On ergodic problem for Hamilton-Jacobi-Isaacs equations

Piernicola Bettiol

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 11, Issue: 4, page 522-541
  • ISSN: 1292-8119

Abstract

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We study the asymptotic behavior of λ v λ as λ → 0 + , where v λ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) λ v λ + H ( x , D v λ ) = 0 , with H ( x , p ) : = min b ∈ B max a ∈ A { - f ( x , a , b ) · p - l ( x , a , b ) } . We discuss the cases in which the state of the system is required to stay in an n-dimensional torus, called periodic boundary conditions, or in the closure of a bounded connected domain Ω ⊂ n with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the boundary) and the case of state constraints boundary conditions. Under the uniform approximate controllability assumption of one player, we extend the uniform convergence result of the value function to a constant as λ → 0 + to differential games. As far as state constraints boundary conditions are concerned, we give an example where the value function is Hölder continuous.

How to cite

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Bettiol, Piernicola. "On ergodic problem for Hamilton-Jacobi-Isaacs equations." ESAIM: Control, Optimisation and Calculus of Variations 11.4 (2010): 522-541. <http://eudml.org/doc/90776>.

@article{Bettiol2010,
abstract = { We study the asymptotic behavior of $\lambda v_\lambda$ as $\lambda\rightarrow 0^+$, where $v_\lambda$ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) \[ \lambda v\_\lambda + H(x,Dv\_\lambda)=0, \] with \[ H(x,p):=\min\_\{b\in B\}\max\_\{a \in A\} \\{-f(x,a,b)\cdot p -l(x,a,b)\\}. \] We discuss the cases in which the state of the system is required to stay in an n-dimensional torus, called periodic boundary conditions, or in the closure of a bounded connected domain $\Omega\subset\{\xR\}^n$ with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the boundary) and the case of state constraints boundary conditions. Under the uniform approximate controllability assumption of one player, we extend the uniform convergence result of the value function to a constant as $\lambda\rightarrow 0^+$ to differential games. As far as state constraints boundary conditions are concerned, we give an example where the value function is Hölder continuous. },
author = {Bettiol, Piernicola},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Hamilton-Jacobi-Isaacs equations; viscosity solutions; asymptotic behavior; differential games; boundary conditions; ergodicity.; ergodicity; Abelian-Tauberian theorem; Skorokhod problem},
language = {eng},
month = {3},
number = {4},
pages = {522-541},
publisher = {EDP Sciences},
title = {On ergodic problem for Hamilton-Jacobi-Isaacs equations},
url = {http://eudml.org/doc/90776},
volume = {11},
year = {2010},
}

TY - JOUR
AU - Bettiol, Piernicola
TI - On ergodic problem for Hamilton-Jacobi-Isaacs equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 4
SP - 522
EP - 541
AB - We study the asymptotic behavior of $\lambda v_\lambda$ as $\lambda\rightarrow 0^+$, where $v_\lambda$ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) \[ \lambda v_\lambda + H(x,Dv_\lambda)=0, \] with \[ H(x,p):=\min_{b\in B}\max_{a \in A} \{-f(x,a,b)\cdot p -l(x,a,b)\}. \] We discuss the cases in which the state of the system is required to stay in an n-dimensional torus, called periodic boundary conditions, or in the closure of a bounded connected domain $\Omega\subset{\xR}^n$ with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the boundary) and the case of state constraints boundary conditions. Under the uniform approximate controllability assumption of one player, we extend the uniform convergence result of the value function to a constant as $\lambda\rightarrow 0^+$ to differential games. As far as state constraints boundary conditions are concerned, we give an example where the value function is Hölder continuous.
LA - eng
KW - Hamilton-Jacobi-Isaacs equations; viscosity solutions; asymptotic behavior; differential games; boundary conditions; ergodicity.; ergodicity; Abelian-Tauberian theorem; Skorokhod problem
UR - http://eudml.org/doc/90776
ER -

