Optimal control of a stochastic heat equation with boundary-noise and boundary-control

Arnaud Debussche; Marco Fuhrman; Gianmario Tessitore

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 1, page 178-205
  • ISSN: 1292-8119

Abstract

top
We are concerned with the optimal control of a nonlinear stochastic heat equation on a bounded real interval with Neumann boundary conditions. The specificity here is that both the control and the noise act on the boundary. We start by reformulating the state equation as an infinite dimensional stochastic evolution equation. The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The C1 regularity of such a solution is then used to construct the optimal feedback for the control problem. In order to overcome the difficulties arising from the degeneracy of the second order operator and from the presence of unbounded terms we study the HJB equation by introducing a suitable forward-backward system of stochastic differential equations as in the appraoch proposed in [Fuhrman and Tessitore, Ann. Probab.30 (2002) 1397-1465; Pardoux and Peng, Lect. Notes Control Inf. Sci.176 (1992) 200-217] for finite dimensional and infinite dimensional semilinear parabolic equations respectively.

How to cite

top

Debussche, Arnaud, Fuhrman, Marco, and Tessitore, Gianmario. "Optimal control of a stochastic heat equation with boundary-noise and boundary-control." ESAIM: Control, Optimisation and Calculus of Variations 13.1 (2007): 178-205. <http://eudml.org/doc/249994>.

@article{Debussche2007,
abstract = { We are concerned with the optimal control of a nonlinear stochastic heat equation on a bounded real interval with Neumann boundary conditions. The specificity here is that both the control and the noise act on the boundary. We start by reformulating the state equation as an infinite dimensional stochastic evolution equation. The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The C1 regularity of such a solution is then used to construct the optimal feedback for the control problem. In order to overcome the difficulties arising from the degeneracy of the second order operator and from the presence of unbounded terms we study the HJB equation by introducing a suitable forward-backward system of stochastic differential equations as in the appraoch proposed in [Fuhrman and Tessitore, Ann. Probab.30 (2002) 1397-1465; Pardoux and Peng, Lect. Notes Control Inf. Sci.176 (1992) 200-217] for finite dimensional and infinite dimensional semilinear parabolic equations respectively. },
author = {Debussche, Arnaud, Fuhrman, Marco, Tessitore, Gianmario},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Boundary noise; optimal boundary control; HJB equation; backward stochastic differential equations.; stochastic heat equation; Neuman boundary condition; boundary control; Hamilton-Jacobi-Bellman equation; forward-backward stochastic differential equation},
language = {eng},
month = {2},
number = {1},
pages = {178-205},
publisher = {EDP Sciences},
title = {Optimal control of a stochastic heat equation with boundary-noise and boundary-control},
url = {http://eudml.org/doc/249994},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Debussche, Arnaud
AU - Fuhrman, Marco
AU - Tessitore, Gianmario
TI - Optimal control of a stochastic heat equation with boundary-noise and boundary-control
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/2//
PB - EDP Sciences
VL - 13
IS - 1
SP - 178
EP - 205
AB - We are concerned with the optimal control of a nonlinear stochastic heat equation on a bounded real interval with Neumann boundary conditions. The specificity here is that both the control and the noise act on the boundary. We start by reformulating the state equation as an infinite dimensional stochastic evolution equation. The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The C1 regularity of such a solution is then used to construct the optimal feedback for the control problem. In order to overcome the difficulties arising from the degeneracy of the second order operator and from the presence of unbounded terms we study the HJB equation by introducing a suitable forward-backward system of stochastic differential equations as in the appraoch proposed in [Fuhrman and Tessitore, Ann. Probab.30 (2002) 1397-1465; Pardoux and Peng, Lect. Notes Control Inf. Sci.176 (1992) 200-217] for finite dimensional and infinite dimensional semilinear parabolic equations respectively.
LA - eng
KW - Boundary noise; optimal boundary control; HJB equation; backward stochastic differential equations.; stochastic heat equation; Neuman boundary condition; boundary control; Hamilton-Jacobi-Bellman equation; forward-backward stochastic differential equation
UR - http://eudml.org/doc/249994
ER -

