Advertisement

Applied Mathematics & Optimization

, Volume 79, Issue 3, pp 547–565 | Cite as

Ergodic Control for Lévy-Driven Linear Stochastic Equations in Hilbert Spaces

  • K. Kadlec
  • B. MaslowskiEmail author
Article
  • 80 Downloads

Abstract

In this paper, controlled linear stochastic evolution equations driven by square integrable Lévy processes are studied in the Hilbert space setting. The control operator may be unbounded which makes the results obtained in the abstract setting applicable to parabolic SPDEs with boundary or point control. The first part contains some preliminary technical results, notably a version of Itô formula which is applicable to weak/mild solutions of controlled equations. In the second part, the ergodic control problem is solved: The feedback form of the optimal control and the formula for the optimal cost are found. As examples, various parabolic type controlled SPDEs are studied.

Keywords

Lévy-driven SPDEs Ergodic control Controlled Lévy driven equations 

Mathematics Subject Classification

60H15 93E20 

Notes

Acknowledgements

The authors are grateful to Szymon Peszat for his valuable comments and remarks. K. Kadlec was supported by the Charles University in Prague, Project GA UK No. 322715 and SVV-2015-260225. B. Maslowski was supported by Grant GACR No. 15-088195.

References

  1. 1.
    Balakrishnan, A.V.: Applied Functional Analysis. Springer, New York (1976)zbMATHGoogle Scholar
  2. 2.
    Chen, G., Triggiani, R.: Proof of extensions of two conjectures on structural damping for elastic systems. Pac. J. Math. 136, 15–55 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    DaPrato, G., Ichikawa, A.: Riccati equations with unbounded coefficients. Ann. Mat. Pura Appl. 140, 209–221 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Adaptive boundary and point control of linear stochastic distributed parameter systems. SIAM J. Control Optim. 32, 648–672 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Adaptive boundary control of linear distributed parameter systems described by analytic semigroups. Appl. Math. Optim. 33, 107–138 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Ergodic control of some stochastic semilinear systems in Hilbert spaces. SIAM J. Control Optim. 36, 1020–1047 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Linear-quadratic control for stochastic equations in a Hilbert space with fractional Brownian motions. SIAM J. Control Optim. 50, 507–531 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Duncan, T. E., Maslowski, B., and Pasik-Duncan, B.: Ergodic control of linear stochastic equations in a Hilbert space with fractional Brownian motions. To appear in Banach Center Publications, vol. 105Google Scholar
  9. 9.
    Duncan, T.E., Goldys, B., Pasik-Duncan, B.: Adaptive control of linear stochastic evolution systems. Stoch. Stoch. Rep. 36, 71–90 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duncan, T.E., Stettner, L., Pasik-Duncan, B.: On ergodic control of stochastic evolution equations. Stoch. Anal. Appl. 15, 723–750 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fuhrman, M., Hu, Y., Tessitore, G.: Stochastic maximum principle for optimal control of SPDEs. C. R. Math. 350, 683–688 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goldys, B., Maslowski, B.: Ergodic control of semilinear stochastic equations and Hamilton–Jacobi equations. J. Math. Anal. Appl. 234, 592–631 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lasiecka, I., Triggiani, R.: Numerical approximations of algebraic Riccati equations modelled by analytic semigroups and applications. Math. Comput. 57(639–662), 513–537 (1991)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lipster, RSh, Shiryayev, A.N.: Theory of Martingales. Kluwer Academic Publishers, Dobrecht (1989)Google Scholar
  15. 15.
    Maslowski, B.: Stability of Semilinear Equations with Boundary and Pointwise Noise, pp. 56–93. Scuola Normale Superiore, Pisa (1995)zbMATHGoogle Scholar
  16. 16.
    Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations Driven by Lévy Processes. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  17. 17.
    Tiefeng, J., Rao, M.B., Wang, X., Deli, L.: Laws of large numbers and moderate deviations for stochastic processes with stationary and independent increments. Stoch. Process. Appl. 44, 205–219 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wang, J.: The Asymptotic Behavior of Locally Square Integrable Martingales. Ann. Probab. 23, 552–585 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles University in PraguePrague 8Czech Republic

Personalised recommendations