zbMATH — the first resource for mathematics

Pareto front characterization for multiobjective optimal control problems using Hamilton-Jacobi approach. (English) Zbl 1429.49003

49J20 Existence theories for optimal control problems involving partial differential equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] User’s Guide for the ROC-HJ Solver, http://uma.ensta-paristech.fr/soft/ROC-HJ (2017).
[2] A. Altarovici, O. Bokanowski, and H. Zidani, A general Hamilton-Jacobi framework for non-linear state-constrained control problems, ESAIM Control Optim. Calc. Var., 19 (2013), pp. 337–357, https://doi.org/10.1051/cocv/2012011. · Zbl 1273.35089
[3] M. Assellaou, O. Bokanowski, A. Désilles, and H. Zidani, A Hamilton-Jacobi-Bellman approach for the optimal control of an abort landing problem, in Proceedings of the IEEE 55th Conference CDC, IEEE, Piscataway, NJ, 2016, pp. 3630–3635, https://doi.org/10.1109/CDC.2016.7798815.
[4] M. Assellaou, O. Bokanowski, A. Désilles, and H. Zidani, Value function and optimal trajectories for a maximum running cost control problem with state constraints. Application to an abort landing problem, ESAIM Math. Model. Numer. Anal., 52 (2018), pp. 305–335, https://doi.org/10.1051/m2an/2017064. · Zbl 1397.49038
[5] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1997. · Zbl 0890.49011
[6] S. Bellaassali and A. Jourani, Necessary optimality conditions in multiobjective dynamic optimization, SIAM J. Control Optim., 42 (2004), pp. 2043–2061. · Zbl 1071.49020
[7] O. Bokanowski, N. Forcadel, and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM J. Control Optim., 48 (2010), pp. 4292–4316, https://doi.org/10.1137/090762075. · Zbl 1214.49025
[8] H. Bonnel and C. Kaya, Optimization over the efficient set of multi-objective convex optimal control problems, J. Optim. Theory Appl., 147 (2010), pp. 93–112. · Zbl 1205.49041
[9] F. H. Clarke, Optimization and Nonsmooth Analysis, Classics Appl. Math. 5, SIAM, Philadelphia, 1990. · Zbl 0696.49002
[10] F. H. Clarke, L. Rifford, and R. Stern, Feedback in state constrained optimal control, ESAIM Control Optim. Calc. Var., 7 (2002), pp. 97–133. · Zbl 1033.49004
[11] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), pp. 1–42. · Zbl 0599.35024
[12] V. de Oliveira and Silva, On sufficient optimality conditions for multiobjective control problems, J. Global Optim., 64 (2016), pp. 721–744. · Zbl 1337.49032
[13] V. de Oliveira, G. Silva, and M. Rojas-Medar, A class of multi-objective control problems, Optimal Control Appl. Methods, 30 (2009), pp. 77–86.
[14] M. Falcone, Numerical solution of dynamic programming equations, in Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston, 1997, pp. 471–504.
[15] M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, Other Titles Appl. Math. 133, SIAM, Philadelphia, 2013.
[16] H. Frankowska and R. B. Vinter, Existence of neighboring feasible trajectories: Applications to dynamics programming for state-constrained optimal control problems, J. Optim. Theory Appl., 104 (2000), pp. 20–40. · Zbl 1050.49022
[17] A. Göpfert, H. Riahi, C. Tammer, and C. Zalinescu, Variational Methods in Partially Ordered Spaces, CMS Books Math./Ouvrages Math. SMC, Springer, New York, 2003.