References

top
  1. P. Albano and P. Cannarsa, Lectures on carleman estimates for elliptic and parabolic operators with applications. Preprint, Università di Roma Tor Vergata.  Zbl0916.49020
  2. S. Albeverio and Y.A. Rozanov, On stochastic boundary conditions for stochastic evolution equations. Teor. Veroyatnost. i Primenen.38 (1993) 3–19.  Zbl0802.60055
  3. E. Alòs and S. Bonaccorsi, Stochastic partial differential equations with Dirichlet white-noise boundary conditions. Ann. Inst. H. Poincaré Probab. Statist.38 (2002) 125–154.  Zbl0998.60065
  4. E. Alòs and S. Bonaccorsi, Stability for stochastic partial differential equations with Dirichlet white-noise boundary conditions. Infin. Dimens. Anal. Quantum Probab. Relat. Top.5 (2002) 465–481.  Zbl1052.60046
  5. J.P. Aubin and H. Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications2. Birkhäuser Boston Inc., Boston, MA (1990).  Zbl0713.49021
  6. A.V. Balakrishnan, Applied functional analysis, Applications of Mathematics3. Springer-Verlag, New York (1976).  Zbl0333.93051
  7. A. Chojnowska-Michalik, A semigroup approach to boundary problems for stochastic hyperbolic systems. Preprint (1978).  Zbl0407.60070
  8. G. Da Prato and J. Zabczyk, Evolution equations with white-noise boundary conditions. Stoch. Stoch. Rep.42 (1993) 167–182.  Zbl0814.60055
  9. G. Da Prato and J. Zabczyk, Ergodicity for infinite-dimensional systems. London Math. Soc. Lect. Notes Ser.229, Cambridge University Press (1996).  Zbl0849.60052
  10. T.E. Duncan, B. Maslowski and B. Pasik-Duncan, Ergodic boundary/point control of stochastic semilinear systems. SIAM J. Control Optim.36 (1998) 1020–1047.  Zbl0924.93045
  11. N. El Karoui, S. Peng and M.C. Quenez, Backward stochastic differential equations in finance. Math. Finance7 (1997) 1–71.  Zbl0884.90035
  12. H.O. Fattorini, Boundary control systems. SIAM J. Control6 (1968) 349–385.  Zbl0164.10902
  13. W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. Appl. Math.25, Springer-Verlag, New York (1993).  Zbl0773.60070
  14. M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control. Ann. Probab.30 (2002) 1397–1465.  Zbl1017.60076
  15. M. Fuhrman and G. Tessitore, Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces. Ann. Probab.32 (2004) 607–660.  Zbl1046.60061
  16. A.V. Fursikov and O.Y. Imanuvilov, Controllability of Evolution Equations. Lect. Notes Ser.34, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul (1996).  Zbl0862.49004
  17. F. Gozzi, Regularity of solutions of second order Hamilton-Jacobi equations and application to a control problem. Comm. Part. Diff. Eq.20 (1995) 775–826.  Zbl0842.49021
  18. F. Gozzi, Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities. J. Math. Anal. Appl.198 (1996) 399–443.  Zbl0858.35129
  19. F. Gozzi, E. Rouy and A. Święch, Second order Hamilton-Jacobi equations in Hilbert spaces and stochastic boundary control. SIAM J. Control Optim.38 (2000) 400–430.  Zbl0994.49019
  20. A. Grorud and E. Pardoux, Intégrales Hilbertiennes anticipantes par rapport à un processus de Wiener cylindrique et calcul stochastique associé. Appl. Math. Optim.25 (1992) 31–49.  
  21. A. Ichikawa, Stability of parabolic equations with boundary and pointwise noise, in Stochastic differential systems (Marseille-Luminy, 1984). Lect. Notes Control Inform. Sci.69 (1985) 55–66.  
  22. I. Lasiecka and R. Triggiani, Differential and algebraic Riccati equations with application to boundary/point control problems: continuous theory and approximation theory. Lect. Notes Control Inform. Sci.164, Springer-Verlag, Berlin (1991).  Zbl0754.93038
  23. B. Maslowski, Stability of semilinear equations with boundary and pointwise noise. Ann. Scuola Norm. Sup. Pisa Cl. Sci.22 (1995) 55–93.  Zbl0830.60056
  24. D. Nualart, The Malliavin Calculus and Related Topics, Probability and its Applications, Springer (1995).  Zbl0837.60050
  25. D. Nualart and E. Pardoux, Stochastic calculus with anticipative integrands. Probab. Th. Rel. Fields78 (1988) 535–581.  Zbl0629.60061
  26. E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation. Syst. Control Lett.14 (1990) 55–61.  Zbl0692.93064
  27. E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic partial differential equations and their applications, B.L. Rozowskii and R.B. Sowers Eds., Springer, Lect. Notes Control Inf. Sci.176 (1992) 200–217.  Zbl0766.60079
  28. Y.A. Rozanov and Yu. A., General boundary value problems for stochastic partial differential equations. Trudy Mat. Inst. Steklov.200 (1991) 289–298.  
  29. R.B. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations. Ann. Probab.22 (1994) (2071–2121).  Zbl0834.60067
  30. A. Święch, “Unbounded” second order partial differential equations in infinite-dimensional Hilbert spaces. Comm. Part. Diff. Eq.19 (1994) 11–12, 1999–2036.  Zbl0812.35154

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.