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Note on the internal stabilization of stochastic parabolic equations with linearly multiplicative Gaussian noise∗

Published online by Cambridge University Press:  04 July 2013

Viorel Barbu
Viorel Barbu*
Affiliation:
Al.I. Cuza University and Octav Mayer Institute of Mathematics (Romanian Academy), Blvd. Carol I, No. 11, 700506 Iaşi, Romania. [email protected]
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Abstract

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.

Keywords

Stochastic equationbrownian motionNavier − Stokes equationfeedback controller
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013 , pp. 1055 - 1063
Copyright
© EDP Sciences, SMAI, 2013

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References

Aniţa, S., Internal stabilization of diffusion equation. Nonlinear Stud. 8 (2001) 193–202. Google Scholar
Barbu, V., Controllability of parabolic and Navier − Stokes equations. Sci. Math. Japon. 56 (2002) 143–211. Google Scholar
V. Barbu, Stabilization of Navier − Stokes Flows, Communication and Control Engineering. Springer, London (2011).
Barbu, V. and Lefter, C., Internal stabilizability of the Navier–Stokes equations. Syst. Control Lett. 48 (2003) 161–167. Google Scholar
Barbu, V., Rascanu, A. and Tessitore, G., Carleman estimates and controllability of linear stochastic heat equations. Appl. Math. Optimiz. 47 (2003) 1197–1209. Google Scholar
Barbu, V., Rodriguez, S.S. and Shirikyan, A., Internal exponential stabilization to a nonstationary solution for 3 − D Navier − Stokes equations. SIAM J. Control Optim. 49 (2011) 1454–1478. Google Scholar
Barbu, V. and Triggiani, R., Internal stabilization of Navier–Stokes equations with finite-dimensional controllers. Indiana Univ. Math. J. 53 (2004) 1443-1494. Google Scholar
G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996).
Lü, Qi, Some results on the controllability of forward stochastic heat equations with control on the drift. J. Funct. Anal. 260 (2011) 832–851. Google Scholar
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (2008).
Goreac, D., Approximate controllability for linear stochastic differential equations in infinite dimensions. Appl. Math. Optim. 53 (2009) 105–132. Google Scholar
Imanuvilov, O., On exact controllability of the Navier–Stokes equations. ESAIM: COCV 3 (1998) 97–131. Google Scholar
R.S. Lipster and A. Shiryaev, Theory of Martingales. Kluwer Academic, Dordrecht (1989).
Tang, S. and Zhang, X., Null controllability for forward and backward stochastic parabolic equations. SIAM J. Control Optim. 48 (2009) 2191–2216.Google Scholar