We study the Bellman equation for undiscounted exit time optimal control problems with fully nonlinear lagrangians and fully nonlinear dynamics using the dynamic programming approach. We allow problems whose non-Lipschitz dynamics admit more than one solution trajectory for some choices of open loop controls and initial positions. We prove a uniqueness theorem which characterizes the value functions of these problems as the unique viscosity solutions of the corresponding Bellman equations that satisfy appropriate boundary conditions. We deduce that the value function for Sussmann's Reflected Brachystochrone Problem for an arbitrary singleton target is the unique viscosity solution of the corresponding Bellman equation in the class of functions which are continuous in the plane, null at the target, and bounded below. Our results also apply to degenerate eikonal equations, and to problems whose targets can be unbounded and whose lagrangians vanish for some points in the state space which are outside the target, including Fuller's Example.

Mots clés : viscosity solutions, dynamical systems, reflected brachystochrone problem

@article{COCV_2001__6__415_0, author = {Malisoff, Michael}, title = {Viscosity solutions of the {Bellman} equation for exit time optimal control problems with {non-Lipschitz} dynamics}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {415--441}, publisher = {EDP-Sciences}, volume = {6}, year = {2001}, zbl = {1006.49023}, mrnumber = {1836050}, language = {en}, url = {http://archive.numdam.org/item/COCV_2001__6__415_0/} }

TY - JOUR AU - Malisoff, Michael TI - Viscosity solutions of the Bellman equation for exit time optimal control problems with non-Lipschitz dynamics JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2001 DA - 2001/// SP - 415 EP - 441 VL - 6 PB - EDP-Sciences UR - http://archive.numdam.org/item/COCV_2001__6__415_0/ UR - https://zbmath.org/?q=an%3A1006.49023 UR - https://www.ams.org/mathscinet-getitem?mr=1836050 LA - en ID - COCV_2001__6__415_0 ER -

Malisoff, Michael. Viscosity solutions of the Bellman equation for exit time optimal control problems with non-Lipschitz dynamics. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 415-441. http://archive.numdam.org/item/COCV_2001__6__415_0/

[1] Bounded-from-below solutions of Hamilton-Jacobi equations. Differential Integral Equations 10 (1997) 419-436. | Zbl 0890.35026

,[2] Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997). | Zbl 0890.49011

and ,[3] On the Bellman equation for some unbounded control problems. NODEA Nonlinear Differential Equations Appl. 4 (1997) 276-285. | MR 1485734 | Zbl 0894.49017

and ,[4] 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). | MR 1678284 | Zbl 1018.49027

, and ,[5] Hamilton-Jacobi equations with singular boundary conditions on a free boundary and applications to differential games. Trans. Amer. Math. Soc. 325 (1991) 205-229. | Zbl 0732.35013

and ,[6] Sur les multi-applications mesurables. RAIRO Oper. Res. 1 (1967). | Numdam | MR 223527 | Zbl 0153.08501

,[7] User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1-67. | Zbl 0755.35015

, and ,[8] On the Bellman equation for infinite horizon problems with unbounded cost functional. Appl. Math. Optim. 41 (1999) 171-197. | MR 1731417 | Zbl 0952.49023

,[9] Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993). | MR 1199811 | Zbl 0773.60070

and ,[10] Real Analysis: Modern Techniques and their Applications. J. Wiley and Sons, New York (1984). | MR 767633 | Zbl 0549.28001

,[11] 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).

,[12] Geometric Control Theory. Cambridge University Press (1997). | MR 1425878 | Zbl 0940.93005

,[13] 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.

,[14] Viscosity solutions of the Bellman equation for exit time optimal control problems with vanishing Lagrangians (submitted). | Zbl 1012.49024

,[15] Pursuit-evasion problems and viscosity solutions of Isaacs equations. SIAM J. Control. Optim. 31 (1993) 604-623. | MR 1214756 | Zbl 0786.35018

,[16] Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex Hamiltonians. Comm. Partial Differential Equations 18 (1993) 1493-1514. | Zbl 0788.49028

,[17] Optimality 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. | Zbl 0955.49016

,[18] Optimal control with discontinuous running cost: Eikonal equation and shape from shading, in Proc. 39th IEEE CDC (to appear).

,[19] 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). | MR 1678285 | Zbl 0947.91014

,[20] A general theorem on local controllability. SIAM J. Control Optim. 25 (1987) 158-194. | MR 872457 | Zbl 0629.93012

,[21] 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.

,[22] Geometry and optimal control, in Mathematical Control Theory, edited by J. Baillieul and J.C. Willems. Springer-Verlag, New York (1998) 140-198. | MR 1661472 | Zbl 1067.49500

,[23] Regular synthesis and sufficient conditions for optimality. SISSA Preprint 68/96/M. SIAM J. Control Optim. (to appear). | MR 1788064 | Zbl 0961.93014

and ,[24] Optimal Control of Differential and Functional Equations. Academic Press, New York (1972). | MR 372708 | Zbl 0253.49001

,[25] Theory of Chattering Control. Birkhäuser, Boston (1994). | MR 1279383

and ,