# 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

top## Abstract

top## How to cite

topMalisoff, Michael. "Viscosity Solutions of the Bellman Equation for Exit Time Optimal Control Problems with Non-Lipschitz Dynamics." ESAIM: Control, Optimisation and Calculus of Variations 6 (2010): 415-441. <http://eudml.org/doc/197298>.

@article{Malisoff2010,

abstract = {
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.
},

author = {Malisoff, Michael},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Viscosity solutions; dynamical systems; Reflected Brachystochrone Problem.; viscosity solutions; reflected brachystochrone problem; Bellman equation; exit time optimal control; dynamic programming; eikonal equations},

language = {eng},

month = {3},

pages = {415-441},

publisher = {EDP Sciences},

title = {Viscosity Solutions of the Bellman Equation for Exit Time Optimal Control Problems with Non-Lipschitz Dynamics},

url = {http://eudml.org/doc/197298},

volume = {6},

year = {2010},

}

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

DA - 2010/3//

PB - EDP Sciences

VL - 6

SP - 415

EP - 441

AB -
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.

LA - eng

KW - Viscosity solutions; dynamical systems; Reflected Brachystochrone Problem.; viscosity solutions; reflected brachystochrone problem; Bellman equation; exit time optimal control; dynamic programming; eikonal equations

UR - http://eudml.org/doc/197298

ER -

## References

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).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.