Skip to main content
SpringerLink
Log in

A Comparison Principle for Hamilton–Jacobi Equations Related to Controlled Gradient Flows in Infinite Dimensions

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We develop new comparison principles for viscosity solutions of Hamilton–Jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures. Our examples are optimal control of Ginzburg–Landau and Fokker–Planck equations. They arise in limit considerations of externally forced non-equilibrium statistical mechanics models, or through the large deviation principle for interacting particle systems. Our approach is based on two key ingredients: an appropriate choice of geometric structure defining the gradient flow, and a free energy inequality resulting from such gradient flow structure. The approach allows us to handle Hamiltonians with singular state dependency in the nonlinear term, as well as Hamiltonians with a state space which does not satisfy the Radon–Nikodym property. In the case where the state space is a Hilbert space, the method simplifies existing theories by avoiding the perturbed optimization principle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosio L., Gigli N., Savaré G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser Verlag, Basel (2005)

    MATH  Google Scholar 

  2. Barles G., Perthame B.: Discontinuous solutions of deterministic optimal stopping time problems. Math. Methods Num. Anal. 21, 557–579 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barles G., Perthame B.: Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26, 1133–1148 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cannarsa P., Tessitore M.E.: Infinite dimensional Hamilton–Jacobi equations and Dirichlet boundary control problems in parabolic type. SIAM J. Control Optim. 34, 1831–1847 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cordero-Erausquin, D., Gangbo, W., Houdre, C.: Inequalities for generalized entropy and optimal transportation. Recent Advances in the Theory and Applications of Mass Transport, pp. 73–94 Contemporary Mathematics, vol. 353. AMS, Providence, RI, 2004

  6. Crandall M.G., Lions P.-L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. A. M. S. 277, 1–42 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  7. Crandall M.G., Lions P.-L.: Hamilton–Jacobi equations in infinite dimensions, PART I. J. Funct. Anal. 62(3), 379–396 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. Crandall M.G., Lions P.-L.: Hamilton–Jacobi equations in infinite dimensions, PART II. J. Funct. Anal. 65(3), 368–405 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Crandall M.G., Lions P.-L.: Hamilton–Jacobi equations in infinite dimensions, PART III. J. Funct. Anal. 68(2), 214–247 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Crandall M.G., Lions P.-L.: Hamilton–Jacobi equations in infinite dimensions, PART IV. J. Funct. Anal. 90(2), 237–283 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Crandall M.G., Lions P.-L.: Hamilton–Jacobi equations in infinite dimensions, PART V. J. Funct. Anal. 97(2), 417–465 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Crandall M.G., Lions P.-L.: Hamilton–Jacobi equations in infinite dimensions, PART VII. J. Funct. Anal. 125(1), 111–148 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Crandall, M.G., Lions, P.-L.: Hamilton–Jacobi equations in infinite dimensions, PART VI: Nonlinear A and Tataru’s method refined. Evolution equations, control theory, and biomathematics (Han sur Lesse, 1991), pp. 51–89, Lecture Notes in Pure and Appl. Math., vol. 155. Dekker, New York, 1994

  14. Crandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. A.M.S., N.S. 27, 1–67 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dawson D.A., Gärtner J.: Large deviation from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20, 247–308 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dembo A., Zeitouni O.: Large Deviations Techniques Techniques and Applications, 2nd Edition. Springer, New York (1998)

    Book  MATH  Google Scholar 

  17. Ethier S.N., Kurtz T.G.: Markov Processes. Wiley, New York (1986)

    Book  MATH  Google Scholar 

  18. Ekeland I.: Nonconvex minimization problems. Bull. A.M.S. (New Series) I(3), 443–474 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Evans, L.C.: Partial differential equations and Monge-Kantorovich mass transfer. Curr. Develop. in Maths. 1997

  20. Fleming W.H.: Exit probabilities and optimal stochastic control. Appl. Math. Optimiz. 4, 329–346 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fleming, W.H.: A stochastic control approach to some large deviation problems Recent mathematical methods in dynamic programming (Rome 1984) Lecture Notes in Math., vol. 1119, pp 52–66. Springer, Berlin, 1985

  22. Fleming W.H., Soner H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (1991)

    MATH  Google Scholar 

  23. Feng, J., Kurtz, T.G.: Large Deviations for Stochastic Processes. Mathematical Surveys and Monograph, vol. 131. American Mathematical Society, Rhode Island, 2006

  24. Gangbo W., McCann J.R.: The geometry of optimal transportation.. Acta Math. 177, 113–161 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Giacomin G., Lebowitz J.L.: Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys. 87, 37 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Gozzi F., Swiech A.: Hamilton–Jacobi–Bellman equations for the optimal control of the Duncan–Mortensen–Zakai equation. J. Funct. Anal. 172(2), 466–510 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ishii H.: Viscosity solutions for a class of Hamilton–Jacobi equations in Hilbert spaces. J. Funct. Anal. 105(2), 301–341 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jordan R., Kinderlehrer D., Otto F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Langer J.S.: Theory of spinodal decomposition in alloys. Ann. Phys. 65, 53 (1971)

    Article  ADS  Google Scholar 

  30. Lions, P.L.: Some properties of the viscosity semigroups for Hamilton–Jacobi equations. Nonlinear differential equations. Pitman Research Notes in Mathematics, vol. 132 (Eds. J.K. Hale and P. Martinez-Amores), 1988

  31. McCann J.R.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309–323 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. P.D.Es 26(1–2), 101–174 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rockafellar R.T.: Characterization of the subdifferentials of convex functions. Pacific J. Math. 17, 497–510 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sato K.: On the generators of non-negative contraction semi-groups in Banach lattices. J. Math. Soc. Japan 20(3), 423–436 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sinestrari E.: Accretive differential operators.. Boll. U.M.I. 13(B(5), 19–31 (1976)

    MathSciNet  MATH  Google Scholar 

  36. Stegall C.: Optimization of functions on certain subsets of Banach spaces. Math. Ann. 236, 171–176 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  37. Stroock D.: An Introduction to the Theory of Large Deviations. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  38. Tataru D.: Viscosity solutions of Hamilton–Jacobi equations with unbounded nonlinear terms. J. Math. Anal. Appl. 163(2), 345–392 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  39. Villani, C.: Topics in Mass Transportation. Graduate Studies in Mathematics, vol 58. American Mathematical Society, Providence, Rhode Island, 2003

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Jin Feng

  2. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01002, USA

    Markos Katsoulakis

Authors
  1. Jin Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Markos Katsoulakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jin Feng.

Additional information

Communicated by F. Otto

Rights and permissions

Reprints and permissions

About this article

Cite this article

Feng, J., Katsoulakis, M. A Comparison Principle for Hamilton–Jacobi Equations Related to Controlled Gradient Flows in Infinite Dimensions. Arch Rational Mech Anal 192, 275–310 (2009). https://doi.org/10.1007/s00205-008-0133-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-008-0133-5

Keywords

Search

Navigation