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 ${C}^{1}$ 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.

Keywords: boundary noise, optimal boundary control, HJB equation, backward stochastic differential equations

@article{COCV_2007__13_1_178_0, author = {Debussche, Arnaud and Fuhrman, Marco and Tessitore, Gianmario}, title = {Optimal control of a stochastic heat equation with boundary-noise and boundary-control}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {178--205}, publisher = {EDP-Sciences}, volume = {13}, number = {1}, year = {2007}, doi = {10.1051/cocv:2007001}, mrnumber = {2282108}, zbl = {1123.60052}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2007001/} }

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 PY - 2007 SP - 178 EP - 205 VL - 13 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007001/ DO - 10.1051/cocv:2007001 LA - en ID - COCV_2007__13_1_178_0 ER -

%0 Journal Article %A Debussche, Arnaud %A Fuhrman, Marco %A Tessitore, Gianmario %T Optimal control of a stochastic heat equation with boundary-noise and boundary-control %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 178-205 %V 13 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007001/ %R 10.1051/cocv:2007001 %G en %F COCV_2007__13_1_178_0

Debussche, Arnaud; Fuhrman, Marco; Tessitore, Gianmario. Optimal control of a stochastic heat equation with boundary-noise and boundary-control. ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 1, pp. 178-205. doi : 10.1051/cocv:2007001. http://www.numdam.org/articles/10.1051/cocv:2007001/

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