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On the free boundary for quenching type parabolic problems via local energy methods
Stabilization of the simplest normal parabolic equation
1. | Department of Mechanics & Mathematics, Moscow State University, Moscow 119991 |
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). |
show all references
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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