Viscosity Solutions of the Bellman Equation for Exit Time Optimal Control Problems with Non-Lipschitz Dynamics
ESAIM: Control, Optimisation and Calculus of Variations (2010)
- Volume: 6, page 415-441
- ISSN: 1292-8119
Access Full Article
topAbstract
topHow to cite
topReferences
top- O. Alvarez, Bounded-from-below solutions of Hamilton-Jacobi equations. Differential Integral Equations10 (1997) 419-436.
- M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997).
- M. Bardi and F. Da Lio, On the Bellman equation for some unbounded control problems. NODEA Nonlinear Differential Equations Appl.4 (1997) 276-285.
- M. Bardi, M. Falcone and P. Soravia, Numerical methods for pursuit-evasion games and viscosity solutions, in Stochastic and Differential Games: Theory and Numerical Methods, edited by M. Bardi, T.E.S. Raghavan and T. Parthasarathy. Birkhäuser, Boston (1999).
- M. Bardi and 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.
- C. Castaing, Sur les multi-applications mesurables. RAIRO Oper. Res.1 (1967).
- M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc.27 (1992) 1-67.
- F. Da Lio, On the Bellman equation for infinite horizon problems with unbounded cost functional. Appl. Math. Optim.41 (1999) 171-197.
- W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993).
- G.B. Folland, Real Analysis: Modern Techniques and their Applications. J. Wiley and Sons, New York (1984).
- H. Ishii, On representation of solutions of Hamilton-Jacobi equations with convex Hamiltonians, in Recent Topics in Nonlinear PDE II, edited by K. Masuda and M. Mimura. Kinokuniya Company, Tokyo (1985).
- V. Jurdjevic, Geometric Control Theory. Cambridge University Press (1997).
- 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 (1999) 877-881.
- M. Malisoff, Viscosity solutions of the Bellman equation for exit time optimal control problems with vanishing Lagrangians (submitted).
- P. Soravia, Pursuit-evasion problems and viscosity solutions of Isaacs equations. SIAM J. Control. Optim.31 (1993) 604-623.
- P. Soravia, Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex Hamiltonians. Comm. Partial Differential Equations18 (1993) 1493-1514.
- P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations I: Equations of unbounded and degenerate control problems without uniqueness. Adv. Differential Equations4 (1999) 275-296.
- P. Soravia, Optimal control with discontinuous running cost: Eikonal equation and shape from shading, in Proc. 39th IEEE CDC (to appear).
- P. Souganidis, Two-player, zero-sum differential games and viscosity solutions, in Stochastic and Differential Games: Theory and Numerical Methods, edited by M. Bardi, T.E.S. Raghavan and T. Parthasarathy. Birkhäuser, Boston (1999).
- H.J. Sussmann, A general theorem on local controllability. SIAM J. Control Optim.25 (1987) 158-194.
- H. Sussmann, From the Brachystochrone problem to the maximum principle, in Proc. of the 35th IEEE Conference on Decision and Control. IEEE Publications, New York (1996) 1588-1594.
- H.J. Sussmann, Geometry and optimal control, in Mathematical Control Theory, edited by J. Baillieul and J.C. Willems. Springer-Verlag, New York (1998) 140-198.
- H.J. Sussmann and B. Piccoli, Regular synthesis and sufficient conditions for optimality. SISSA Preprint 68/96/M. SIAM J. Control Optim. (to appear).
- J. Warga, Optimal Control of Differential and Functional Equations. Academic Press, New York (1972).
- M.I. Zelikin and V.F. Borisov, Theory of Chattering Control. Birkhäuser, Boston (1994).