Differential continuation for regular optimal control problems

Optimization Methods and Software

Volumn 27, Issue 2, 2012, Pages 177-196

  Caillau, J B ^a   Cots, O ^a   Gergaud, J ^b  

^a UNIVERSITÉ DE BOURGOGNE   (France)

^b UNIVERSITÉ DE TOULOUSE   (France)

Author keywords

automatic differentiation; conjugate points; differential homotopy; optimal control; second order conditions

Indexed keywords

AUTOMATIC DIFFERENTIATIONS; CONJUGATE POINTS; DIFFERENTIAL HOMOTOPY; OPTIMAL CONTROLS; SECOND-ORDER CONDITIONS;

OPTIMIZATION; SOFTWARE ENGINEERING;

OPTIMAL CONTROL SYSTEMS;

EID: 84859178813     PISSN: 10556788     EISSN: 10294937     Source Type: Journal    
DOI: 10.1080/10556788.2011.593625     Document Type: Article

References (31)

1. A.A. Agrachev and Y.L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, vol. 87, Springer, Berlin, 2004.
2. E.L. Allgower and K. Georg, Introduction to Numerical Continuation Methods, SIAM, Philadelphia, PA, 2003.
3. M.C. Bartholomew-Biggs, L.C.W. Dixon, S.E. Hersom, and Z.A. Maany, The solution of some difficult problems in low-thrust interplanetary trajectory optimization, Optimal Control Appl. Meth. 9 (1998), pp. 229-251.
4. N. Bérend, J.F. Bonnans, M. Haddou, J. Laurent-Varin, and C. Talbot, An interior-point approach to trajectory optimization, J. Guidance Control Dyn. 30(5) (2007), pp. 1228-1238.
5. R. Bertrand and R. Epenoy, New smoothing techniques for solving bang-bang optimal control problems. Numerical results and statistical interpretation, Optimal Control Appl. Meth. 23(4) (2002), pp. 171-197.
6. C. Bischof, A. Carle, P. Kladem, A. Mauer, Adifor 2.0: Automatic differentiation of Fortran 77 programs, IEEE Comput. Sci. Eng. 3 (1996), pp. 18-32.
7. H.G. Bock, Numerical treatment of inverse problems in chemical reaction kinetics, Modell. Chem. Reaction Syst. 18 (1981), pp. 102-125.
8. B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Mathematics and Applications, vol. 40, Springer, Berlin, 2003.
9. B. Bonnard, J.-B. Caillau, and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control, ESAIM Control Optim. Calc. Var. 13(2) (2007), pp. 207-236.
10. B. Bonnard, J.-B. Caillau, and O. Cots, Energy minimization in two-level dissipative quantum control: The integrable case. AIMS Procs, Proceedings of the 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Dresden, May 2010, to appear.
11. B. Bonnard, J.-B. Caillau, and E. Trélat, cotcot: Conditions of order two: conjugate times. Available at apo.enseeiht.fr/cotcot.
12. B. Bonnard, O. Cots, N. Shcherbakova, and D. Sugny, The energy minimization problem for two-level dissipative quantum systems, J. Math. Phys. 51 (2010), 092705.
13. B. Bonnard, N. Shcherbakova, and D. Sugny, The smooth continuation method in optimal control with an application to quantum systems, ESAIM Control Optim. Calc. Var. 17(1) (2010), pp. 267-292.
14. H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1992.
15. J.-B. Caillau and J. Noailles, Continuous optimal control sensitivity analysis with AD, Automatic Differentiation: From Simulation to Optimization, Springer, Berlin, 2002, pp. 109-117. Proceedings of AD'2000, Nice, June 2000.
16. J.-B. Caillau, J. Gergaud, and J. Noailles, 3D geosynchronous transfer of a satellite: Continuation on the thrust, J. Optim. Theory Appl. 118(3) (2003), pp. 541-565.
17. J.-B. Caillau, O. Cots, and J. Gergaud, hampath: path following for Hamiltonian systems. Available at apo.enseeiht.fr/hampath.
18. M.P. do Carmo, Riemannian Geometry, Birkhäuser, Basel, 1992.
19. B.S. Garbow, K.E. Hillstrom, J.J. Moré, and J.L. Nazareth, Minpack Project, Argonne National Laboratories, 1980.
20. C.B. García and W.I. Zangwill, An approach to homotopy and degree theory, Math. Oper. Res. 4(4) (1979), pp. 390-405.
21. J. Gergaud and T. Haberkorn, Homotopy method for minimum consumption orbit transfer problem, ESAIM Control Optim. Calc. Var. 12(2) (2006), pp. 294-310.
22. T. Haberkorn, P. Martinon, and J. Gergaud, Low thrust minimum-fuel orbital transfer: a homotopic approach, AAIA 27(6) (2004), pp. 1046-1060.
23. E. Hairer, S.P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer, Berlin, 1993.
24. L. Hascoët and V. Pascual, TAPENADE 2.1 users's guide, INRIA Technical Report no. 300, 2004, pp. 1-78.
25. K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints, Comput. Optim. Appl. 5(3) (1996), pp. 253-283.
26. P. Martinon and J. Gergaud, Using switching detection and variational equations for the shooting method, Optim. Control Appl. Meth. 28(2) (2007), pp. 95-116.
27. H.J. Oberle and K. Taubert, Existence and multiple solutions of the minimum-fuel orbit transfer problem, J. Optim. Theory Appl. 95(2) (1997), pp. 243-262.
28. A.V. Sarychev, The index of second variation of a control system, Mat. Sb. 41 (1982), pp. 338-401.
29. C.J. Silva and E. Trélat, Smooth regularization of bang-bang optimal control problems, IEEE Trans. Autom. Control 55(11) (2010), pp. 2488-2499.
30. 1-minimization in optimal control and applications to robotics, Optim. Control Appl. Meth. 27 (2006), pp. 301-321.
31. L.T. Watson, M. Sosonkina, R.C. Melville, A.P. Morgan, and H.F. Walker, Algorithm 777: HOMPACK90: A suite of Fortran 90 Codes for Globally Convergent Homotopy Algorithms, ACM Trans. Math. Softw. 23(4) (1997), pp. 514-549.