The parabolic equations driven by linearly multiplicative Gaussian noise are stabilizable in probability by linear feedback controllers with support in a suitably chosen open subset of the domain. This procedure extends to Navier - Stokes equations with multiplicative noise. The exact controllability is also discussed.
Mots clés : stochastic equation, brownian motion, Navier − Stokes equation, feedback controller
@article{COCV_2013__19_4_1055_0, author = {Barbu, Viorel}, title = {Note on the internal stabilization of stochastic parabolic equations with linearly multiplicative gaussian noise}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1055--1063}, publisher = {EDP-Sciences}, volume = {19}, number = {4}, year = {2013}, doi = {10.1051/cocv/2012044}, mrnumber = {3182680}, zbl = {1283.35062}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2012044/} }
TY - JOUR AU - Barbu, Viorel TI - Note on the internal stabilization of stochastic parabolic equations with linearly multiplicative gaussian noise JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 1055 EP - 1063 VL - 19 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012044/ DO - 10.1051/cocv/2012044 LA - en ID - COCV_2013__19_4_1055_0 ER -
%0 Journal Article %A Barbu, Viorel %T Note on the internal stabilization of stochastic parabolic equations with linearly multiplicative gaussian noise %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 1055-1063 %V 19 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012044/ %R 10.1051/cocv/2012044 %G en %F COCV_2013__19_4_1055_0
Barbu, Viorel. Note on the internal stabilization of stochastic parabolic equations with linearly multiplicative gaussian noise. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1055-1063. doi : 10.1051/cocv/2012044. http://www.numdam.org/articles/10.1051/cocv/2012044/
[1] Internal stabilization of diffusion equation. Nonlinear Stud. 8 (2001) 193-202.
,[2] Controllability of parabolic and Navier − Stokes equations. Sci. Math. Japon. 56 (2002) 143-211. | MR
,[3] Stabilization of Navier − Stokes Flows, Communication and Control Engineering. Springer, London (2011). | MR
,[4] Internal stabilizability of the Navier-Stokes equations. Syst. Control Lett. 48 (2003) 161-167. | MR
and ,[5] Carleman estimates and controllability of linear stochastic heat equations. Appl. Math. Optimiz. 47 (2003) 1197-1209. | MR
, and ,[6] Internal exponential stabilization to a nonstationary solution for 3 − D Navier − Stokes equations. SIAM J. Control Optim. 49 (2011) 1454-1478. | MR
, and ,[7] Internal stabilization of Navier-Stokes equations with finite-dimensional controllers. Indiana Univ. Math. J. 53 (2004) 1443-1494. | MR
and ,[8] Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996). | MR
and ,[9] Some results on the controllability of forward stochastic heat equations with control on the drift. J. Funct. Anal. 260 (2011) 832-851. | MR
,[10] Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (2008). | MR
and ,[11] Approximate controllability for linear stochastic differential equations in infinite dimensions. Appl. Math. Optim. 53 (2009) 105-132. | MR
,[12] On exact controllability of the Navier-Stokes equations. ESAIM: COCV 3 (1998) 97-131. | Numdam | MR
,[13] Theory of Martingales. Kluwer Academic, Dordrecht (1989). | MR
and ,[14] Null controllability for forward and backward stochastic parabolic equations. SIAM J. Control Optim. 48 (2009) 2191-2216. | MR
and ,Cité par Sources :