[18] A. Guigue, Set-valued return function and generalized solutions for multiobjective optimal control problems (MOC), SIAM J. Control Optim., 51 (2013), pp. 2379–2405. · Zbl 1273.49022
[19] Guigue, A., Approximation of the Pareto optimal set for multiobjective optimal control problems using viability kernels, ESAIM Control Optim. Calc. Var., 20 (2014), pp. 95–115, https://doi.org/10.1051/cocv/2013056. · Zbl 1307.90159
[20] N. Hayek, Infinite horizon multiobjective optimal control problems in the discrete time case, Optimization, 60 (2011), pp. 509–529, https://doi.org/10.1080/02331930903480352. · Zbl 1251.90353
[21] C. Hermosilla, P. Wolenski, and H. Zidani, The Mayer and minimum time problems with stratified state constraints, Set-Valued Var. Anal., 26 (2017), pp. 643–662, https://doi.org/10.1007/s11228-017-0413-z. · Zbl 1404.35469
[22] C. Hermosilla and H. Zidani, Infinite horizon problems on stratifiable state-constraints sets, J. Differential Equations, 258 (2015), pp. 1420–1460, https://doi.org/10.1016/j.jde.2014.11.001. · Zbl 1319.49041
[23] J. Jahn, Vector Optimization: Theory, Applications and Extensions, Springer, Berlin, 2004. · Zbl 1055.90065
[24] C. Y. Kaya and H. Maurer, A numerical method for nonconvex multi-objective optimal control problems, Comput. Optim. Appl., 57 (2014), pp. 685–702. · Zbl 1301.49082
[25] B. T. Kien, N.-C. Wong, and J.-C. Yao, Necessary conditions for multiobjective optimal control problems with free end-time, SIAM J. Control Optim., 47 (2008), pp. 2251–2274. · Zbl 1171.49021
[26] A. Kumar and A. Vladimirsky, An efficient method for multiobjective optimal control and optimal control subject to integral constraints, J. Comput. Math., 28 (2010), pp. 517–551, http://www.jstor.org/stable/43693920. · Zbl 1240.90345
[27] J. Lin, Maximal vectors and multi-objective optimization, J. Optim. Theory Appl., 18 (1976), pp. 41–64. · Zbl 0298.90056
[28] F. Logist, B. Houska, M. Diehl, and J. van Impe, Fast Pareto set generation for nonlinear optimal control problems with multiple objectives, Struct. Multidiscip. Optim., 42 (2010), pp. 591–603. · Zbl 1274.90530
[29] F. Logist, M. Vallerio, B. Houska, M. Diehl, and J. van Impe, Multi-objective optimal control of chemical processes using Acado toolkit, J. Comput. Chem. Engrg., 37 (2012), pp. 191–199.
[30] R. Marler and J. Arora, Survey of multi-objective optimization methods for engineering, Struct. Multidiscip. Optim., 26 (2004), pp. 369–395. · Zbl 1243.90199
[31] K. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic, Norwell, MA, 1999. · Zbl 0949.90082
[32] M. Motta, On nonlinear optimal control problems with state constraints, SIAM J. Control Optim., 33 (1995), pp. 1411–1424. · Zbl 0861.49018
[33] M. Motta and F. Rampazzo, Multivalued dynamics on a closed domain with absorbing boundary. Applications to optimal control problems with integral constraints, Nonlinear Anal., 41 (2000), pp. 631–647. · Zbl 0961.34003
[34] T.-N. Ngo and N. Hayek, Necessary conditions of Pareto optimality for multiobjective optimal control problems under constraints, Optimization, 66 (2017), pp. 149–177. · Zbl 1367.49012
[35] S. Osher and C.-W. Shu, Highorder essentially nonoscillatory schemes for Hamilton–Jacobi equations, SIAM J. Numer. Anal., 28 (1991), pp. 907–922. · Zbl 0736.65066
[36] V. Pareto, Cours d’Economie Politique, Rouge, Lausanni, Switzerland, 1896.
[37] H. M. Soner, Optimal control with state-space constraint I, SIAM J. Control Optim., 24 (1986), pp. 552–561, https://doi.org/10.1137/0324032. · Zbl 0597.49023
[38] H. M. Soner, Optimal control with state-space constraint. II, SIAM J. Control Optim., 24 (1986), pp. 1110–1122, https://doi.org/10.1137/0324067. · Zbl 0619.49013
[39] W. Stadler, Initiators of multicriteria optimization, in Recent Advances and Historical Development of Vector Optimization, W. Jahn, J; Krabs, ed., Lecture Notes in Econom. and Math. Systems 294, 1987, pp. 3–25.
[40] W. Stadler, Fundamentals of multicriteria optimization, in Multicriteria Optimization in Engineering and in the Sciences, W. Stadler, ed., Plenum Press, New York, 1988, pp. 1–25. · Zbl 0669.00028
[41] R. Takei, W. Chen, Z. Clawson, S. Kirov, and A. Vladimirsky, Optimal control with budget constraints and resets, SIAM J. Control Optim., 53 (2015), pp. 712–744, https://doi.org/10.1137/110853182. · Zbl 1322.90091
[42] R. Vinter, Optimal Control, Birkhäuser, Boston, 2000. · Zbl 0952.49001
[43] A. P. Wierzbicki, A mathematical basis for satisficing decision making, Math. Model., 3 (1982), pp. 391–405. · Zbl 0521.90097
[44] P. L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl., 14 (1974), pp. 319–377. · Zbl 0268.90057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.