Hamilton–Jacobi–Bellman Equations

  • Adriano Festa
  • Roberto Guglielmi
  • Christopher Hermosilla
  • Athena Picarelli
  • Smita Sahu
  • Achille Sassi
  • Francisco J. Silva
Part of the Lecture Notes in Mathematics book series (LNM, volume 2180)


In this chapter we present recent developments in the theory of Hamilton–Jacobi–Bellman (HJB) equations as well as applications. The intention of this chapter is to exhibit novel methods and techniques introduced few years ago in order to solve long-standing questions in nonlinear optimal control theory of Ordinary Differential Equations (ODEs).


  1. 1.
    Abgrall, R.: Numerical discretization of the first-order Hamilton–Jacobi equation on triangular meshes. Commun. Pure Appl. Math. 49, 1339–1373 (1998)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Abgrall, R.: Construction of simple, stable and convergent high order scheme for steady first order Hamilton-Jacobi equation. SIAM J. Sci. Comput. 31, 2419–2446 (2009)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Achdou, Y., Barles, G., Ishii, H., Litvinov, G.L.: Hamilton–Jacobi Equations: Approximations, Numerical Analysis and Applications. Lecture Notes in Mathematics, vol. 2074. Springer/Fondazione C.I.M.E., Heidelberg/Florence (2013). Lecture Notes from the CIME Summer School held in Cetraro, August 29–September 3, 2011, Edited by Paola Loreti and Nicoletta Anna Tchou, Fondazione CIME/CIME Foundation SubseriesGoogle Scholar
  4. 4.
    Achdou, Y., Camilli, F., Dolcetta, I.C.: Mean field games: numerical methods for the planning problem. SIAM J. Control Optim. 50, 79–109 (2012)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Achdou, Y., Camilli, F., Dolcetta, I.C.: Mean field games: convergence of a finite difference method. SIAM J. Numer. Anal. 51(5), 2585–2612 (2013)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Achdou, Y., Dolcetta, I.C.: Mean field games: numerical methods. SIAM J. Numer. Anal. 48–3, 1136–1162 (2010)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Achdou, Y., Perez, V.: Iterative strategies for solving linearized discrete mean field games systems. Netw. Heterog. Media 7(2), 197–217 (2012)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Achdou, Y., Porretta, A.: Convergence of a finite difference scheme to weak solutions of the system of partial differential equation arising in mean field games. SIAM J. Numer. Anal. 54(1), 161–186 (2016)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Allsopp, T., Mason, A., Philpott, A.: Optimising yacht routes under uncertainty. In: Proceedings of the 2000 Fall National Conference of the Operations Research Society of Japan, vol. 176, p. 183 (2000)Google Scholar
  10. 10.
    Altarovici, A., Bokanowski, O., Zidani, H.: A general Hamilton–Jacobi framework for nonlinear state-constrained control problems. ESAIM Control Optim. Calc. Var. 19(2), 337–357 (2012)MATHCrossRefGoogle Scholar
  11. 11.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lecture notes in Mathematics ETH Zürich, 2nd edn. Birkhäuser, Bassel (2008)Google Scholar
  12. 12.
    Aubert, G., Kornprobst, P.: Partial differential equations and the calculus of variations. In: Mathematical Problems in Image Processing. Applied Mathematical Sciences, vol. 147, 2nd edn. Springer, New York (2006).Google Scholar
  13. 13.
    Aubin, J.P.: Viability solutions to structured Hamilton–Jacobi equations under constraints. SIAM J. Control Optim. 49, 1881–1915 (2011)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Aubin, J.P., Cellina, A.: Differential Inclusions: Set-Valued Maps and Viability Theory. Springer, New York (1984)MATHCrossRefGoogle Scholar
  15. 15.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)MATHGoogle Scholar
  16. 16.
    Aubin, J.P., Frankowska, H.: The viability kernel algorithm for computing value functions of infinite horizon optimal control problems. J. Math. Anal. Appl. 201, 555–576 (1996)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Systems & Control: Foundations & Applications. Birkhäuser, Boston (1997). With appendices by Maurizio Falcone and Pierpaolo SoraviaGoogle Scholar
  18. 18.
