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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

Andrei Fursikov 1, and  Lyubov Shatina 2,

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

Received  February 2017 Revised  October 2017 Published  December 2017

Fund Project: The research of the first author was supported by the Ministry of Education and Science of the Russian Federation (grant 14.Z50.31.0037). The second author was supported by RFBR grants 15-01-03576 and 15-01-08023.

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 [10], [15]. The proof for the other two is given here.

Keywords: Equations of normal type, stabilization by starting control.
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 & Continuous Dynamical Systems, 2018, 38 (3) : 1187-1242. doi: 10.3934/dcds.2018050
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.  Google Scholar

[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.  Google Scholar

[3]

J. M. Coron, Control and Nonlinearity, Math. Surveys and Monographs, AMS, Providence, RI, 2007.  Google Scholar

[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.   Google Scholar

[5]

G. Eskin, Lectures on Linear Partial Differential Equations, Amer. Math. Society, Providence RI, 2011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete and Cont. Dyn. Syst., 10 (2004), 289-314.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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)  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

V. I. Yudovich, Non-stationary flow of ideal incompressible fluid, Computational Mathematics and Mathematical Physics, 3 (1963), 1032-1066.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

J. M. Coron, Control and Nonlinearity, Math. Surveys and Monographs, AMS, Providence, RI, 2007.  Google Scholar

[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.   Google Scholar

[5]

G. Eskin, Lectures on Linear Partial Differential Equations, Amer. Math. Society, Providence RI, 2011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete and Cont. Dyn. Syst., 10 (2004), 289-314.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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)  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

V. I. Yudovich, Non-stationary flow of ideal incompressible fluid, Computational Mathematics and Mathematical Physics, 3 (1963), 1032-1066.   Google Scholar

Figure 1.  Signs of $c(m)d(l)c(m+l)$
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Figure 2.  Signs of $c(m)c(l)d(m+l)$
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Figure 3.  Signs of $A(k)A(k+l)B(l)$
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Figure 4.  Signs of $(-A(k)A(l)B(k+l))$
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