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.

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.

Classification : 49L25,  93Cxx
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] O. Alvarez, Bounded-from-below solutions of Hamilton-Jacobi equations. Differential Integral Equations 10 (1997) 419-436. | Zbl 0890.35026

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

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

[4] 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). | MR 1678284 | Zbl 1018.49027

[5] 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. | Zbl 0732.35013

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

[7] 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. | Zbl 0755.35015

[8] F. Da Lio, 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] W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993). | MR 1199811 | Zbl 0773.60070

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

[11] 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).

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

[13] 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.

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

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

[16] P. Soravia, 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] 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 Equations 4 (1999) 275-296. | Zbl 0955.49016

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

[19] 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). | MR 1678285 | Zbl 0947.91014

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

[21] 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.

[22] 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. | MR 1661472 | Zbl 1067.49500

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

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

[25] M.I. Zelikin and V.F. Borisov, Theory of Chattering Control. Birkhäuser, Boston (1994). | MR 1279383