    Bardi, M., Falcone, M.: An approximation scheme for the minimum time function. SIAM J. Control Optim. 28(4), 950–965 (1990)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Barles, G.: Solutions de Viscosité des équatiuons de Hamilton–Jacobi. Mathematiques et Applications (Berlin), vol. 17. Springer, Paris (1994)Google Scholar
  20. 20.
    Barles, G., Souganidis, P.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4, 271–283 (1991)MathSciNetMATHGoogle Scholar
  21. 21.
    Barnard, R.C., Wolenski, P.R.: Flow invariance on stratified domains. Set Valued Var. Anal. 21(2), 377–403 (2013)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Barron, E.N.: Viscosity solutions and analysis in l . In: Proceedings the NATO Advanced Study Institute, pp. 1–60 (1999)Google Scholar
  23. 23.
    Barron, E.N., Ishii, H.: The bellman equation for minimizing the maximum cost. Nonlinear Anal. Theor. 13(9), 1067–1090 (1989)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Benamou, J.D., Carlier, G.: Augmented lagrangian methods for transport optimization, mean field games and degenerate elliptic equations. J. Optim. Theory Appl. 167(1), 1–26 (2015)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Bokanowski, O., Cheng, Y., Shu, C.W.: A discontinuous Galerkin scheme for front propagation with obstacle. Numer. Math. 126(2), 1–31 (2013)MathSciNetMATHGoogle Scholar
  26. 26.
    Bokanowski, O., Falcone, M., Ferretti, R., Grüne, L., Kalise, D., Zidani, H.: Value iteration convergence of ε-monotone schemes for stationary Hamilton-Jacobi equations. Discret. Continuous Dyn. Syst. Ser. A 35, 4041–4070 (2015)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Bokanowski, O., Falcone, M., Sahu, S.: An efficient filtered scheme for some first order time-dependent Hamilton–Jacobi equations. SIAM J. Sci. Comput. 38(1), A171–A195 (2016)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    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)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Bokanowski, O., Forcadel, N., Zidani, H.: Deterministic state-constrained optimal control problems without controllability assumptions. ESAIM Control Optim. Calc. Var. 17(04), 995–1015 (2011)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Bokanowski, O., Picarelli, A., Zidani, H.: State constrained stochastic optimal control problems via reachability approach. SIAM J. Control Optim. 54(5), 2568–2593 (2016)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Branicky, M., Borkar, V., Mitter, S.: A unified framework for hybrid control: model and optimal control theory. IEEE Trans. Autom. Control 43(1), 31–45 (1998)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Breuß, M., Cristiani, E., Durou, J.D., Falcone, M., Vogel, O.: Perspective shape from shading: ambiguity analysis and numerical approximations. SIAM J. Imaging Sci. 5(1), 311–342 (2012)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    C., F., Ledyaev, Y., Stern, R., Wolenski, P.: Nonsmooth Analysis and Control Theory, vol. 178. Springer, Berlin (1998)Google Scholar
  34. 34.
    Camilli, F., Falcone, M.: An approximation scheme for the optimal control of diffusion processes. RAIRO Modél. Math. Anal. Numér. 29(1), 97–122 (1995)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Camilli, F., Grüne, L.: Numerical approximation of the maximal solutions for a class of degenerate Hamilton–Jacobi equations. SIAM J. Numer. Anal. 38(5), 1540–1560 (2000)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Camilli, F., Siconolfi, A.: Maximal subsolutions for a class of degenerate Hamilton–Jacobi problems. Indiana Univ. Math. J. 48(3), 1111–1131 (1999)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Camilli, F., Silva, F.J.: A semi-discrete in time approximation for a first order-finite mean field game problem. Netw. Heterog. Media 7(2), 263–277 (2012)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Cannarsa, P., Scarinci, T.: Conjugate times and regularity of the minimum time function with differential inclusions. In: Analysis and Geometry in Control Theory and Its Applications. Springer INdAM Series, vol. 11, pp. 85–110. Springer, Cham (2015)Google Scholar
  39. 39.
    Cannarsa, P., Sinestrari, C.: Semiconcave functions, Hamilton–Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and Their Applications. Birkauser, Boston (2004)MATHGoogle Scholar
  40. 40.
