Figure 1.
Signs of $c(m)d(l)c(m+l)$
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March
2018, 38(3): 1187-1242.
doi: 10.3934/dcds.2018050
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 [
Mathematics Subject Classification:
Primary: 93D15, 42A16; Secondary: 93C20, 35Q30.
Citation:
Andrei Fursikov, Lyubov Shatina. Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system. Discrete and Continuous Dynamical Systems,
2018, 38
(3)
: 1187-1242.
doi: 10.3934/dcds.2018050
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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.
|
Figure 2.
Signs of $c(m)c(l)d(m+l)$
Figure 3.
Signs of $A(k)A(k+l)B(l)$
Figure 4.
Signs of $(-A(k)A(l)B(k+l))$
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