no. 3
The Lebeau–Robbiano inequality for the one-dimensional fourth order elliptic operator and its application
Gao, Peng1, 2
1 School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, P.R. China
2 Institute of Mathematics, Jilin University, Changchun 130012, P.R. China.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 811-831.

In this paper, we establish the Lebeau–Robbiano inequality for the one-dimensional fourth order elliptic operator by using a point-wise estimate. Based on this inequality, we obtain the null controllability of one-dimensional stochastic fractional order Cahn–Hilliard equation.

Reçu le :
DOI : 10.1051/cocv/2015030
Classification : 93B05, 35K35, 60H15
Mots clés : Lebeau–Robbiano inequality, null controllability, stochastic fractional order Cahn–Hilliard equation
Gao, Peng 1, 2

1 School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, P.R. China
2 Institute of Mathematics, Jilin University, Changchun 130012, P.R. China. 
@article{COCV_2016__22_3_811_0,
     author = {Gao, Peng},
     title = {The {Lebeau{\textendash}Robbiano} inequality for the one-dimensional fourth order elliptic operator and its application},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {811--831},
     publisher = {EDP-Sciences},
     volume = {22},
     number = {3},
     year = {2016},
     doi = {10.1051/cocv/2015030},
     mrnumber = {3527945},
     zbl = {1342.93025},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2015030/}
}
TY  - JOUR
AU  - Gao, Peng
TI  - The Lebeau–Robbiano inequality for the one-dimensional fourth order elliptic operator and its application
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2016
SP  - 811
EP  - 831
VL  - 22
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2015030/
DO  - 10.1051/cocv/2015030
LA  - en
ID  - COCV_2016__22_3_811_0
ER  - 
%0 Journal Article
%A Gao, Peng
%T The Lebeau–Robbiano inequality for the one-dimensional fourth order elliptic operator and its application
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2016
%P 811-831
%V 22
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2015030/
%R 10.1051/cocv/2015030
%G en
%F COCV_2016__22_3_811_0
Gao, Peng. The Lebeau–Robbiano inequality for the one-dimensional fourth order elliptic operator and its application. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 811-831. doi : 10.1051/cocv/2015030. http://www.numdam.org/articles/10.1051/cocv/2015030/

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992). | MR

Y. Hu and S. Peng, Adapted solution of backward stochastic evolution equations. Stoch. Anal. Appl. 9 (1991) 445–459. | DOI | MR | Zbl

D. Jerison and G. Lebeau, Nodal sets of sums of eigenfunctions, in: Harmonic Analysis and Partial Differential Equations. Chicago Lectures in Math. Univ. Chicago Press, Chicago, IL (1999) 223–239. | MR | Zbl

G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20 (1995) 335–336. | DOI | MR | Zbl

G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity. Arch. Rational Mech. Anal. 141 (1998) 297–329. | DOI | MR | Zbl

J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin, Heidelberg, New York (1971). | MR | Zbl

Q. Lü, Observation and control for stochastic partial differential equations. Ph.D. thesis, School of Mathematics. Sichuan University, Chengdu (2010).

Q. Lü, Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations. Acta Math. Sin. (Engl. Ser.) 26 (2010) 2377–2386. | DOI | MR | Zbl

Q. Lü, Some results on the controllability of forward stochastic heat equations with control on the drift. J. Funct. Anal. 260 (2011) 832–851. | DOI | MR | Zbl

Q. Lü and G. Wang, On the existence of time optimal controls with constraints of the rectangular type for heat equations. SIAM J. Control Optim. 49 (2011) 1124–1149. | DOI | MR | Zbl

Q. Lü and Z. Yin, The L ∞ -null controllability of parabolic equation with equivalued surface boundary conditions. Asymptot. Anal. 83 (2013) 355–378. | MR | Zbl

N.I. Mahmudov and M.A. Mckibben, On backward stochastic evolution equations in Hilbert spaces and optimal control. Nonlinear Anal. 67 (2007) 1260–1274. | DOI | MR | Zbl

M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations, 2nd edn. Vol. 13 of Texts in Applied Mathematics. Springer-Verlag, New York (2004). | MR | Zbl

G. Wang, L ∞ -null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim. 47 (2008) 1701–1720. | DOI | MR | Zbl

Z.C. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation. Taiwanese J. Math. 16 (2012) 1991–2017. | MR | Zbl

Cité par Sources :