Abstract

We study the solutions of the Hamilton-Jacobi equation that arise in connection with minimal time control, in a new global framework. These solutions, for which we establish existence using the minimal time function as a function of two variables, turn out to be closely related to time-geodesic trajectories.

Author information

Contact details are reproduced from the original publication and may be historical.

F. H. Clarke

Institut Girard Desargues, Université Lyon I, 21 Av. Claude Bernard, 69622 Villeurbanne,
France
and: Institut Universitaire de France

clarke@igd.univ.lyon1.fr

C. Nour

Institut Girard Desargues, Université Lyon I, 21 Av. Claude Bernard, 69622 Villeurbanne,
France

chadi@igd.univ.lyon1.fr

Keywords

  • minimal time function
  • viscosity solutions
  • geodesic trajectories
  • proximal analysis
  • monotonicity of trajectories
  • nonsmooth analysis

References

  1. J. P. Aubin, A. Cellina: Differential Inclusions, Springer, New York (1984).https://doi.org/10.1007/978-3-642-69512-4Google Scholar
  2. M. Bardi, I. Capuzzo-Dolcetta: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston (1997).https://doi.org/10.1007/978-0-8176-4755-1Google Scholar
  3. A. Bressan: On two conjectures by Hájek, Funckcial Ekvac. 23 (1980) 221–227.Google Scholar
  4. P. Cannarsa, C. Sinestrari: Convexity properties of the minimum time function, Calc. Var. 3 (1995) 273–298.https://doi.org/10.1007/BF01189393Google Scholar
  5. P. Cannarsa, C. Sinestrari: Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control Problems, book in preparation.Google Scholar
  6. C. Carathéodory: Calculus of Variations and Partial Differential Equations of the First Order, Teubner, Berlin (1935) (in German); Second (revised) English edition: Chelsea, New York (1982).Google Scholar
  7. F. H. Clarke: Optimization and Nonsmooth Analysis, Wiley-Interscience, New York (1983); Classics in Applied Mathematics 5, SIAM (1990).Google Scholar
  8. F. H. Clarke, Yu. Ledyaev: Mean value inequalities in Hilbert space, Trans. Amer. Math. Soc. 344 (1994) 307–324.https://doi.org/10.1090/S0002-9947-1994-1227093-8Google Scholar
  9. F. H. Clarke, Yu. Ledyaev, R. Stern, P. Wolenski: Qualitative properties of trajectories of control systems: A survey, J. Dynam. Control Systems 1 (1995) 1–48.https://doi.org/10.1007/BF02254655Google Scholar
  10. F. H. Clarke, Yu. Ledyaev, R. Stern, P. Wolenski: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics 178, Springer, New York (1998).Google Scholar
  11. M. G. Crandall, P. L. Lions: Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983) 1–42.https://doi.org/10.1090/S0002-9947-1983-0690039-8Google Scholar
  12. O. Hájek: On the differentiability of the minimal time function, Funckcial Ekvac. 20 (1977) 97–114.https://doi.org/10.1007/BF03284497Google Scholar
  13. P. D. Loewen: Optimal Control Via Nonsmooth Analysis, CRM Proceeding and Lecture Notes 2, AMS, Providence (1993).https://doi.org/10.1090/crmp/002Google Scholar
  14. C. Nour: The Hamilton-Jacobi Equation in Optimal Control: Duality and Geodesics, PhD thesis, Université Claude Bernard Lyon I (2003); Directeur: F. H. Clarke.Google Scholar
  15. N. N. Petrov: On the Bellman function for the time process problem, J. Appl. Math. Mech. 34 (1970) 785–791.https://doi.org/10.1016/0021-8928(70)90060-2Google Scholar
  16. P. Soravia: Hölder continuity of the minimum-time function for Isaacs equations, J. Optim. Theory Appl. 75 (1992) 401–421.https://doi.org/10.1007/BF00941476Google Scholar
  17. P. Soravia: Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Differ. Equations 18 (1993) 1493–1514.https://doi.org/10.1080/03605309308820983Google Scholar
  18. H. J. Sussmann: A general theorem on local controllability, SIAM J. Control Optim. 25 (1987) 158–133.https://doi.org/10.1137/0325011Google Scholar
  19. V. M. Veliov: Lipschitz continuity of the value function in optimal control, Journal Optimization Theory Appl. 94 (1997) 335–363.https://doi.org/10.1023/A:1022683628650Google Scholar
  20. R. B. Vinter: Optimal Control, Birkhäuser, Boston (2000).Google Scholar
  21. P. Wolenski, Y. Zhuang: Proximal analysis and the minimal time function, SIAM J. Control Optim. 36 (1998) 1048–1072.https://doi.org/10.1137/S0363012996299338Google Scholar

Suggested citation

F. H. Clarke, C. Nour. “The Hamilton-Jacobi Equation of Minimal Time Control.” Journal of Convex Analysis 11 (2004), No. 2, 413–436. https://doi.org/10.68381/jca11026

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2004. Rights now held by Banach Press LLC.