    Capuzzo-Dolcetta, I., Lions, P.L.: Hamilton–Jacobi equations with state constraints. Trans. Am. Math. Soc. 318(2), 643–683 (1990)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Cardaliaguet, P.: Notes on Mean Field Games: From P.-L. Lions’ lectures at Collège de France (2013)Google Scholar
  42. 42.
    Cardaliaguet, P., Quincampoix, M., Saint-Pierre, P.: Optimal times for constrained nonlinear control problems without local controllability. Appl. Math. Optim 36, 21–42 (1997)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Cardaliaguet, P., Quincampoix, M., Saint-Pierre, P.: Numerical schemes for discontinuous value functions of optimal control. Set-Valued Anal. 8, 111–126 (2000)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Carlini, E., Falcone, M., Ferretti, R.: An efficient algorithm for Hamilton-Jacobi equations in high dimension. Comput. Vis. Sci. 7(1), 15–29 (2004)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Carlini, E., Ferretti, R., Russo, G.: A weighted essentially nonoscillatory, large time-step scheme for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 27(3), 1071–1091 (2005)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Carlini, E., Silva, F.J.: On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications, arXiv preprint 1708.02042 (2017)Google Scholar
  47. 47.
    Carlini, E., Silva, F.J.: Semi-lagrangian schemes for mean field game models. In: 2013 IEEE 52nd Annual Conference on Decision and Control (CDC), pp. 3115–3120 (2013)Google Scholar
  48. 48.
    Carlini, E., Silva, F.J.: A fully discrete semi-lagrangian scheme for a first order mean field game problem. SIAM J. Numer. Anal. 52(1), 45–67 (2014)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Carlini, E., Silva, F.J.: A semi-lagrangian scheme for a degenerate second order mean field game system. Discrete. Continuous Dyn. Syst. 35(9), 4269–4292 (2015)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Clarke, F., Stern, R.: Hamilton–Jacobi characterization of the state constrained value. Nonlinear Anal. Theor. 61(5), 725–734 (2005)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Cockburn, B., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems,. J. Comput. Phys. 84, 90–113 (1989)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Crandall, M.G., Lions, P.L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Crandall, M.G., Lions, P.L.: Two approximations of solutions of Hamilton–Jacobi equations. Comput. Methods Appl. Mech. Eng. 195, 1344–1386 (1984)MATHGoogle Scholar
  54. 54.
    Debrabant, K., Jakobsen, E.R.: Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Math. Comput. 82(283), 1433–1462 (2013)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Dharmatti, S., Ramaswamy, M.: Hybrid control systems and viscosity solutions. SIAM J. Control Optim. 44(4), 1259–1288 (2005)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Dolcetta, I.C.: On a discrete approximation of the Hamilton–Jacobi equation of dynamic programming. Appl. Math. Optim. 10, 367–377 (1983)MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Dolcetta, I.C., Ishii, H.: Approximate solutions of the Bellman equation of deterministic control theory. Appl. Math. Optim. 11, 161–181 (1984)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Van den Dries, L., Miller, C.: Geometric categories and o-minimal structures. Duke Math. J. 84(2), 497–540 (1996)MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Durou, J.D., Falcone, M., Sagona, M.: A survey of numerical methods for shape from shading. Comput. Vis. Image Underst. 109(1), 22–43 (2008)CrossRefGoogle Scholar
  60. 60.
    Epstein, C.L., Gage, M.: The curve shortening flow. In: Wave motion: Theory, Modelling, and Computation (Berkeley, Calif., 1986). Mathematical Sciences Research Institute Publications, vol. 7, pp. 15–59. Springer, New York (1987)Google Scholar
  61. 61.
    Falcone, M., Ferretti, R.: Discrete time high-order schemes for viscosity solutions of Hamilton–Jacobi–Bellman equations. Numer. Math. 67(3), 315–344 (1994)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Falcone, M., Ferretti, R.: Convergence analysis for a class of high-order semi-Lagrangian advection schemes. SIAM J. Numer. Anal. 35(3), 909–940 (1998)MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. MOS-SIAM Series on Optimization. SIAM, Philadelphia, PA (2013)MATHCrossRefGoogle Scholar
  64. 64.
