Carleman Estimates and Controllability of Linear Stochastic Heat Equations

Abstract

Abstract. This work is concerned with Carleman inequalities and controllability properties for the following stochastic linear heat equation (with Dirichlet boundary conditions in the bounded domain D⊂R d and multiplicative noise): $ \left\{ \begin{array}{@{}l} {\displaystyle d_{t}y^{u}-\Delta y^{u}+$ay$^{u}\,$dt$=f\,$dt$+1_{{\cal D}_{0} }u\,$dt$+$by$\,d \beta_t\qquad\mbox{in}\quad ]0,T]\times {\cal D},}\\ \\ \vspace{-9pt} y^{u}=0\qquad\mbox{on}\quad ]0,T]\times\partial {\cal D},\\ \\ \vspace{-9pt} y^{u}(0)=y_{0}\qquad\mbox{in } {\cal D}, \end{array} \right. $ and for the corresponding backward dual equation: $ \left\{ \begin{array}{@{}l} {\displaystyle d_{t}p^{v}+\Delta p^{v}\,$dt$-$ap$^{v}\,$dt$+$bk$^{v}\,$dt$ =1_{{\cal D}_{0}} v\,$dt$+k^{v}\,d\beta_t\qquad\mbox{in}\quad [0,T[\,\times\, {\cal D},}\\ \\ \vspace{-9pt} p^{v}=0\qquad\mbox{on}\quad [0,T[\,\times\partial{\cal D},\\ \\ \vspace{-9pt} p^{v}(T)=\eta\qquad\mbox{in } {\cal D}. \end{array} \right. $ We prove the null controllability of the backward equation and obtain partial results for the controllability of the forward equation. \par