
-
Previous Article
A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation
- DCDS Home
- This Issue
-
Next Article
Soliton solutions for the elastic metric on spaces of curves
Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system
1. | Department of Mechanics & Mathematics, Moscow State University, Moscow 119991, Russia, Voronezh State University, Voronezh, Russia |
2. | Department of Mechanics & Mathematics, Moscow State University, Moscow 119991, Russia |
Let $ y(t,x;y_0) $ be a solution to the semilinear parabolic equation of normal type generated by the 3D Helmholtz system with periodic boundary conditions and arbitrary initial datum $ y_0(x) $. The problem of stabilization to zero of the solution $ y(t,x;y_0) $ by starting control is studied. This problem is reduced to establishing three inequalities connected with starting control, one of which has been proved in [
References:
[1] |
V. Barbu, I. Lasiecka and R. Triggiani,
Abstract setting of tangential boundary stabilization of Navier-Stokes equations by high-andlow-gain feedback controllers, Nonlinear Analysis, 64 (2006), 2704-2746.
doi: 10.1016/j.na.2005.09.012. |
[2] |
J. M. Coron,
On null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domains, SIAM J.Control Optim., 37 (1999), 1874-1896.
doi: 10.1137/S036301299834140X. |
[3] |
J. M. Coron,
Control and Nonlinearity, Math. Surveys and Monographs, AMS, Providence, RI, 2007. |
[4] |
J. M. Coron and A. V. Fursikov,
Global exact controllability of the 2D Navier-Stokes equations on manifold without boundary, J.Russian Math. Phys., 4 (1996), 429-448.
|
[5] |
G. Eskin,
Lectures on Linear Partial Differential Equations, Amer. Math. Society, Providence RI, 2011. |
[6] |
A. V. Fursikov, On one semilinear parabolic equation of normal type, in Proceeding volume "Mathematics and life sciences" De Gruyter, 1 (2013), 147-160. |
[7] |
A. V. Fursikov,
The simplest semilinear parabolic equation of normal type, Mathematical Control and Related Fields(MCRF), 2 (2012), 141-170.
doi: 10.3934/mcrf.2012.2.141. |
[8] |
A. V. Fursikov, On the normal semilinear parabolic equations corresponding to 3D NavierStokes system, in CSMO 2011, (eds. D. Homberg and F. Troltzsch), IFIP AICT, 391 (2013), 338-347. |
[9] |
A. V. Fursikov,
On parabolic system of normal type corresponding to 3D Helmholtz system, Advances in Mathematical Analysis of PDEs. AMS Transl. Series 2, 232 (2014), 99-118.
|
[10] |
A. V. Fursikov,
Stabilization of the simplest normal parabolic equation by starting control, Communications on Pure and Applied Analysis, 13 (2014), 1815-1854.
doi: 10.3934/cpaa.2014.13.1815. |
[11] |
A. V. Fursikov,
Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete and Cont. Dyn. Syst., 10 (2004), 289-314.
|
[12] |
A. V. Fursikov and A. V. Gorshkov,
Certain questions of feedback stabilization for NavierStokes equations, Evolution Equations and Control Theory (EECT), 1 (2012), 109-140.
doi: 10.3934/eect.2012.1.109. |
[13] |
A. V. Fursikov and O. Yu Immanuvilov,
Yu Immanuvilov, Exact controllability of Navier-Stokes and Boussinesq
equations, Russian Math. Survveys, 54 (1999), 565-618.
|
[14] |
A. V. Fursikov and A. A. Kornev, Feedback stabilization for Navier-Stokes equations: Theory and calculations, Mathematical Aspects of Fluid Mechanics (LMS Lecture Notes Series), 402, Cambridge University Press, (2012), 130-172. |
[15] |
A. V. Fursikov and L. S. Shatina, On an estimate related to the stabilization on a normal parabolic equation by starting control,
Fundamental and Applied Mathematics, 19 (2014), 197-230 (in Russian) |
[16] |
M. Krstic,
On global stabilizationof Burgers' equation by boundary control, Systems of Control Letters, 37 (1999), 123-141.
doi: 10.1016/S0167-6911(99)00013-4. |
[17] |
J.-P. Raymond,
Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl., 87 (2007), 627-669.
doi: 10.1016/j.matpur.2007.04.002. |
[18] |
V. I. Yudovich,
Non-stationary flow of ideal incompressible fluid, Computational Mathematics and Mathematical Physics, 3 (1963), 1032-1066.
|
show all references
References:
[1] |
V. Barbu, I. Lasiecka and R. Triggiani,
Abstract setting of tangential boundary stabilization of Navier-Stokes equations by high-andlow-gain feedback controllers, Nonlinear Analysis, 64 (2006), 2704-2746.
doi: 10.1016/j.na.2005.09.012. |
[2] |
J. M. Coron,
On null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domains, SIAM J.Control Optim., 37 (1999), 1874-1896.
doi: 10.1137/S036301299834140X. |
[3] |
J. M. Coron,
Control and Nonlinearity, Math. Surveys and Monographs, AMS, Providence, RI, 2007. |
[4] |
J. M. Coron and A. V. Fursikov,
Global exact controllability of the 2D Navier-Stokes equations on manifold without boundary, J.Russian Math. Phys., 4 (1996), 429-448.
