April  2019, 39(4): 2173-2185. doi: 10.3934/dcds.2019091

Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback

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

Fund Project: 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.

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.

Citation: Ionuţ Munteanu. Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2173-2185. doi: 10.3934/dcds.2019091
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.  Google Scholar

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V. Barbu, Boundary stabilization of equilibrium solutions to parabolic equations, IEEE Trans. Autom. Control, 58 (2013), 2416-2420.  doi: 10.1109/TAC.2013.2254013.  Google Scholar

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V. Barbu and G. Da Prato, Internal stabilization by noise of the Navier–Stokes equation, SIAM J. Control Optimiz., 49(1) (2012), 1-0.   Google Scholar

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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.  Google Scholar

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V. BarbuI. 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.  Google Scholar

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T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Inc., New York, 2006. Google Scholar

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H. ChoiR. TemamP. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[10]

G. Da PratoA. Debussche and R. Temam, Stochastic Burgers' equation, Nonlin. Diff. Equations Appl., 1 (1994), 389-402.  doi: 10.1007/BF01194987.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

H. LiuP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

I. Munteanu, Stabilisation of parabolic semilinear equations, Int. J. Control, 90 (2017), 1063-1076.  doi: 10.1080/00207179.2016.1200747.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[5]

V. BarbuI. 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.  Google Scholar

[6]

T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Inc., New York, 2006. Google Scholar

[7]

H. ChoiR. TemamP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

G. Da PratoA. Debussche and R. Temam, Stochastic Burgers' equation, Nonlin. Diff. Equations Appl., 1 (1994), 389-402.  doi: 10.1007/BF01194987.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

H. LiuP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

I. Munteanu, Stabilisation of parabolic semilinear equations, Int. J. Control, 90 (2017), 1063-1076.  doi: 10.1080/00207179.2016.1200747.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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