Optimal Control of First-Order Hamilton–Jacobi Equations with Linearly Bounded Hamiltonian

Applied Mathematics and Optimization

Volumn 70, Issue 2, 2014, Pages 185-224

  Graber, P Jameson ^a  

^a INRIA SACLAY   (France)

Author keywords

Front propagation; Hamilton Jacobi equations; Mean field games; Nonlinear PDE; Optimal control; Viscosity solutions

Indexed keywords

DIFFERENTIAL EQUATIONS; HAMILTONIANS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS;

FRONT PROPAGATION; JACOBI EQUATION; MEAN FIELD GAMES; NON LINEAR PDE; OPTIMAL CONTROLS; VISCOSITY SOLUTIONS;

NONLINEAR EQUATIONS;

EID: 84939414771     PISSN: 00954616     EISSN: 14320606     Source Type: Journal    
DOI: 10.1007/s00245-014-9239-3     Document Type: Article

References (26)

1. Altarovici, A., Bokanowski, O., Zidani, H. et al.: A general Hamilton–Jacobi framework for nonlinear state-constrained control problems. Control Optim. Calculus Var. ESAIM 19(2), 337–357 (2013)
2. Ambrosio, L., Crippa, G.: Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields. Transport Equations and Multi-D Hyperbolic Conservation Laws, pp. 3–57. Springer, Berlin (2008)
3. Ambrosio, L., Dal Maso, G.: A general chain rule for distributional derivatives. Proc. Am. Math. Soc. 108(3), 691–702 (1990)
4. Bressan, A., Burago, M., Friend, A., Jou, J.: Blocking strategies for a fire control problem. Anal. Appl. 06(03), 229–246 (2008)
5. Bardi, M., Capuzzo-Dolcetta, I: Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations, Birkhäuser (1997)
6. Blackwell, D., Dubins, L.E.: An extension of Skorohods almost sure representation theorem. Proc. Am. Math. Soc. 89(4), 691–692 (1983)
7. Bressan, A., De Lellis, C.: Existence of optimal strategies for a fire confinement problem. Commun. Pure Appl. Math. 62(6), 789–830 (2009)
8. Bokanowski, O., Forcadel, N., Zidani, H.: Reachability and minimal times for state constrained nonlinear problems without any controllability assumption. SIAM J. Control Optim. 48(7), 4292–4316 (2010)
9. Bressan, A.: Differential inclusions and the control of forest fires. J. Differ. Equ. 243(2), 179–207 (2007). Special Issue in Honor of Arrigo Cellina and Jim Yorke
10. Bressan, A.: Dynamic blocking problems for a model of fire propagation, advances in applied mathematics, modeling, and computational science. In: Melnik, R., Kotsireas, I.S. (eds.) Fields Institute Communications, vol.66, pp. 11–40. Springer, Berlin (2013)
11. Bressan, A., Wang, T.: The minimum speed for a blocking problem on the half plane. J. Math. Anal. Appl. 356(1), 133–144 (2009)
12. Bressan, A., Wang, T.: Equivalent formulation and numerical analysis of a fire confinement problem. ESAIM Control Optim. Calculus Var. 16, 974–1001 (2010)
13. Cardaliaguet, P: Weak solutions for first order mean field games with local coupling, arXiv, preprint arXiv:1305.7015 (2013)
14. Cardaliaguet, P., Carlier, G., Nazaret, B.: Geodesics for a class of distances in the space of probability measures. Calculus Var. Partial Differ. Equ. 7, 1–26 (2012)
15. Cardaliaguet, P., Silvestre, L.: Hölder continuity to Hamilton–Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side. Commun. Partial Differ. Equ. 37(9), 1668–1688 (2012)
16. De Lellis, C., Robyr, R.: Hamilton Jacobi equations with obstacles. Arch. Ration. Mech. Anal. 200, 1051–1073 (2011)
17. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, vol. 5. CRC Press, Boca Raton (1992)
18. Greene, R.E., Wu, H.: On the subharmonicity and plurisubharmonicity of geodesically convex functions. Indiana Univ. Math. J. 22(7), 641–653 (1973)
19. Greene, R.E., Wu, H.: Integrals of subharmonic functions on manifolds of nonnegative curvature. Invent. Math. 27(4), 265–298 (1974)
20. Ishii, Hitoshi: On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. Funkcial. Ekvac 38(1), 101–120 (1995)
21. Juutinen, P., Lindqvist, P., Manfredi, J.J.: On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J. Math. Anal. 33(3), 699–717 (2001)
22. Lasry, J.M., Lions, P.L.: Jeux à champ moyen. i-le cas stationnaire. Comptes Rendus Math. 343(9), 619–625 (2006)
23. Lasry, J.M., Lions, P.L.: Jeux à champ moyen. ii-horizon fini et contrôle optimal. Comptes Rendus Math. 343(10), 679–684 (2006)
24. Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)
25. Reshetnyak, Y.G.: Weak convergence of completely additive vector functions on a set. Sib. Math. J. 9(6), 1039–1045 (1968)
26. Stampacchia, G.: Le problème de dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, vol. 15, pp. 189–257. Annales de l’institut Fourier, Institut Fourier (1965)