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 and Continuous Dynamical Systems, 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.

[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. 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.

[6]

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

[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.

[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 PratoA. 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. 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.

[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.

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.

[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. 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.

[6]

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

[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.

[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 PratoA. 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. 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.

[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.

[1]

Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations and Control Theory, 2022, 11 (1) : 301-324. doi: 10.3934/eect.2021014

[2]

Wasim Akram, Debanjana Mitra. Local stabilization of viscous Burgers equation with memory. Evolution Equations and Control Theory, 2022, 11 (3) : 939-973. doi: 10.3934/eect.2021032

[3]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003

[4]

Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016

[5]

Jong Uhn Kim. On the stochastic Burgers equation with a polynomial nonlinearity in the real line. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 835-866. doi: 10.3934/dcdsb.2006.6.835

[6]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140

[7]

Ran Wang, Jianliang Zhai, Shiling Zhang. Large deviation principle for stochastic Burgers type equation with reflection. Communications on Pure and Applied Analysis, 2022, 21 (1) : 213-238. doi: 10.3934/cpaa.2021175

[8]

Anya Désilles, Hélène Frankowska. Explicit construction of solutions to the Burgers equation with discontinuous initial-boundary conditions. Networks and Heterogeneous Media, 2013, 8 (3) : 727-744. doi: 10.3934/nhm.2013.8.727

[9]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285

[10]

Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021

[11]

Alexei A. Ilyin. Lower bounds for the spectrum of the Laplace and Stokes operators. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 131-146. doi: 10.3934/dcds.2010.28.131

[12]

Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057

[13]

Wei Mao, Yanan Jiang, Liangjian Hu, Xuerong Mao. Stabilization by intermittent control for hybrid stochastic differential delay equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 569-581. doi: 10.3934/dcdsb.2021055

[14]

Boumediène Chentouf, Baowei Feng. On the stabilization of a flexible structure via a nonlinear delayed boundary control. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022032

[15]

Chun-Hsiung Hsia, Xiaoming Wang. On a Burgers' type equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1121-1139. doi: 10.3934/dcdsb.2006.6.1121

[16]

Diogo Poças, Bartosz Protas. Transient growth in stochastic Burgers flows. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2371-2391. doi: 10.3934/dcdsb.2018052

[17]

Xenia Kerkhoff, Sandra May. Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021054

[18]

Boumedièene Chentouf, Sabeur Mansouri. Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1127-1141. doi: 10.3934/dcdss.2021090

[19]

Hui Yin, Huijiang Zhao. Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space. Kinetic and Related Models, 2009, 2 (3) : 521-550. doi: 10.3934/krm.2009.2.521

[20]

Joachim von Below, Gaëlle Pincet Mailly, Jean-François Rault. Growth order and blow up points for the parabolic Burgers' equation under dynamical boundary conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 825-836. doi: 10.3934/dcdss.2013.6.825

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (265)
  • HTML views (100)
  • Cited by (1)

Other articles
by authors

[Back to Top]