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Semigeodesics and the minimal time function

Published online by Cambridge University Press:  15 December 2005

Chadi Nour
Chadi Nour*
Affiliation:
Computer Science and Mathematics Division, Lebanese American University, Byblos Campus, P.O. Box 36, Byblos, Lebanon; [email protected] This paper was written while the author was an ATER at Institut Girard Desargues, Université Lyon I.
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Abstract

We study the Hamilton-Jacobi equation of the minimal time function in a domain which contains the target set. We generalize the results of Clarke and Nour [J. Convex Anal., 2004], where the target set is taken to be a single point. As an application, we give necessary and sufficient conditions for the existence of solutions to eikonal equations.

Keywords

Minimal time functionHamilton-Jacobi equationsviscosity solutionsminimal trajectorieseikonal equationsmonotonicity of trajectoriesproximal analysisnonsmoothanalysis.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 1 , January 2006 , pp. 120 - 138
Copyright
© EDP Sciences, SMAI, 2006

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References

Alvarez, O., Koike, S. and Nakayama, I., Uniqueness of lower semicontinuous viscosity solutions for the minimum time problem. SIAM J. Control Optim. 38 (2000) 470–481. CrossRef
J.P. Aubin and A. Cellina, Differential inclusions. Springer-Verlag, New York (1984).
M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. With appendices by Maurizio Falcone and Pierpaolo Soravia. Birkhäuser Boston, Inc., Boston, MA (1997).
Barron, E.N. and Jensen, R., Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equations 15 (1990) 1713–1742. CrossRef
Cannarsa, P. and Sinestrari, C., Convexity properties of the minimum time function. Calc. Var. 3 (1995) 273–298. CrossRef
P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations and optimal control problems. Birkhäuser Boston (2004).
Cardaliaguet, P., Quincampoix, M. and Saint-Pierre, P., Optimal times for constrained nonlinear control problems without local controllability. Appl. Math. Optim. 36 (1997) 21–42. CrossRef
Clarke, F.H. and Ledyaev, Yu., Mean value inequalities in Hilbert space. Trans. Amer. Math. Soc. 344 (1994) 307–324. CrossRef
Clarke, F.H., Ledyaev, Yu., Stern, R. and Wolenski, P., Qualitative properties of trajectories of control systems: A survey. J. Dynam. Control Syst. 1 (1995) 1–48. CrossRef
F.H. Clarke, Yu. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory. Graduate Texts Math. 178 (1998). Springer-Verlag, New York.
Clarke, F.H. and Nour, C., The Hamilton-Jacobi equation of minimal time control. J. Convex Anal. 11 (2004) 413–436.
Crandall, M.G., Ishi, H. and Lions, P.L., User's guide to the viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67. CrossRef
Crandall, M.G. and Lions, P.L., Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1–42. CrossRef
W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, New York (1993).
Frankowska, H., Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257–272. CrossRef
C. Nour, The Hamilton-Jacobi equation in optimal control: duality and geodesics. Ph.D. Thesis, Université Claude Bernard Lyon I (2003).
C. Nour, The bilateral minimal time function. J. Convex Anal., to appear.
Soravia, P., Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex Hamiltonians. Comm. Partial Differ. Equ. 18 (1993) 1493–1514. CrossRef
Sussmann, H.J., A general theorem on local controllability. SIAM J. Control Optim. 25 (1987) 158–133. CrossRef
Veliov, V.M., Lipschitz continuity of the value function in optimal control. J. Optim. Theory Appl. 94 (1997) 335–363. CrossRef
R.B. Vinter, Optimal control. Birkhäuser Boston, Inc., Boston, MA (2000).
Wolenski, P. and Zhuang, Y., Proximal analysis and the minimal time function. SIAM J. Control Optim. 36 (1998) 1048–1072. CrossRef