Bounded-from-below solutions of the hamilton-Jacobi equation for optimal control problems with exit times: Vanishing lagrangians, eikonal equations, and shape-from-shading

Nonlinear Differential Equations and Applications

Volumn 11, Issue 1, 2004, Pages 95-122

  Malisoff, Michael ^a  

^a LOUISIANA STATE UNIVERSITY   (United States)

Author keywords

Dynamic programming; Exit time problems; Optimal control; Viscosity solutions

Indexed keywords

EID: 1542613204     PISSN: 10219722     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00030-003-1051-8     Document Type: Article

References (37)

1. O. ALVAREZ, Bounded-from-below solutions of Hamilton-Jacobi equations, Differential and Integral Equations 10 (1997), 419-436.
2. Z. ARTSTEIN, Stabilization with relaxed controls, Nonlinear Analysis 7 (1983), 1163-1173.
3. M. BARDI, I. CAPUZZO DOLCETTA. Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston, 1997.
4. M. BARDI, F. DA LIO, On the Bellman equation for some unbounded control problems, NoDEA Nonlinear Differential Equations Appl. 4 (1997), 491-510.
5. M. BARDI, M. FALCONE, P. SORAVIA, Numerical methods for pursuit-evasion games and viscosity solutions, in Stochastic and Differential Games: Theory and Numerical Methods, M. Bardi, T.E.S. Raghavan, and T. Parthasarathy, Eds., Birkhäuser, Boston, 1999.
6. M. BARDI, S. KOIKE, P. SORAVIA, Pursuit-evasion games with state constraints: dynamic programming and discrete time approximations, Discrete Contin. Dynamic Systems 6 (2000), 361-380.
7. M. BARDI, P. SORAVIA, Hamilton-Jacobi equations with singular boundary conditions on a free boundary and applications to differential games, Trans. Amer. Math. Soc. 325 (1991), 205-229.
8. F. CAMILLI, A. SICONOLFI, Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems, Indiana Univ. Math. Journal 48 (1999), 1111-1131.
9. F. CLARKE, Y. LEDYAEV, R. STERN, P. WOLENSKI, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.
10. M. CRANDALL, H. ISHII, P.-L. LIONS, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1-67.
11. F. DA LIO, On the Bellman equation for infinite horizon problems with unbounded cost functional, Appl. Math. Optim. 41 (1999), 171-197.
12. W. DESCH, H. LONGEMANN, E. RYAN, E.D. SONTAG, Meagre functions and asymptotic behavior of dynamical systems, Nonlinear Analysis TMA 44 (2001), 1087-1109.
13. W. FLEMING, H. SONER, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 1993.
14. H. ISHII, On representation of solutions of Hamilton-Jacobi equations with convex Hamiltonians, in Recent Topics in Nonlinear PDE II, K. Masuda and M. Mimura, Eds., Kinokuniya Company, Tokyo, 1985.
15. V. JURDJEVIC, Geometric Control Theory, Cambridge University Press, 1997.
16. P.-L. LIONS, E. ROUY, A. TOURIN, Shape from shading, viscosity solutions and edges, Numer. Math. 64 (1993), 323-353.
17. M. MALISOFF, On the Bellman equation of optimal control problems with exit times and unbounded cost functionals, in Proc. 38th IEEE Conf. on Decision and Control, Phoenix, AZ, December 1999, 23-28.
18. M. MALISOFF, A remark on the Bellman equation for optimal control problems with exit times and noncoercing dynamics, in Proc. 38th IEEE Conf. on Decision and Control, Phoenix, AZ, December 1999, 877-881.
19. M. MALISOFF, Viscosity solutions of the Bellman equation for exit time optimal control problems with vanishing Lagrangians, SIAM J. Control Optim. 40 (2002), 1358-1383.
20. M. MALISOFF, Viscosity solutions of the Bellman equation for exit time optimal control problems with non-Lipschitz dynamics, ESAIM: Control, Optimisation and Calculus of Variations 6 (2001), 415-441.
21. M. MALISOFF, Further results on the Bellman equation for optimal control problems with exit times and nonnegative Lagrangians, Systems and Control Letters 50 (2003), 65-79.
22. M. MALISOFF, H. SUSSMANN, Further results on the Bellman equation for optimal control problems with exit times and nonnegative Lagrangians: The case of Fuller's Problem, in Proc. 39th IEEE Conference on Decision and Control, Sydney, Australia, December 2000, 2308-2310.
23. J. MUNKRES, Topology: A First Course, Prentice-Hall, Englewood Cliffs, NJ, 1975.
24. B. PICCOLI, H. SUSSMANN, Regular synthesis and sufficient conditions for optimality, SIAM J. Control Optim. 39 (2000), 359-410.
25. F. RAMPAZZO, C. SARTORI, Hamilton-Jacobi-Bellman equations with fast gradient dependence, Indiana University Math. J. 49 (2000), 1043-1077.
26. E. ROUY, A. TOURIN, A viscosity solutions approach to shape from shading, SIAM J. Numer. Anal. 29 (1992), 867-884.
27. P. SORAVIA, Pursuit-evasion problems and viscosity solutions of Isaacs equations, SIAM J. Control. Optim. 31 (1993), 604-623.
28. P. SORAVIA, Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Diff. Eq. 18 (1993), 1493-1514.
29. P. SORAVIA, Optimally principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations I: Equations of unbounded and degenerate control problems without uniqueness, Adv. Differential Equations 4 (1999), 275-296.
30. P. SORAVIA, Optimal control with discontinuous running cost: eikonal equation and shape from shading, in Proc. 39th IEEE Conference on Decision and Control, Sydney, Australia, December 2000, 79-84.
31. P. SOUGANIDIS, Two-player, zero-sum differential games and viscosity solutions, in Stochastic and Differential Games: Theory and Numerical Methods, M. Bardi, T.E.S. Raghavan, and T. Parthasarathy, Eds., Birkhäuser, Boston, 1999.
32. H. SUSSMANN, A general theorem on local controllability, SIAM J. Control Optim. 25 (1987), 158-194.
33. H. SUSSMANN, From the Brachystochrone problem to the maximum principle, in Proceedings of the 35th IEEE Conference on Decision and Control, IEEE Publications, New York, 1996, 1588-1594.
34. H. SUSSMANN, Geometry and optimal control, in Mathematical Control Theory, J. Baillieul and J. C. Willems, Eds., Springer-Verlag, New York, 1998, 140-198.
35. H. SUSSMANN, J. WILLEMS, The Brachystrochrone problem and modern control theory, in Contemporary Trends in Nonlinear Geometric Control Theory and its Applications, A. Anzaldo-Meneses, B. Bonnard, J.-P. Gauthier, and F. Monroy-Perez, Eds., World Scientific Publishers, Singapore, 2001.
36. J. WARGA, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.
37. M. ZELIKIN, V. BORISOV, Theory of Chattering Control, with Applications to Astronautics, Robotics, Economics, and Engineering, Birkhäuser, Boston, 1994.