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Dynamic programming for the stochastic burgers equation

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Abstract

We solve a control problem for the stochastic Burgers equation using the dynamic programming approach. The cost functional involves exponentially growing functions and the analog of the kinetic energy; the case of a distributed parameter control is considered. The Hamilton-Jacobi equation is solved by a compactness method and a-priori estimates are obtained thanks to the regularizing properties of the transition semigroup associated to the stochastic Burgers equation; a fixed point argument does not seem to apply here.

References

  1. [1]

    J. M. Bismut,Large deviations and the Malliavin Calculus, Birkhäuser, Basel, Boston, Berlin, 1984.

  2. [2]

    P. CannarsaG. Da Prato,Some results on nonlinear optimal control problems and Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal.,90 (1990), pp. 27–47.

  3. [3]

    P. Cannarsa—G. Da Prato,Direct solution of a second order Hamilton-Jacobi equation in Hilbert spaces, in: Stochastic partial differential equations and applications (G. Da Prato—L. Tubaro Eds.), pp. 72–85, Pitman Research Notes in Mathematics Series n. 268, 1992.

  4. [4]

    D. H. ChambersR. J. AdrianP. MoinD. S. StewartH. J. Sung,Karhunuen-Loeve expansion of Burgers model of turbulence, Phys. Fluids (31), p. 2573, 1988.

  5. [5]

    H. ChoiR. TemanP. MoinJ. Kim,Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. Fluid Mech.,253 (1993), pp. 509–543.

  6. [6]

    G. Da Prato,Some results on Bellman equation in Hilbert spaces, SIAM J. Control and Optimization,23, 1 (1985), pp. 61-71.

  7. [7]

    G. Da PratoA. Debussche,Control of the stochastic Burgers model of turbulence, SIAM J. Control Optimiz,37, No. 4 (1999), pp. 1123–1149.

  8. [8]

    G. Da PratoA. Debussche,Differentiability of the transition semigroup of stochastic Burgers equation, Rend. Acc. Naz. Lincei, s. 9, v.9 (1998), pp. 267–277.

  9. [9]

    G. Da Prato—A. Debussche—R. Teman,Stochastic Burgers equation, NoDEA (1994), pp. 389–402.

  10. [10]

    G. Da Prato—J. Zabczyk,Stochastic Evolution Equations in Infinite Dimensions, Cambridge University Press, 1992.

  11. [11]

    K. D. Elworthy,Stochastic flows on Riemannian manifolds, in: Diffusion processes and related problems in analysis, Vol. II (M. A. Pinsky and V. Wihstutz, eds.), pp. 33–72, Birkhäuser, 1992.

  12. [12]

    F. Gozzi,Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem, Commun. in partial differential equations,20 (5–6) (1995), pp. 775–826.

  13. [13]

    F. Gozzi,Global Regular Solutions of Second Order Hamilton-Jacobi Equations in Hilbert spaces with locally Lipschitz nonlineartities, J. Math. Anal. Appl.,198 (1996), pp. 399–443.

  14. [14]

    F. GozziE. Rouy,Regular solutions of second order stationary Hamilton-Jacobi equations, J. Differential Equations,130 (1996), pp. 201–234.

  15. [15]

    F. Gozzi—E. Rouy—A. Swiech,Second order Hamilton-Jacobi equations in Hilbert spaces and stochastic boundary control, SIAM J. Control and Optimis, to appear.

  16. [16]

    P. L. Lions,Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions, Part I:The case of bounded stochastic evolution Acta Math.,161 (1988), pp. 243–278; Part II:Optimal control of Zakai's equations, in: Stochastic partial differential equations and applications (G. Da Prato—L. Tubaro eds.) Lecture Notes in Mathematics No. 1390, Springer-Verlag, pp. 147–170, 1990; Part III:Uniqueness of viscosity solutions for general second order equations, J. Funct. Anal.86 (1991), pp. 1–18.

  17. [17]

    A. Swiech,Viscosity solutions of fully nonlinear partial differential equations with «unbounded» terms in infinite dimensions, Ph. D. Thesis, University of California at Santa Barbara, 1993.

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Entrata in Redazione il 10 dicembre 1998.

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Da Prato, G., Debussche, A. Dynamic programming for the stochastic burgers equation. Annali di Matematica pura ed applicata 178, 143–174 (2000). https://doi.org/10.1007/BF02505893

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Keywords

  • Control Problem
  • Dynamic Programming
  • Mild Solution
  • Burger Equation
  • Galerkin Approximation