valued Anal. 5 (1997) 227-246. | MR | Zbl
[11] and , Evolutionary algorithms in control systems engineering: a survey. Control Engrg. Pract. 10 (2002) 1223-1241.
[12] , An approximation method for multiobjective optimal control problems application to a robotic trajectory planning problem. Submitted to Optim. Engrg. (2010).
[13] , Set-valued return function and generalized solutions for multiobjective optimal control problems (moc). Submitted to SIAM J. Control Optim. (2011). | MR | Zbl
[14] , , and , A discrete dynamic programming approximation to the multiobjective deterministic finite horizon optimal control problem. SIAM J. Control Optim. 48 (2009) 2581-2599. | MR | Zbl
[15] , , and , Pareto optimality and multiobjective trajectory planning for a 7-dof redundant manipulator. IEEE Trans. Robotics 26 (2010) 1094-1099.
[16] and , Numerical solution to the optimal feedback control of continuous casting process. J. Glob. Optim. 39 (1998) 171-195. | MR | Zbl
[17] and , An efficient method for multiobjective optimal control and optimal control subject to integral constraints. J. Comp. Math. 28 (2010) 517-551. | MR | Zbl
[18] and , A review of applications of multiple-criteria decision-making techniques to fisheries. Mar. Resour. Econ. 14 (1998) 41-63. | Zbl
[19] , Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston (1999). | MR | Zbl
[20] , and , Theory of Multiobjective Optimization. Academic Press, Inc., Orlando (1985). | MR | Zbl
[21] , Sensitivity analysis in multiobjective optimization. J. Optim. Theory Appl. 56 (1988) 479-499. | MR | Zbl
[22] , Optimal Control. Birkauser, Boston (2000). | MR | Zbl
[23] , Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14 (1974) 319-377. | MR | Zbl
Cité par Sources :
