This paper is devoted to the null controllability of a cascade system of $ m $ coupled backward stochastic heat equations governed by a unique distributed control force, where $ m\geq 2 $. This task is obtained by means of proving a suitable observability estimate for the dual system of the controlled system.
Citation: |
[1] |
E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj and L. Maniar, Null controllability of degenerate parabolic cascade systems,, Portugal. Math., 68 (2011), 345-367.
doi: 10.4171/PM/1895.![]() ![]() ![]() |
[2] |
E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj and L. Maniar, Carleman estimates and nul controllability of coupled degenerate systems,, Evol. Equ. Control Theory, 2 (2013), 441-459.
doi: 10.3934/eect.2013.2.441.![]() ![]() ![]() |
[3] |
E. M. Ait Ben Hassi, M. Fadili and L. Maniar, On algebraic condition for null controllability of some coupled degenerate systems,, Mathematical Control and Related Fields, 9 (2019), 77-95.
doi: 10.3934/mcrf.2019004.![]() ![]() ![]() |
[4] |
F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems,, J. Evol. Equ., 9 (2009), 267-291.
doi: 10.1007/s00028-009-0008-8.![]() ![]() ![]() |
[5] |
V. Barbu, Controllability and Stabilization of Parabolic Equations, , Birkhäuser, 2018.
doi: 10.1007/978-3-319-76666-9.![]() ![]() ![]() |
[6] |
V. Barbu, A. R$\breve{a}$scanu and G. Tessitore, Carleman estimate and controllability of linear stochastic heat equations,, Appl. Math. Optim., 47 (2003), 97-120.
doi: 10.1007/s00245-002-0757-z.![]() ![]() ![]() |
[7] |
M. Baroun, M. Fadili, A. Khchine and L. Maniar, Carleman estimates and controllability of stochastic degenerate parabolic heat equations, preprint, arXiv: 2202.10070, (2022).
![]() |
[8] |
M. Fadili and L. Maniar, Null controllability of $n$-coupled degenerate parabolic systems with $m$-controls,, J. Evol. Equ., 17 (2017), 1311-1340.
doi: 10.1007/s00028-017-0385-3.![]() ![]() ![]() |
[9] |
M. González-Burgos and L. de Teresa, Controllability results for cascade systems of $m$-coupled parabolic PDEs by one control force, Port. Math., 67 (2010), 91-113.
doi: 10.4171/PM/1859.![]() ![]() ![]() |
[10] |
V. Hern$\acute{a}$ndez-Santamaría, K. L. Balc'h and L. Peralta, Global null-controllability for stochastic semilinear parabolic equations, preprint, arXiv: 2010.08854v1, (2020).
![]() |
[11] |
O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations,, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274.
doi: 10.2977/prims/1145476103.![]() ![]() ![]() |
[12] |
H. Li and Q. Lü, Null controllability for some systems of two backward stochastic heat equations with one control force,, Chin. Ann. Math., 33B (2012), 909-918.
doi: 10.1007/s11401-012-0743-y.![]() ![]() |
[13] |
X. Liu, Global Carleman estimate for stochastic parabolic equations, and its application,, ESAIM Control Optim. Calc. Var., 20 (2014), 823-839.
doi: 10.1051/cocv/2013085.![]() ![]() ![]() |
[14] |
X. Liu and Y. Yu, Carleman estimates of some stochastic degenerate parabolic equations and application,, SIAM J. Control Optim, 57, 5 (2019), 3527-3552.
doi: 10.1137/18M1221448.![]() ![]() ![]() |
[15] |
É. Pardoux and S. Peng, Adapted solution of backward stochastic equations,, System Control Lett., 14 (1990), 55-61.
doi: 10.1016/0167-6911(90)90082-6.![]() ![]() ![]() |
[16] |
E. Pardoux and A. R$\breve{a}$scanu, Backward stochastic variational inequalities,, Stochastics Stochastic Rep., 67 (1999), 159-167.
doi: 10.1080/17442509908834208.![]() ![]() ![]() |
[17] |
S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations,, SIAM J. Control Optim., 48 (2009), 2191-2216.
doi: 10.1137/050641508.![]() ![]() ![]() |
[18] |
J. Zabczyk, Mathematical Control Theory, , Birkhäuser, Boston, 1995.
![]() |