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

Guigue, Alexis

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 95-115.

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.

[1] J.-P. Aubin, Viability theory. Birkhauser, Boston (1991). | MR | Zbl

[2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhauser, Boston (1990). | MR | Zbl

[3] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Boston (1997). | MR | Zbl

[4] 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. | MR | Zbl

[5] P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Numerical schemes for discontinuous value functions of optimal control. Set-Valued Anal. 8 (2000) 111-126. | MR | Zbl

[6] 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. | MR | Zbl

[7] V. Coverstone-Carroll, J.W. Hartmann and W.J. Mason, Optimal multi-objective low-thrust spacecraft trajectories. Comput. Methods Appl. Mech. Engrg. 186 (2000) 387-402. | Zbl

[8] A.J. Diaz De Leon and J.C. Seijo, A multi-criteria non-linear optimization model for the control and management of a tropical fishery. Mar. Resour. Econ. 7 (1992) 23-40.

[9] K. Deb, Multi-objective optimization using evolutionary algorithms. John Wiley and Sons, Chichister (2001). | MR | Zbl

[10] L. Doyen and P. Saint-Pierre, Scale of viability and minimal time of crisis. Set-Valued Anal. 5 (1997) 227-246. | MR | Zbl

[11] P.J. Fleming and R.C. Purshouse, Evolutionary algorithms in control systems engineering: a survey. Control Engrg. Pract. 10 (2002) 1223-1241.

[12] A. Guigue, An approximation method for multiobjective optimal control problems application to a robotic trajectory planning problem. Submitted to Optim. Engrg. (2010).

[13] A. Guigue, Set-valued return function and generalized solutions for multiobjective optimal control problems (moc). Submitted to SIAM J. Control Optim. (2011). | MR | Zbl

[14] A. Guigue, M. Ahmadi, M.J.D. Hayes and R.G. Langlois, 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] A. Guigue, M. Ahmadi, R.G. Langlois and M.J.D. Hayes, Pareto optimality and multiobjective trajectory planning for a 7-dof redundant manipulator. IEEE Trans. Robotics 26 (2010) 1094-1099.

[16] B.-Z. Guo and B. Sun, Numerical solution to the optimal feedback control of continuous casting process. J. Glob. Optim. 39 (1998) 171-195. | MR | Zbl

[17] A. Kumar and A. Vladimirsky, An efficient method for multiobjective optimal control and optimal control subject to integral constraints. J. Comp. Math. 28 (2010) 517-551. | MR | Zbl

[18] S. Mardle and S. Pascoe, A review of applications of multiple-criteria decision-making techniques to fisheries. Mar. Resour. Econ. 14 (1998) 41-63. | Zbl

[19] K.M. Miettinen, Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston (1999). | MR | Zbl

[20] Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization. Academic Press, Inc., Orlando (1985). | MR | Zbl

[21] T. Tanino, Sensitivity analysis in multiobjective optimization. J. Optim. Theory Appl. 56 (1988) 479-499. | MR | Zbl

[22] R. Vinter, Optimal Control. Birkauser, Boston (2000). | MR | Zbl

[23] P.L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14 (1974) 319-377. | MR | Zbl