    Falcone, M., Giorgi, T., Loreti, P.: Level sets of viscosity solutions: some applications to fronts and rendez-vous problems. SIAM J. Appl. Math. 54, 1335–1354 (1994)MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Ferretti, R.: Convergence of semi-Lagrangian approximations to convex Hamilton–Jacobi equations under (very) large Courant numbers. SIAM J. Numer. Anal. 40(6), 2240–2253 (2002)MathSciNetMATHCrossRefGoogle Scholar
  66. 66.
    Ferretti, R., Zidani, H.: Monotone numerical schemes and feedback construction for hybrid control systems. J. Optim. Theory Appl. 165(2), 507–531 (2015)MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    Festa, A., Falcone, M.: An approximation scheme for an Eikonal equation with discontinuous coefficient. SIAM J. Numer. Anal. 52(1), 236–257 (2014)MathSciNetMATHCrossRefGoogle Scholar
  68. 68.
    Fjordholm, U.S.: High-order accurate entropy stable numerical schemes for hyperbolic conservation laws. Ph.D. Thesis, ETH Zurich Switzerland (2013)Google Scholar
  69. 69.
    Fjordholm, U.S., Mishra, S., Tadmor, E.: Arbitrarily high order accurate entropy stable essentially non-oscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50, 423–444 (2012)MATHCrossRefGoogle Scholar
  70. 70.
    Frankowska, H., Mazzola, M.: Discontinuous solutions of Hamilton–Jacobi–Bellman equation under state constraints. Calc. Var. Partial Differ. Equ. 46(3–4), 725–747 (2013)MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    Frankowska, H., Nguyen, L.: Local regularity of the minimum time function. J. Optim. Theory Appl. 164(1), 68–91 (2015)MathSciNetMATHCrossRefGoogle Scholar
  72. 72.
    Frankowska, H., Plaskacz, S.: Semicontinuous solutions of Hamilton–Jacobi–Bellman equations with degenerate state constraints. J. Math. Anal. Appl. 251(2), 818–838 (2000)MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    Frankowska, H., Vinter, R.B.: Existence of neighboring feasible trajectories: Applications to dynamic programming for state-constrained optimal control problems. J. Optim. Theory Appl. 104(1), 20–40 (2000)MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    Froese, B.D., Oberman, A.M.: Convergent filtered schemes for the Monge-Ampère partial differential equation. SIAM J. Numer. Anal. 51, 423–444 (2013)MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Gomes, D., Saúde, J.: Mean field games models—a brief survey. Dyn. Games Appl. 4(2), 110–154 (2014)MathSciNetMATHCrossRefGoogle Scholar
  76. 76.
    Gottlieb, S., Shu, C.W.: Total variation diminishing Runge-Kutta schemes. Math. Comput. 67(221), 73–85 (1998)MathSciNetMATHCrossRefGoogle Scholar
  77. 77.
    Gozzi, F., Loreti, P.: Regularity of the minimum time function and minimum energy problems: the linear case. SIAM J. Control Optim. 37(4), 1195–1221 (1999)MathSciNetMATHCrossRefGoogle Scholar
  78. 78.
    Guéant, O.: Mean field games equations with quadratic Hamiltonian: a specific approach. Math. Models Methods Appl. Sci. 22(9), 1250,022, 37 (2012)Google Scholar
  79. 79.
    Guéant, O.: New numerical methods for mean field games with quadratic costs. Netw. Heterog. Media 7, 315–336 (2012)MathSciNetMATHCrossRefGoogle Scholar
  80. 80.
    Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. phys. 49, 357–393 (1983)MathSciNetMATHCrossRefGoogle Scholar
  81. 81.
    Harten, A.: On a class of high resolution total-variation finite difference schemes. SIAM J. Numer. Anal. 21, 1–23 (1984)MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    Harten, A., Engquist, B., Osher, S., Chakravarty, S.: Uniformly high order essentially non-oscillatory schemes. J. Comput. phys. 4, 231–303 (1987)MATHCrossRefGoogle Scholar
  83. 83.
    Hermosilla, C., Zidani, H.: Infinite horizon problems on stratifiable state-constraints sets. J. Differ. Equ. 258(4), 1430–1460 (2015)MathSciNetMATHCrossRefGoogle Scholar
  84. 84.
    Horn, B.: Obtaining shape from shading information. In: The Psychology of Computer Vision, pp. 115–155. McGraw-Hill, New York (1975)Google Scholar
  85. 85.
