Discontinuous solutions of Hamilton–Jacobi–Bellman equation under state constraints

  • Hélène FrankowskaEmail author
  • Marco Mazzola


This article is devoted to the Hamilton–Jacobi partial differential equation
$$\left\{\begin{array}{lll}\frac{\partial V}{\partial t} = H\left(t, x, - \frac{\partial V}{\partial x}\right) & \hbox{on} & [0, 1]\times {\overline{\Omega}}\\V(1, x) = g(x) & \hbox{on}& {\overline{\Omega}},\end{array}\right.$$
where the Hamiltonian \({{H:[0, 1] \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}}}\) is convex and positively homogeneous with respect to the last variable, \({{\Omega \subset \mathbb{R}^n}}\) is open and \({{g : \mathbb{R}^n \to \mathbb{R} \cup \{+ \infty\}}}\) is lower semicontinuous. Such Hamiltonians do arise in the optimal control theory. We apply the method of generalized characteristics to show uniqueness of lower semicontinuous solution of this first order PDE. The novelty of our setting lies in the fact that we do not ask regularity of the boundary of Ω and extend the Soner inward pointing condition in a nontraditional way to get uniqueness in the class of lower semicontinuous functions.

Mathematics Subject Classification (2000)

34A60 35D05 35F20 49L25 


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  1. 1.
    Aubin J.-P.: Viability Theory. Birkhäuser, Boston (1991)zbMATHGoogle Scholar
  2. 2.
    Aubin J.-P.: Optima and Equilibria. Springer, Berlin (1993)zbMATHGoogle Scholar
  3. 3.
    Aubin J.-P., Frankowska H.: Set-Valued Analysis. Birkhäuser, Boston (1990)zbMATHGoogle Scholar
  4. 4.
    Bardi M., Capuzzo-Dolcetta I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, Boston (1997)zbMATHCrossRefGoogle Scholar
  5. 5.
    Barles G.: Discontinuous viscosity solutions of first order of Hamilton–Jacobi equations: a guided visit. Nonlinear Anal. TMA. 20, 1123–1134 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Barron E.N., Jensen R.: Semicontinuous viscosity solutions for Hamilton–Jacobi equations with convex Hamiltonian. Comm. Partial Diff. Equ. 15, 1713–1742 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Barron E.N., Jensen R.: Optimal control and semicontinuous viscosity solutions. Proc. Am. Math. Soc. 113, 397–402 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bettiol P., Bressan A., Vinter R.B.: On trajectories satisfying a state onstraint: W 1,1 estimates and counter-examples. SIAM J. Control Optim. 48, 4664–4679 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bettiol P., Frankowska H., Vinter R.B.: L estimates on trajectories confined to a closed subset. J. Diff. Equ. 252, 1912–1933 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cannarsa P., Frankowska H.: Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim. 29, 1322–1347 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Cardaliaguet P., Quincampoix M., Saint-Pierre P.: Optimal times for constrained non-linear control problems without local controllability. Appl. Math. Optim. 36, 21–42 (1997)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Capuzzo-Dolcetta I., Lions P.-L.: Hamilton Jacobi equations with state constraints. Trans. Am. Math. Soc. 318, 643–685 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Castaing, C., Valadier, M. (ed.): Convex Analysis and Measurable Multifunctions. Springer, Berlin (1977)Google Scholar
  14. 14.
    Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)zbMATHGoogle Scholar
  15. 15.
    Crandall M.G., Evans L.C., Lions P.L.: Some properties of viscosity solutions of Hamilton–Jacobi equation. Trans. Amer. Math. Soc. 282, 487–502 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Crandall M.G., Lions P.L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Amer. Math. Soc. 277, 1–42 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Evans L.C., Souganidis P.E.: Differential games and representation formulas for solutions of Hamilton–Jacobi equations. Indiana Univ. Math. J. 33, 773–797 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Filippov A.F.: Classical solutions of differential equations with multivalued right-hand side. SIAM J. Control Optim. 5, 609–621 (1967)zbMATHCrossRefGoogle Scholar
  19. 19.
