Differential continuation for regular optimal control problems

References

• Agrachev , A. A. and Sachkov , Y. L. 2004 . “ Control Theory from the Geometric Viewpoint ” . In Encyclopaedia of Mathematical Sciences , Vol. 87 , Berlin : Springer .
   Google Scholar
• Allgower , E. L. and Georg , K. 2003 . Introduction to Numerical Continuation Methods , Philadelphia , PA : SIAM .
   Google Scholar
• Bartholomew-Biggs , M. C. , Dixon , L. C.W. , Hersom , S. E. and Maany , Z. A. 1998 . The solution of some difficult problems in low-thrust interplanetary trajectory optimization . Optimal Control Appl. Meth , 9 : 229 – 251 .
   Google Scholar
• Bérend , N. , Bonnans , J. F. , Haddou , M. , Laurent-Varin , J. and Talbot , C. 2007 . An interior-point approach to trajectory optimization . J. Guidance Control Dyn. , 30 ( 5 ) : 1228 – 1238 .
   Web of Science ®Google Scholar
• Bertrand , R. and Epenoy , R. 2002 . New smoothing techniques for solving bang-bang optimal control problems. Numerical results and statistical interpretation . Optimal Control Appl. Meth. , 23 ( 4 ) : 171 – 197 .
   Web of Science ®Google Scholar
• Bischof , C. , Carle , A. , Kladem , P. and Mauer , A. 1996 . Adifor 2.0: Automatic differentiation of Fortran 77 programs . IEEE Comput. Sci. Eng. , 3 : 18 – 32 .
   Google Scholar
• Bock , H. G. 1981 . Numerical treatment of inverse problems in chemical reaction kinetics . Modell. Chem. Reaction Syst. , 18 : 102 – 125 .
   Google Scholar
• Bonnard , B. and Chyba , M. 2003 . “ Singular Trajectories and Their Role in Control Theory ” . In Mathematics and Applications , Vol. 40 , Berlin : Springer .
   Google Scholar
• Bonnard , B. , Caillau , J.-B. and Trélat , E. 2007 . Second order optimality conditions in the smooth case and applications in optimal control . ESAIM Control Optim. Calc. Var. , 13 ( 2 ) : 207 – 236 .
   Web of Science ®Google Scholar
• Bonnard , B. , Caillau , J.-B. and Cots , O. 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 . May , Dresden . to appear
   Google Scholar
• B. Bonnard, J.-B. Caillau, and E. Trélat, cotcot: Conditions of order two: conjugate times. Available at apo.enseeiht.fr/cotcot.
   Google Scholar
• Bonnard , B. , Cots , O. , Shcherbakova , N. and Sugny , D. 2010 . The energy minimization problem for two-level dissipative quantum systems . J. Math. Phys. , 51 : 092705
   Web of Science ®Google Scholar
• Bonnard , B. , Shcherbakova , N. and Sugny , D. 2010 . The smooth continuation method in optimal control with an application to quantum systems . ESAIM Control Optim. Calc. Var. , 17 ( 1 ) : 267 – 292 .
   Web of Science ®Google Scholar
• Brezis , H. 1992 . Analyse Fonctionnelle. Théorie et Applications , Paris : Masson .
   Google Scholar
• Caillau , J.-B. and Noailles , J. 2002 . “ Continuous optimal control sensitivity analysis with AD ” . In Automatic Differentiation: From Simulation to Optimization , 109 – 117 . Berlin : Springer . Proceedings of AD’2000, Nice, June 2000
   Google Scholar
• Caillau , J.-B. , Gergaud , J. and Noailles , J. 2003 . 3D geosynchronous transfer of a satellite: Continuation on the thrust . J. Optim. Theory Appl. , 118 ( 3 ) : 541 – 565 .
   Web of Science ®Google Scholar
• J.-B. Caillau, O. Cots, and J. Gergaud, hampath: path following for Hamiltonian systems. Available at apo.enseeiht.fr/hampath.
   Google Scholar
• Do Carmo , M. P. 1992 . Riemannian Geometry , Basel : Birkhäuser .
   Google Scholar
• B.S. Garbow, K.E. Hillstrom, J.J. Moré, and J.L. Nazareth, Minpack Project, Argonne National Laboratories, 1980
   Google Scholar
• García , C. B. and Zangwill , W. I. 1979 . An approach to homotopy and degree theory . Math. Oper. Res. , 4 ( 4 ) : 390 – 405 .
   Google Scholar
• Gergaud , J. and Haberkorn , T. 2006 . Homotopy method for minimum consumption orbit transfer problem . ESAIM Control Optim. Calc. Var. , 12 ( 2 ) : 294 – 310 .
   Web of Science ®Google Scholar
• Haberkorn , T. , Martinon , P. and Gergaud , J. 2004 . Low thrust minimum-fuel orbital transfer: a homotopic approach . AAIA , 27 ( 6 ) : 1046 – 1060 .
   Web of Science ®Google Scholar
• Hairer , E. , Nørsett , S. P. and Wanner , G. 1993 . Solving Ordinary Differential Equations I, Nonstiff Problems , Berlin : Springer .
   Google Scholar
• L. Hascoët and V. Pascual, TAPENADE 2.1 users's guide, INRIA Technical Report no. 300, 2004, pp. 1–78.
   Google Scholar
• Malanowski , K. and Maurer , H. 1996 . Sensitivity analysis for parametric control problems with control-state constraints . Comput. Optim. Appl. , 5 ( 3 ) : 253 – 283 .
   Google Scholar
• Martinon , P. and Gergaud , J. 2007 . Using switching detection and variational equations for the shooting method . Optim. Control Appl. Meth. , 28 ( 2 ) : 95 – 116 .
   Web of Science ®Google Scholar
• Oberle , H. J. and Taubert , K. 1997 . Existence and multiple solutions of the minimum-fuel orbit transfer problem . J. Optim. Theory Appl. , 95 ( 2 ) : 243 – 262 .
   Web of Science ®Google Scholar
• Sarychev , A. V. 1982 . The index of second variation of a control system . Mat. Sb. , 41 : 338 – 401 .
   Google Scholar
• Silva , C. J. and Trélat , E. 2010 . Smooth regularization of bang-bang optimal control problems . IEEE Trans. Autom. Control , 55 ( 11 ) : 2488 – 2499 .
   Web of Science ®Google Scholar
• Vossen , G. and Maurer , H. 2006 . On L1-minimization in optimal control and applications to robotics . Optim. Control Appl. Meth. , 27 : 301 – 321 .
   Web of Science ®Google Scholar
• Watson , L. T. , Sosonkina , M. , Melville , R. C. , Morgan , A. P. and Walker , H. F. 1997 . Algorithm 777: HOMPACK90: A suite of Fortran 90 Codes for Globally Convergent Homotopy Algorithms . ACM Trans. Math. Softw. , 23 ( 4 ) : 514 – 549 .
   Web of Science ®Google Scholar