References

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  1. O. Alvarez and M. Bardi, A general convergence result for singular perturbations of fully nonlinear degenerate parabolic PDEs. University of Padova, Preprint (2002).  
  2. O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch. Rational Mech. Anal.170 (2003) 17–61.  
  3. M. Arisawa, Ergodic problem for the Hamilton-Jacobi-Bellman equation I. Ann. Inst. H. Poincaré Anal. Non Linéaire14 (1997) 415–438.  
  4. M. Arisawa, Ergodic problem for the Hamilton-Jacobi-Bellman equation II. Ann. Inst. H. Poincaré Anal. Non Linéaire15 (1998) 1–24.  
  5. M. Arisawa and P.L. Lions, Continuity of admissible trajectories for state constraints control problems. Discrete Cont. Dyn. Systems2 (1996) 297–305.  
  6. M. Arisawa and P.L. Lions, On ergodic stochastic control. Commun. Partial Differ. Equations23 (1998) 2187–2217.  
  7. J.P. Aubin and A. Cellina, Differential inclusions. Set-valued maps and viability theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin 264 (1984) XIII+342.  
  8. M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of the Hamilton-Jacobi equations. Birkhäuser, Boston (1997).  
  9. M. Bardi, S. Koike and P. Soravia, Pursuit-evasion game with state constraints: dynamic programming and discrete-time approximations. Discrete Cont. Dyn. Systems6 (2000) 361–380.  
  10. G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. (French) [Viscosity solutions of Hamilton-Jacobi equations.] Mathématiques & Applications [Mathematics & Applications]. Springer-Verlag, Paris 17 (1994) X+194.  
  11. P. Bettiol, Weak Solutions in Hamilton-Jacobi and Control Theory. Ph.D. Thesis University of Padova (2002).  
  12. P. Bettiol, P. Cardaliaguet and M. Quincampoix, Zero-sum state constrained Differential Games: Victory domains and Existence of value function for Bolza Problem. Preprint SISSA/ISAS Ref. 85/2004/M.  
  13. I. Capuzzo-Dolcetta and P.L. Lions, Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc.318 (1990) 643–687.  
  14. P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Pursuit differential games with state constraints. SIAM J. Control Optim.39 (2001) 1615–1632.  
  15. P. Cardaliaguet and S. Plaskacz, Invariant solutions of differential games and Hamilton-Jacobi equations for time-measurable hamiltonians. SIAM J. Control Optim.38 (2000) 1501–1520.  
  16. I.P. Cornfeld, S.V. Fomin and Ya.G. Sinaĭ, Ergodic theory. Springer-Verlag, New York (1982). X+486.  
  17. M.G. Crandall and P.L. Lions, Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre. (French. English summary.) C. R. Acad. Sci. Paris Sér. I Math.292 (1981) 183–186.  
  18. M.G. Crandall and P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.277 (1983) 1–42.  
  19. M.G. Crandall, L.C. Evans and P.L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.282 (1984) 487–502.  
  20. L.C. Evans, Partial differential equations. Graduate Studies in Mathematics, 19 AMS, Rhodeisland (1998).  
  21. L.C. Evans and H. Ishii, Differential games and nonlinear first order PDE on bounded domains. Manuscripta Math.49 (1984) 109–139.  
  22. H. Federer, Curvature measures. Trans. Amer. Math. Soc.93 (1959) 418–491.  
  23. H. Frankowska, S. Plaskacz and T. Rzeżuchowski, Measurable viability theorems and the Hamilton-Jacobi-Bellman equation. J. Differential Equations116 (1995) 265–305.  
  24. H. Frankowska and F. Rampazzo, Filippov's and Filippov-Ważewski's theorems on closed domains. J. Differential Equations161 (2000) 449–478.  
  25. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin (2001). XIV+517.  
  26. H. Ishii, Lecture notes on viscosity solutions. Brown University, Providence, RI (1988).  
  27. S. Koike, On the state constraint problem for differential games. Indiana Univ. Math. J.44 (1995) 467–487.  
  28. P.L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, Mass.-London 69 (1982) IV+317.  
  29. P.L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications. Comm. Partial Differ. Equ.8 (1983) 1101–1174.  
  30. P.L. Lions, Neumann type boundary conditions for Hamilton-Jacobi equations. Duke Math. J.52 (1985), 793–820.  
  31. P.L. Lions and A.S. Sznitman, Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math.37 (1984) 511–537.  
  32. P. Loreti and M.E. Tessitore, Approximation and regularity results on constrained viscosity solutions of Hamilton-Jacobi-Bellman equations. J. Math. Systems Estim. Control4 (1994) 467–483.  
  33. B. Simon, Functional integration and quantum physics.Pure Appl. Math. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London 86 (1979) IX+296.  
  34. M.H. Soner, Optimal control with state-space constraint. I. SIAM J. Control Optim.24 (1986) 552–561.  

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