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Viscosity Solutions of the Bellman Equation for Exit Time Optimal Control Problems with Non-Lipschitz Dynamics

Published online by Cambridge University Press:  15 August 2002

Michael Malisoff
Michael Malisoff*
Affiliation:
Department of Systems Science and Mathematics, Washington University in Saint Louis, One Brookings Drive, Campus Box 1040, Saint Louis, MO 63130-4899, U.S.A.; [email protected].
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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.

Keywords

Viscosity solutionsdynamical systemsReflected Brachystochrone Problem.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 415 - 441
Copyright
© EDP Sciences, SMAI, 2001

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References

Alvarez, O., Bounded-from-below solutions of Hamilton-Jacobi equations. Differential Integral Equations 10 (1997) 419-436.
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997).
Bardi, M. and Da Lio, F., On the Bellman equation for some unbounded control problems. NODEA Nonlinear Differential Equations Appl. 4 (1997) 276-285. CrossRef
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).
Bardi, M. and Soravia, P., Hamilton-Jacobi equations with singular boundary conditions on a free boundary and applications to differential games. Trans. Amer. Math. Soc. 325 (1991) 205-229. CrossRef
C. Castaing, Sur les multi-applications mesurables. RAIRO Oper. Res. 1 (1967).
Crandall, M.G., Ishii, H. and Lions, P.-L., User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1-67. CrossRef
Da Lio, F., On the Bellman equation for infinite horizon problems with unbounded cost functional. Appl. Math. Optim. 41 (1999) 171-197. CrossRef
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).
Soravia, P., Pursuit-evasion problems and viscosity solutions of Isaacs equations. SIAM J. Control. Optim. 31 (1993) 604-623. CrossRef
Soravia, P., Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex Hamiltonians. Comm. Partial Differential Equations 18 (1993) 1493-1514. CrossRef
Soravia, P., 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.
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).
Sussmann, H.J., A general theorem on local controllability. SIAM J. Control Optim. 25 (1987) 158-194. CrossRef
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).