Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback

Ionuţ Munteanu

doi:10.3934/dcds.2019091

Article Contents

2019, Volume 39, Issue 4: 2173-2185. Doi: 10.3934/dcds.2019091

This issue Previous Article Quasiregular semigroups with examples Next Article Existence of the normalized solutions to the nonlocal elliptic system with partial confinement

Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback

Ionuţ Munteanu

Alexandru Ioan Cuza University, Department of Mathematics, Octav Mayer Institute of Mathematics (Romanian Academy), 700506 Iaşi, Romania

Received: May 2018

Revised: September 2018

Published: January 2019

This work was supported by a grant of the "Alexandru Ioan Cuza" University of Iasi, within the Research Grants program, Grant UAIC, code GI-UAIC-2018-03

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In the present paper it is designed a simple, finite-dimensional, linear deterministic stabilizing boundary feedback law for the stochastic Burgers equation with unbounded time-dependent coefficients. The stability of the system is guaranteed no matter how large the level of the noise is.

Keywords:

Mathematics Subject Classification: 60H15, 76F70, 93B52, 60H20.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	V. Barbu and M. Rockner, Global solutions to random 3D vorticity equations for small initial data, J. Diff. Equations, 263 (2017), 5395-5411. doi: 10.1016/j.jde.2017.06.020.
[2]	V. Barbu, Boundary stabilization of equilibrium solutions to parabolic equations, IEEE Trans. Autom. Control, 58 (2013), 2416-2420. doi: 10.1109/TAC.2013.2254013.
[3]	V. Barbu and G. Da Prato, Internal stabilization by noise of the Navier–Stokes equation, SIAM J. Control Optimiz., 49(1) (2012), 1-0.
[4]	V. Barbu, Viorel, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), x+128 pp. doi: 10.1090/memo/0852.
[5]	V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal., 64 (2006), 2704-2746. doi: 10.1016/j.na.2005.09.012.
[6]	T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Inc., New York, 2006.
[7]	H. Choi, R. Temam, P. Moin and J. Kim, Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. Fluid Mech., 253 (1993), 509-543. doi: 10.1017/S0022112093001880.
[8]	G. Da Prato and A. Debussche, Dynamic programming for the stochastic Burgers equation, Ann. Mat. Pura Appl., 178 (2000), 143-174. doi: 10.1007/BF02505893.
[9]	G. Da Prato and A. Debussche, Control of the stochastic Burgers model of turbulence, SIAM J. Control Optimiz., 37 (1999), 1123-1149. doi: 10.1137/S0363012996311307.
[10]	G. Da Prato, A. Debussche and R. Temam, Stochastic Burgers' equation, Nonlin. Diff. Equations Appl., 1 (1994), 389-402. doi: 10.1007/BF01194987.
[11]	M. Foondun and E. Nualart, On the behaviour of stochastic heat equations on bounded domains, Latin Amer. J. Probab. Math. Statistics, 12 (2015), 551-571.
[12]	I. Gyongy and D. Nualart, On the stochastic Burgers equation in the real line, Annals of Probab., 27 (1999), 782-802. doi: 10.1214/aop/1022677386.
[13]	M. Krstic, On global stabilization of Burgers' equation by boundary control, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171), (1998). doi: 10.1109/CDC.1998.758248.
[14]	W.-J. Liu and M. Krstic, Backstepping boundary control of Burgers equation with actuator dynamics, Syst. Control Lett., 41 (2000), 291-303. doi: 10.1016/S0167-6911(00)00068-2.
[15]	H. Liu, P. Hu and I. Munteanu, Boundary feedback stabilization of Fisher's equation, Syst. Control Lett., 97 (2016), 55-60. doi: 10.1016/j.sysconle.2016.09.003.
[16]	J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl., 342 (2008), 753-760. doi: 10.1016/j.jmaa.2007.11.019.
[17]	I. Munteanu, Boundary stabilization of the stochastic heat equation by proportional feedbacks, Automatica, 87 (2018), 152-158. doi: 10.1016/j.automatica.2017.10.003.
[18]	I. Munteanu, Boundary stabilisation to non-stationary solutions for deterministic and stochastic parabolic-type equations, Int. J. Control, 2017. doi: 10.1080/00207179.2017.1407878.
[19]	I. Munteanu, Stabilisation of parabolic semilinear equations, Int. J. Control, 90 (2017), 1063-1076. doi: 10.1080/00207179.2016.1200747.
[20]	I. Munteanu, Boundary stabilization of the phase field system by finite-dimensional feedback controllers, J. Math. Anal. Appl., 412 (2014), 964-975. doi: 10.1016/j.jmaa.2013.11.018.
[21]	I. Munteanu, Boundary stabilization of the Navier–Stokes equation with fading memory, Int. J. Control, 88 (2015), 531-542. doi: 10.1080/00207179.2014.964780.
[22]	I. Munteanu, Stabilization of semilinear heat equations, with fading memory, by boundary feedbacks, J. Diff. Equations, 259 (2015), 454-472. doi: 10.1016/j.jde.2015.02.010.
[23]	I. Munteanu, Boundary stabilization of a 2-D periodic MHD channel flow, by proportional feedbacks, ESAIM: COCV, 23 (2017), 1253-1266. doi: 10.1051/cocv/2016025.
[24]	I. Munteanu, Stabilization of a 3-D periodic channel flow by explicit normal boundary feedbacks, J. Dynam. Control Syst., 23 (2017), 387-403. doi: 10.1007/s10883-016-9332-9.