    Frankowska H.: Optimal trajectories associated to a solution of contingent Hamilton–Jacobi equations. Appl. Math. Optim. 19, 291–311 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Frankowska H.: Lower semicontinuous solutions of Hamilton–Jacobi–Bellman equation. In: Proceedings of IEEE CDC Conference. Brighton, England (1991)Google Scholar
  21. 21.
    Frankowska H.: Lower semicontinuous solutions of Hamilton–Jacobi–Bellman equations. SIAM J. Control Optim. 31, 257–272 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Frankowska H., Plaskacz S., Rzeżuchowski T.: Measurable viability theorems and Hamilton–Jacobi–Bellman equation. J. Diff. Equ. 116, 265–305 (1995)zbMATHCrossRefGoogle Scholar
  23. 23.
    Frankowska, H., Plaskacz, S.: Hamilton–Jacobi equations for infinite horizon control problems with state constraints. In: Proceedings of International Conference “Calculus of Variations and Related Topics”, Haifa, March 25–April 1, 1998 (1999)Google Scholar
  24. 24.
    Frankowska H., Plaskacz S.: Semicontinuous solutions of Hamilton–Jacobi–Bellman equations with degenerate state constraints. JMAA 251, 818–838 (2000)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Frankowska H., Rampazzo F.: Filippov’s and Filippov-Wazewski’s theorems on closed domains. J. Diff. Equ. 161, 449–478 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Frankowska H., Vinter R.B.: Existence of neighbouring feasible trajectories: applications to dynamic programming for state constrained optimal control problems. J. Optim. Theory Appl. 104, 21–40 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Ishii H.: Hamilton–Jacobi equations with discontinuous Hamiltonians on arbitrary open subsets. Bull. Fat. Sci. Eng. Chuo Univ. 28, 33–77 (1985)Google Scholar
  28. 28.
    Ishii H.: Perron’s method for Hamilton–Jacobi equations. Duke Math. J. 55, 369–384 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Ishii H.: Representation of solutions of Hamilton–Jacobi equations. Nonlinear Anal. TMA. 12, 121–146 (1988)zbMATHCrossRefGoogle Scholar
  30. 30.
    Ishii H., Koike S.: A new formulation of state constraint problems for first order PDE’s. SIAM J. Control Optim. 34, 554–571 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Lions P.-L.: Generalized Solutions of Hamilton–Jacobi Equations. Pitman, Boston (1982)zbMATHGoogle Scholar
  32. 32.
    Lions P.-L., Perthame B.: Remarks on Hamilton–Jacobi equations with measurable time-dependent Hamiltonians. Nonlinear Anal. TMA. 11, 613–621 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Lions P.-L., Souganidis P.E.: Differential games, optimal control and directional derivatives of viscosity solutions of Bellman’s and Isaac’s equations. SIAM J. Control Optim. 23, 566–583 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Loreti P., Tessitore M.E.: Approximation and regularity results on constrained viscosity solutions of Hamilton–Jacobi–Bellman equations. J. Math. Syst., Estim. Control 4, 467–483 (1994)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Rampazzo F.: Faithful representations for convex Hamilton–Jacobi equations. SIAM J. Control Optim. 44, 867–884 (2006)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Rockafellar, T., Wets, R.: Variational Analysis. Grundlehren Math. Wiss. 317, Springer, Berlin (1998)Google Scholar
  37. 37.
    Siconolfi A.: Almost continuous solutions of geometric Hamilton–Jacobi equations. Ann. I. H. Poincaré AN. 20, 237–269 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Soner H.M.: Optimal control with state-space constraints. SIAM J. Control Optim. 24, 552–561 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Subbotin A.I.: A generalization of the basic equation of the theory of the differential games. Soviet. Math. Dokl. 22, 358–362 (1980)zbMATHGoogle Scholar

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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.CNRS and Institut de Mathématiques de JussieuUniversité Pierre et Marie CurieParisFrance
  2. 2.ITN Marie Curie Network SADCO at Institut de Mathématiques de JussieuUniversité Pierre et Marie CurieParisFrance

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