    Ishii, H.: Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations. Indiana Univ. Math. J. 33(5), 721–748 (1984)MathSciNetMATHCrossRefGoogle Scholar
  86. 86.
    Ishii, H., Koike, S.: A new formulation of state constraint problems for first-order PDEs. SIAM J. Control Optim. 34(2), 554–571 (1996)MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    Ishii, H., Ramaswamy, M.: Uniqueness results for a class of Hamilton-Jacobi equations with singular coefficients. Commun. Partial Differ. Equ. 20(11–12), 2187–2213 (1995)MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    Kaloshin, V.: A geometric proof of the existence of whitney stratifications. Mosc. Math. J 5(1), 125–133 (2005)MathSciNetMATHGoogle Scholar
  89. 89.
    Kurganov, A., Tadmor, E.: New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations. J. Comput. Phys. 160, 720–742 (2000)MathSciNetMATHCrossRefGoogle Scholar
  90. 90.
    Kurzhanski, A.B., Mitchell, I.M., Varaiya, P.: Optimization techniques for state-constrained control and obstacle problems. J. Optim. Theory Appl. 128, 499–521 (2006)MathSciNetMATHCrossRefGoogle Scholar
  91. 91.
    Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability under state constraints. SIAM J. Control Optim. 45, 1369–1394 (2006)MathSciNetMATHCrossRefGoogle Scholar
  92. 92.
    Lachapelle, A., Salomon, J., Turinici, G.: Computation of mean field equilibria in economics. Math. Mod. Meth. Appl. Sci. 20(4), 567–588 (2010)MathSciNetMATHCrossRefGoogle Scholar
  93. 93.
    Lasry, J.M., Lions, P.L.: Jeux à champ moyen I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343, 619–625 (2006)MATHCrossRefGoogle Scholar
  94. 94.
    Lasry, J.M., Lions, P.L.: Jeux à champ moyen II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343, 679–684 (2006)MATHCrossRefGoogle Scholar
  95. 95.
    Lasry, J.M., Lions, P.L.: Mean field games. Jpn. J. Math. 2, 229–260 (2007)MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    Leoni, G.: A First Course in Sobolev Spaces, vol. 105. American Mathematical Society, Providence (2009)MATHGoogle Scholar
  97. 97.
    Lepsky, O., Hu, C., Shu, C.W.: Analysis of the discontinuous Galerkin method for Hamilton-Jacobi equations. Appl. Num. Math. 33, 423–434 (2000)MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    Li, F., Shu, C.W.: Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations. Appl. Math. Lett. 18(11), 1204–1209 (2005)MathSciNetMATHCrossRefGoogle Scholar
  99. 99.
    Lin, C.T., Tadmor, E.: L 1-stability and error estimates for approximate Hamilton-Jacobi solutions. Numer. Math. 87(4), 701–735 (2001)MathSciNetMATHCrossRefGoogle Scholar
  100. 100.
    Lions, P.L.: Cours au Collège de France (2007–2008).
  101. 101.
    Lions, P.L., Rouy, E., Tourin, A.: Shape-from-shading, viscosity solutions and edges. Numer. Math. 64(3), 323–353 (1993)MathSciNetMATHCrossRefGoogle Scholar
  102. 102.
    Lions, P.L., Souganidis, P.E.: Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations. Numer. Math. 69(4), 441–470 (1995)MathSciNetMATHCrossRefGoogle Scholar
  103. 103.
    Loreti, P.: Some properties of constrained viscosity solutions of Hamilton–Jacobi–Bellman equations. SIAM J. Control Optim. 25, 1244–1252 (1987)MathSciNetMATHCrossRefGoogle Scholar
  104. 104.
    Loreti, P., Tessitore, E.: Approximation and regularity results on constrained viscosity solutions of Hamilton-Jacobi-Bellman equations. J. Math. Syst. Estimation Control 4, 467–483 (1994)MathSciNetMATHGoogle Scholar
  105. 105.
    Lygeros, J.: On reachability and minimum cost optimal control. Automatica 40, 917–927 (2004)MathSciNetMATHCrossRefGoogle Scholar
  106. 106.
