A Semigroup Approach to Stochastic Dynamical Boundary Value Problems

  • S. Bonaccorsi
  • G. Ziglio
Part of the IFIP International Federation for Information Processing book series (IFIPAICT, volume 202)


In many physical applications, the evolution of the system is endowed with dynamical boundary conditions, i.e., with boundary operators containing time derivatives. In this paper we discuss a generalization of such systems, where stochastic perturbations affect the way the system evolves in the interior of the domain as well as on the boundary.


Stochastic differential equations boundary noise semigroup theory dynamical boundary conditions 


  1. [1]
    V. Casarino, K.-J. Engel, R. Nagel and G. Nickel, 2003. A semigroup approach to boundary feedback systems. Integral Equations Operator Theory 47(3): 289–306.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    I. Chueshov and B. Schmalfuss, 2004. Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations, 17(7–8): 751–780.MathSciNetGoogle Scholar
  3. [3]
    G. Da Prato and J. Zabczyk, 1992. Stochastic equations in infinite dimensions. Cambridge University Press.Google Scholar
  4. [4]
    G. Da Prato and J. Zabczyk, 1993. Evolution equations with white-noise boundary conditions. Stochastics Stochastics Rep. 42(3–4): 167–182.MATHMathSciNetGoogle Scholar
  5. [5]
    G. Da Prato and J. Zabczyk, 1996. Ergodicity for infinite-dimensional systems. Cambridge University Press.Google Scholar
  6. [6]
    Debussche, A. and Fuhrman, M. and Tessitore, G., Optimal control of a stochastic heat equation with boundary noise and boundary control, preprint 2004.Google Scholar
  7. [7]
    T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Ergodic boundary/point control of stochastic semilinear systems, S1AMJ. Control Optim. 36 (1998), no. 3, 1020–1047 (electronic).MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    K.-J. Engel, 1999. Spectral theory and generator property for one-sided coupled operator matrices. Semigroup Forum 58(2): 267–295.MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    G. Greiner, 1987. Perturbing the boundary conditions of a generator. Houston J. Math. 13: 213–229.MATHMathSciNetGoogle Scholar
  10. [10]
    M. Kumpf and G. Nickel, 2004. Dynamic boundary conditions and boundary control for the one-dimensional heat equation. J. Dynam. Control Systems 10(2): 213–225.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Lions, J.-L. and Magenes, E., 1972. Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York.MATHGoogle Scholar
  12. [12]
    R. Nagel, 1990. The spectrum of unbounded operator matrices with nondiagonal domain. J. Funct. Anal. 89(2): 291–302.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© International Federation for Information Processing 2006

Authors and Affiliations

  • S. Bonaccorsi
    • 1
  • G. Ziglio
    • 1
  1. 1.Department of MathematicsUniversity of TrentoTrentoItaly

Personalised recommendations