Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T06:11:40.140Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels

Published online by Cambridge University Press:  10 October 2013

Alexis Guigue
Alexis Guigue*
Affiliation:
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C., V6T 1Z2, Canada. [email protected]
Get access

Abstract

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.

Keywords

Multiobjective optimal controlPareto optimalityviability theoryconvergent numerical approximationdynamic programming
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014 , pp. 95 - 115
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.-P. Aubin, Viability theory. Birkhauser, Boston (1991).
J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhauser, Boston (1990).
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Boston (1997).
P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, 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.
Cardaliaguet, P., Quincampoix, M. and Saint-Pierre, P., Numerical schemes for discontinuous value functions of optimal control. Set-Valued Anal. 8 (2000) 111–126. Google Scholar
P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, 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.
Coverstone-Carroll, V., Hartmann, J.W. and Mason, W.J., Optimal multi-objective low-thrust spacecraft trajectories. Comput. Methods Appl. Mech. Engrg. 186 (2000) 387–402. Google Scholar
Diaz de Leon, A.J. and Seijo, J.C., A multi-criteria non-linear optimization model for the control and management of a tropical fishery. Mar. Resour. Econ. 7 (1992) 23–40. Google Scholar
K. Deb, Multi-objective optimization using evolutionary algorithms. John Wiley and Sons, Chichister (2001).
Doyen, L. and Saint-Pierre, P., Scale of viability and minimal time of crisis. Set-Valued Anal. 5 (1997) 227–246. Google Scholar
Fleming, P.J. and Purshouse, R.C., Evolutionary algorithms in control systems engineering: a survey. Control Engrg. Pract. 10 (2002) 1223–1241. Google Scholar
A. Guigue, An approximation method for multiobjective optimal control problems application to a robotic trajectory planning problem. Submitted to Optim. Engrg. (2010).
A. Guigue, Set-valued return function and generalized solutions for multiobjective optimal control problems (moc). Submitted to SIAM J. Control Optim. (2011).
Guigue, A., Ahmadi, M., Hayes, M.J.D. and Langlois, R.G., A discrete dynamic programming approximation to the multiobjective deterministic finite horizon optimal control problem. SIAM J. Control Optim. 48 (2009) 2581–2599. Google Scholar
Guigue, A., Ahmadi, M., Langlois, R.G. and Hayes, M.J.D., Pareto optimality and multiobjective trajectory planning for a 7-dof redundant manipulator. IEEE Trans. Robotics 26 (2010) 1094–1099. Google Scholar
Guo, B.-Z. and Sun, B., Numerical solution to the optimal feedback control of continuous casting process. J. Glob. Optim. 39 (1998) 171–195. Google Scholar
Kumar, A. and Vladimirsky, A., An efficient method for multiobjective optimal control and optimal control subject to integral constraints. J. Comp. Math. 28 (2010) 517–551. Google Scholar
Mardle, S. and Pascoe, S., A review of applications of multiple-criteria decision-making techniques to fisheries. Mar. Resour. Econ. 14 (1998) 41–63. Google Scholar
K.M. Miettinen, Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston (1999).
Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization. Academic Press, Inc., Orlando (1985).
Tanino, T., Sensitivity analysis in multiobjective optimization. J. Optim. Theory Appl. 56 (1988) 479–499. Google Scholar
R. Vinter, Optimal Control. Birkauser, Boston (2000).
Yu, P.L., Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14 (1974) 319–377. Google Scholar