|
[5] |
G. Eskin,
Lectures on Linear Partial Differential Equations, Amer. Math. Society, Providence RI, 2011. |
[6] |
A. V. Fursikov, On one semilinear parabolic equation of normal type, in Proceeding volume "Mathematics and life sciences" De Gruyter, 1 (2013), 147-160. |
[7] |
A. V. Fursikov,
The simplest semilinear parabolic equation of normal type, Mathematical Control and Related Fields(MCRF), 2 (2012), 141-170.
doi: 10.3934/mcrf.2012.2.141. |
[8] |
A. V. Fursikov, On the normal semilinear parabolic equations corresponding to 3D NavierStokes system, in CSMO 2011, (eds. D. Homberg and F. Troltzsch), IFIP AICT, 391 (2013), 338-347. |
[9] |
A. V. Fursikov,
On parabolic system of normal type corresponding to 3D Helmholtz system, Advances in Mathematical Analysis of PDEs. AMS Transl. Series 2, 232 (2014), 99-118.
|
[10] |
A. V. Fursikov,
Stabilization of the simplest normal parabolic equation by starting control, Communications on Pure and Applied Analysis, 13 (2014), 1815-1854.
doi: 10.3934/cpaa.2014.13.1815. |
[11] |
A. V. Fursikov,
Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete and Cont. Dyn. Syst., 10 (2004), 289-314.
|
[12] |
A. V. Fursikov and A. V. Gorshkov,
Certain questions of feedback stabilization for NavierStokes equations, Evolution Equations and Control Theory (EECT), 1 (2012), 109-140.
doi: 10.3934/eect.2012.1.109. |
[13] |
A. V. Fursikov and O. Yu Immanuvilov,
Yu Immanuvilov, Exact controllability of Navier-Stokes and Boussinesq
equations, Russian Math. Survveys, 54 (1999), 565-618.
|
[14] |
A. V. Fursikov and A. A. Kornev, Feedback stabilization for Navier-Stokes equations: Theory and calculations, Mathematical Aspects of Fluid Mechanics (LMS Lecture Notes Series), 402, Cambridge University Press, (2012), 130-172. |
[15] |
A. V. Fursikov and L. S. Shatina, On an estimate related to the stabilization on a normal parabolic equation by starting control,
Fundamental and Applied Mathematics, 19 (2014), 197-230 (in Russian) |
[16] |
M. Krstic,
On global stabilizationof Burgers' equation by boundary control, Systems of Control Letters, 37 (1999), 123-141.
doi: 10.1016/S0167-6911(99)00013-4. |
[17] |
J.-P. Raymond,
Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl., 87 (2007), 627-669.
doi: 10.1016/j.matpur.2007.04.002. |
[18] |
V. I. Yudovich,
Non-stationary flow of ideal incompressible fluid, Computational Mathematics and Mathematical Physics, 3 (1963), 1032-1066.
|




[1] |
Andrei Fursikov. Stabilization of the simplest normal parabolic equation. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1815-1854. doi: 10.3934/cpaa.2014.13.1815 |
[2] |
Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085 |
[3] |
Cătălin-George Lefter, Elena-Alexandra Melnig. Feedback stabilization with one simultaneous control for systems of parabolic equations. Mathematical Control and Related Fields, 2018, 8 (3&4) : 777-787. doi: 10.3934/mcrf.2018034 |
[4] |
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 |
[5] |
Shirshendu Chowdhury, Debanjana Mitra, Michael Renardy. Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control. Evolution Equations and Control Theory, 2018, 7 (3) : 447-463. doi: 10.3934/eect.2018022 |
[6] |
Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6359-6376. doi: 10.3934/dcdsb.2021022 |
[7] |
Andrei Fursikov. The simplest semilinear parabolic equation of normal type. Mathematical Control and Related Fields, 2012, 2 (2) : 141-170. doi: 10.3934/mcrf.2012.2.141 |
[8] |
Jan Prüss, Gieri Simonett, Rico Zacher. On normal stability for nonlinear parabolic equations. Conference Publications, 2009, 2009 (Special) : 612-621. doi: 10.3934/proc.2009.2009.612 |
[9] |
Shui-Nee Chow, Kening Lu, Yun-Qiu Shen. Normal forms for quasiperiodic evolutionary equations. Discrete and Continuous Dynamical Systems, 1996, 2 (1) : 65-94. doi: 10.3934/dcds.1996.2.65 |
[10] |
Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control and Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353 |
[11] |
Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 |
[12] |
Yong Ren, Qi Zhang. Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3811-3829. doi: 10.3934/dcdsb.2021207 |
[13] |
Sorin Micu, Jaime H. Ortega, Lionel Rosier, Bing-Yu Zhang. Control and stabilization of a family of Boussinesq systems. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 273-313. doi: 10.3934/dcds.2009.24.273 |
[14] |
Serge Nicaise, Fredi Tröltzsch. Optimal control of some quasilinear Maxwell equations of parabolic type. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1375-1391. doi: 10.3934/dcdss.2017073 |
[15] |
Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879 |
[16] |
John A. D. Appleby, Xuerong Mao, Alexandra Rodkina. On stochastic stabilization of difference equations. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 843-857. doi: 10.3934/dcds.2006.15.843 |
[17] |
Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169 |
[18] |
Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 883-901. doi: 10.3934/dcdsb.2021072 |
[19] |
Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063 |
[20] |
Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control and Related Fields, 2017, 7 (1) : 53-72. doi: 10.3934/mcrf.2017004 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]