# American Institute of Mathematical Sciences

September  2014, 13(5): 1815-1854. doi: 10.3934/cpaa.2014.13.1815

## Stabilization of the simplest normal parabolic equation

 1 Department of Mechanics & Mathematics, Moscow State University, Moscow 119991

Received  November 2013 Revised  April 2014 Published  June 2014

The simplest semilinear parabolic equation of normal type with periodic boundary condition is considered, and the problem of stabilization to zero of its solution with arbitrary initial condition by starting control supported in a prescribed subset is investigated. This problem is reduced to one inequality for starting control, and the proof of this inequality is given.
Citation: 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
##### References:
 [1] V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM: Control, Optimization and Calculus of Variation, 9 (2003), 197-205. doi: 10.1051/cocv:2003009. [2] V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimentional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494. doi: 10.1512/iumj.2004.53.2445. [3] 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. [4] V. Barbu, S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations, SIAM J. Control Optim., 49 (2011), 1454-1478. doi: 10.1137/100785739. [5] J. M. Coron, On null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domains, SIAM J. ControlOptim., 37 (1999), 1874-1896. doi: 10.1137/S036301299834140X. [6] J. M. Coron, Control and Nonlinearity, Math.Surveys and Monographs, 136 AMS, Providence, RI, 2007. [7] A. V. Fursikov, On one semilinear parabolic equation of normal type, in Proceeding volume "Mathematics and life sciences", De Gruyter, 1 (2012), 147-160. [8] 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. [9] A. V. Fursikov, On the normal semilinear parabolic equations corresponding to 3D Navier-Stokes system, in CSMO 2011, IFIP AICT 391 (eds D.Homberg and F.Troltzsch), (2013), 338-347. [10] A. V. Fursikov, On parabolic system of normal type corresponding to 3D Helmholtz system, work in progress. [11] A. V. Fursikov, Stabilizability of quasi linear parabolic equation by feedback boundary control, Sbornik: Mathematics, 192 (2001), 593-639. doi: 10.1070/sm2001v192n04ABEH000560. [12] A. V. Fursikov, Stabilizability of Two-Dimensional Navier-Stokes equations with help of a boundary feedback control, J. of Math. Fluid Mech., 3 (2001), 259-301. doi: 10.1007/PL00000972. [13] A. V. Fursikov, Real process corresponding to 3D-Navier Stokes system and its feedback stabilization from boundary, Amer. Math. Soc. Transl. Series 2, 206, Advances in Math.Sciences-51. PDE M. Vishik seminar (2002), 95-123. [14] A. V. Fursikov, Real processes and realizability of a stabilization method for Navier-Stokes equations by boundary feedback control, Nonlinear Problems in Mathematical Physics and Related Topics II, In Honor of Professor O. A. Ladyzhenskaya, Kluwer/Plenum Publishers, New-York, Boston, Dordrecht, London, Moscow, (2002), 137-177. doi: 10.1007/978-1-4615-0701-7_8. [15] A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete and Cont. Dyn. Syst., 10 (2004), 289-314. doi: 10.3934/dcds.2004.10.289. [16] A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Translations of Mathematical Monographs, 187, Amer. Math. Society, Providence, Rhode Island, 2000. [17] A. V. Fursikov and A. V. Gorshkov, Certain questions of feedback stabilization for Navier-Stokes equations, Evolution equations and control theory (EECT), 1 (2012), 109-140. doi: 10.3934/eect.2012.1.109. [18] A. V. Fursikov and A. A. Kornev, Feedback stabilization for Navier-Stokes equations: theory and calculations, in Mathematical Aspects of Fluid Mechanics (LMS Lecture Notes Series), 402, Cambridge University Press, (2012), 130-172. [19] M. Krstic, On global stabilization of Burgers' equation by boundary control, Systems & Control Letters, 37 (1999), 123-141. doi: 10.1016/S0167-6911(99)00013-4. [20] J.-P. Raymond, Feedback boundary stabilization of the twodimensional Navier-Stokes equations, SIAM J. Control Optimization, 45 (2006), 709-728. doi: 10.1137/050628726. [21] 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. [22] J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers, Discrete and Continuous Dynamical Systems-A, 27 (2010), 1159-1187. doi: 10.3934/dcds.2010.27.1159. [23] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhauser Verlag AG, Basel, Boston, Berlin, 2009. doi: 10.1007/978-3-7643-8994-9. [24] V. I. Yudovich, Non-stationary flows of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz., 3 (1963), 1032-1066 (Russian).