    Lygeros, J., Tomlin, C., Sastry, S.: Controllers for reachability specifications for hybrid systems. Automatica 35, 349–370 (1999)MathSciNetMATHCrossRefGoogle Scholar
  107. 107.
    Margellos, K., Lygeros, J.: Hamilton-Jacobi formulation for reach-avoid differential games. IEEE Trans. Automat Control 56, 1849–1861 (2011)MathSciNetMATHCrossRefGoogle Scholar
  108. 108.
    Motta, M.: On nonlinear optimal control problems with state constraints. SIAM J. Control Optim. 33(5), 1411–1424 (1995)MathSciNetMATHCrossRefGoogle Scholar
  109. 109.
    Motta, M., Rampazzo, F.: Multivalued dynamics on a closed domain with absorbing boundary. Applications to optimal control problems with integral constraints. Nonlinear Anal. 41, 631–647 (2000)MATHGoogle Scholar
  110. 110.
    Nour, C., Stern, R.: The state constrained bilateral minimal time function. Nonlinear Anal. Theor. 69(10), 3549–3558 (2008)MathSciNetMATHCrossRefGoogle Scholar
  111. 111.
    Oberman, A.M., Salvador, T.: Filtered schemes for Hamilton-Jacobi equations: a simple construction of convergent accurate difference schemes. J. Comput. Phys. 284, 367–388 (2015)MathSciNetMATHCrossRefGoogle Scholar
  112. 112.
    Osher, S., Sethian, J.A.: Fronts propagation with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)MathSciNetMATHCrossRefGoogle Scholar
  113. 113.
    Osher, S., Shu, C.W.: High order essentially nonoscillatory schemes for Hamilton–Jacobi equations. SIAM J. Numer. Anal. 4, 907–922 (1991)MathSciNetMATHCrossRefGoogle Scholar
  114. 114.
    Rockafellar, R.T.: Proximal subgradients, marginal values, and augmented lagrangians in nonconvex optimization. Math. Oper. Res. 6(3), 424–436 (1981)MathSciNetMATHCrossRefGoogle Scholar
  115. 115.
    Sahu, S.: High-order filtered scheme for front propagation problems. Bull. Braz. Math. Soc. 47(2), 727–744 (2016)MathSciNetMATHCrossRefGoogle Scholar
  116. 116.
    Sethian, J.A.: Evolution, implementation, and application of level set and fast marching methods for advancing fronts. J. Comput. Phys. 169(2), 503–555 (2001)MathSciNetMATHCrossRefGoogle Scholar
  117. 117.
    Soner, H.: Optimal control with state-space constraint I. SIAM J. Control Optim. 24(3), 552–561 (1986)MathSciNetMATHCrossRefGoogle Scholar
  118. 118.
    Soravia, P.: Estimates of convergence of fully discrete schemes for the Isaacs equation of pursuit-evasion differential games via maximum principle. SIAM J. Control Optim. 36(1), 1–11 (1998)MathSciNetMATHCrossRefGoogle Scholar
  119. 119.
    Stern, R.: Characterization of the state constrained minimal time function. SIAM J. Control Optim. 43(2), 697–707 (2004)MathSciNetMATHCrossRefGoogle Scholar
  120. 120.
    Wolenski, P., Zhuang, Y.: Proximal analysis and the minimal time function. SIAM J. Control Optim. 36(3), 1048–1072 (1998)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Adriano Festa
    • 1
  • Roberto Guglielmi
    • 2
  • Christopher Hermosilla
    • 3
  • Athena Picarelli
    • 4
  • Smita Sahu
    • 5
  • Achille Sassi
    • 6
  • Francisco J. Silva
    • 7
  1. 1.LMI LabINSA RouenSaint-Étienne-du-RouvrayFrance
  2. 2.Dyrecta LabConversanoItaly
  3. 3.Departamento de MatemáticasUniversidad Técnica Federico Santa MaríaValparaísoChile
  4. 4.Mathematical InstituteUniversity of OxfordOxfordUK
  5. 5.Department of Mathematical SciencesDurham UniversityDurhamUK
  6. 6.Applied Mathematics DepartmentENSTA ParisTechPalaiseauFrance
  7. 7.XLIM-DMI, UMR CNRS 7252, Faculté des Sciences et TechniquesUniversité de LimogesLimogesFrance

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