Semigeodesics and the minimal time function
ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 1, pp. 120-138.

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.

DOI: 10.1051/cocv:2005032
Classification: 49J52, 49L20, 49L25
Keywords: minimal time function, Hamilton-Jacobi equations, viscosity solutions, minimal trajectories, eikonal equations, monotonicity of trajectories, proximal analysis, nonsmooth analysis
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     title = {Semigeodesics and the minimal time function},
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Nour, Chadi. Semigeodesics and the minimal time function. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 1, pp. 120-138. doi : 10.1051/cocv:2005032. http://www.numdam.org/articles/10.1051/cocv:2005032/

[1] O. Alvarez, S. Koike and I. Nakayama, Uniqueness of lower semicontinuous viscosity solutions for the minimum time problem. SIAM J. Control Optim. 38 (2000) 470-481. | MR | Zbl

[2] J.P. Aubin and A. Cellina, Differential inclusions. Springer-Verlag, New York (1984). | MR | Zbl

[3] 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). | MR | Zbl

[4] E.N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equations 15 (1990) 1713-1742. | MR | Zbl

[5] P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function. Calc. Var. 3 (1995) 273-298. | MR | Zbl

[6] P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations and optimal control problems. Birkhäuser Boston (2004). | MR | Zbl

[7] P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Optimal times for constrained nonlinear control problems without local controllability. Appl. Math. Optim. 36 (1997) 21-42. | MR | Zbl

[8] F.H. Clarke and Yu. Ledyaev, Mean value inequalities in Hilbert space. Trans. Amer. Math. Soc. 344 (1994) 307-324. | MR | Zbl

[9] F.H. Clarke, Yu. Ledyaev, R. Stern and P. Wolenski, Qualitative properties of trajectories of control systems: A survey. J. Dynam. Control Syst. 1 (1995) 1-48. | MR | Zbl

[10] F.H. Clarke, Yu. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory. Graduate Texts Math. 178 (1998). Springer-Verlag, New York. | MR | Zbl

[11] F.H. Clarke and C. Nour, The Hamilton-Jacobi equation of minimal time control. J. Convex Anal. 11 (2004) 413-436. | MR | Zbl

[12] M.G. Crandall, H. Ishi and P.L. Lions, User's guide to the viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1-67. | MR | Zbl

[13] M.G. Crandall and P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1-42. | MR | Zbl

[14] W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, New York (1993). | MR | Zbl

[15] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257-272. | MR | Zbl

[16] C. Nour, The Hamilton-Jacobi equation in optimal control: duality and geodesics. Ph.D. Thesis, Université Claude Bernard Lyon I (2003).

[17] C. Nour, The bilateral minimal time function. J. Convex Anal., to appear. | MR | Zbl

[18] P. Soravia, Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex Hamiltonians. Comm. Partial Differ. Equ. 18 (1993) 1493-1514. | MR | Zbl

[19] H.J. Sussmann, A general theorem on local controllability. SIAM J. Control Optim. 25 (1987) 158-133. | MR | Zbl

[20] V.M. Veliov, Lipschitz continuity of the value function in optimal control. J. Optim. Theory Appl. 94 (1997) 335-363. | MR | Zbl

[21] R.B. Vinter, Optimal control. Birkhäuser Boston, Inc., Boston, MA (2000). | MR | Zbl

[22] P. Wolenski and Y. Zhuang, Proximal analysis and the minimal time function. SIAM J. Control Optim. 36 (1998) 1048-1072. | MR | Zbl

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