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##### References:
 [1] V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM: Control, Optimization and Calculus of Variation, 9 (2003), 197-205. doi: 10.1051/cocv:2003009. [2] V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimentional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494. doi: 10.1512/iumj.2004.53.2445. [3] 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. [4] V. Barbu, S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations, SIAM J. Control Optim., 49 (2011), 1454-1478. doi: 10.1137/100785739. [5] J. M. Coron, On null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domains, SIAM J. ControlOptim., 37 (1999), 1874-1896. doi: 10.1137/S036301299834140X. [6] J. M. Coron, Control and Nonlinearity, Math.Surveys and Monographs, 136 AMS, Providence, RI, 2007. [7] A. V. Fursikov, On one semilinear parabolic equation of normal type, in Proceeding volume "Mathematics and life sciences", De Gruyter, 1 (2012), 147-160. [8] 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. [9] A. V. Fursikov, On the normal semilinear parabolic equations corresponding to 3D Navier-Stokes system, in CSMO 2011, IFIP AICT 391 (eds D.Homberg and F.Troltzsch), (2013), 338-347. [10] A. V. Fursikov, On parabolic system of normal type corresponding to 3D Helmholtz system, work in progress. [11] A. V. Fursikov, Stabilizability of quasi linear parabolic equation by feedback boundary control, Sbornik: Mathematics, 192 (2001), 593-639. doi: 10.1070/sm2001v192n04ABEH000560. [12] A. V. Fursikov, Stabilizability of Two-Dimensional Navier-Stokes equations with help of a boundary feedback control, J. of Math. Fluid Mech., 3 (2001), 259-301. doi: 10.1007/PL00000972. [13] A. V. Fursikov, Real process corresponding to 3D-Navier Stokes system and its feedback stabilization from boundary, Amer. Math. Soc. Transl. Series 2, 206, Advances in Math.Sciences-51. PDE M. Vishik seminar (2002), 95-123. [14] A. V. Fursikov, Real processes and realizability of a stabilization method for Navier-Stokes equations by boundary feedback control, Nonlinear Problems in Mathematical Physics and Related Topics II, In Honor of Professor O. A. Ladyzhenskaya, Kluwer/Plenum Publishers, New-York, Boston, Dordrecht, London, Moscow, (2002), 137-177. doi: 10.1007/978-1-4615-0701-7_8. [15] A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete and Cont. Dyn. Syst., 10 (2004), 289-314. doi: 10.3934/dcds.2004.10.289. [16] A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Translations of Mathematical Monographs, 187, Amer. Math. Society, Providence, Rhode Island, 2000. [17] A. V. Fursikov and A. V. Gorshkov, Certain questions of feedback stabilization for Navier-Stokes equations, Evolution equations and control theory (EECT), 1 (2012), 109-140. doi: 10.3934/eect.2012.1.109. [18] A. V. Fursikov and A. A. Kornev, Feedback stabilization for Navier-Stokes equations: theory and calculations, in Mathematical Aspects of Fluid Mechanics (LMS Lecture Notes Series), 402, Cambridge University Press, (2012), 130-172. [19] M. Krstic, On global stabilization of Burgers' equation by boundary control, Systems & Control Letters, 37 (1999), 123-141. doi: 10.1016/S0167-6911(99)00013-4. [20] J.-P. Raymond, Feedback boundary stabilization of the twodimensional Navier-Stokes equations, SIAM J. Control Optimization, 45 (2006), 709-728. doi: 10.1137/050628726. [21] 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. [22] J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers, Discrete and Continuous Dynamical Systems-A, 27 (2010), 1159-1187. doi: 10.3934/dcds.2010.27.1159. [23] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhauser Verlag AG, Basel, Boston, Berlin, 2009. doi: 10.1007/978-3-7643-8994-9. [24] V. I. Yudovich, Non-stationary flows of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz., 3 (1963), 1032-1066 (Russian).
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