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Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms
Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints
1. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China, China |
References:
[1] |
R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", 2nd edition, (1979). Google Scholar |
[2] |
X. Li and J. Yong, "Optimal Control Theory for Infinite, Dimensional System, (1995). Google Scholar |
[3] |
G. Wang, Pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint,, Nonlinear Anal., 51 (2002), 509.
doi: 10.1016/S0362-546X(01)00843-4. |
[4] |
G. Wang and L. Wang, Maximum principle of state-constrained optimal control governed by fluid dynamic systems,, Nonlinear Anal., 52 (2003), 1911.
doi: 10.1016/S0362-546X(02)00282-1. |
[5] |
H. Liu, Optimal control problems with state constraint governed by Navier-Stokes equations,, Nonlinear Anal., 73 (2010), 3924.
doi: 10.1016/j.na.2010.08.026. |
[6] |
G. Wang, Optimal controls of 3-dimensional Navier-Stokes equations with state constraints,, SIAM J. Control Optim., 41 (2002), 583.
doi: 10.1137/S0363012901385769. |
[7] |
E. Casas, J.-P. Raymond and H. Zidani, Pontryagin's principle for local solutions of control problems with mixed control-state constraints,, SIAM J. Control Optim., 39 (2000), 1182.
|
[8] |
J.-P. Raymond and H. Zidani, Pontryagin's principles for state-constrained control problems governed by semilinear parabolic equations with unbounded controls,, SIAM J. Control Optim., 36 (1998), 1853. Google Scholar |
[9] |
F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey,, Math. Control Relat. Fields, 1 (2011), 267.
doi: 10.3934/mcrf.2011.1.267. |
[10] |
S. W. Hansen and O. Yu Imanuvilov, Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions,, Math. Control Relat. Fields, 1 (2011), 189.
doi: 10.3934/mcrf.2011.1.189. |
[11] |
V. Barbu, The time optimal control of Navier-Stokes equations,, Systems Control Lett., 30 (1997), 93.
doi: 10.1016/S0167-6911(96)00083-7. |
[12] |
L. Baudouin, E. Crépeau and J. Valein, Global Carleman estimate on a network for the wave equation and application to an inverse problem,, Math. Control Relat. Fields, 1 (2011), 307.
doi: 10.3934/mcrf.2011.1.307. |
[13] |
I. Lasiecka, R. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part I: $H^1(\Omega)$-estimates,, J. Inv. Ill-Posed Problems, 12 (2004), 43.
|
[14] |
M. Badra, Global Carleman inequalities for Stokes and penalized Stokes equations,, Math. Control Relat. Fields, 1 (2011), 149.
doi: 10.3934/mcrf.2011.1.149. |
[15] |
S. Ervedoza and E. Zuazua, Observability of heat processes by transmutation without geometric restrictions,, Math. Control Relat. Fields, 1 (2011), 177.
doi: 10.3934/mcrf.2011.1.177. |
[16] |
R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", CBMS-NSF Regional Conference Series in Applied Mathematics, (1983). Google Scholar |
[17] |
H. O. Fattorini and S. Sritharan, Necessary and sufficient conditions for optimal controls in viscous flow problems,, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 211.
doi: 10.1017/S0308210500028444. |
[18] |
E. Casas, J.-P. Raymond and H. Zidani, Optimal control problem governed by semilinear elliptic equations with integral control constraints and pointwise state constraints,, in, 126 (1998), 89.
|
[19] |
L. Cesari, "Optimization, Theory and Applications,", Springer-Verlag, (1983). Google Scholar |
[20] |
V. Barbu, Optimal control of Navier-Stokes equations with periodic inputs,, Nonlinear Anal., 31 (1998), 15.
doi: 10.1016/S0362-546X(96)00306-9. |
[21] |
X. J. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems,, SIAM J. Control Optim., 29 (1991), 895.
doi: 10.1137/0329049. |
show all references
References:
[1] |
R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", 2nd edition, (1979). Google Scholar |
[2] |
X. Li and J. Yong, "Optimal Control Theory for Infinite, Dimensional System, (1995). Google Scholar |
[3] |
G. Wang, Pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint,, Nonlinear Anal., 51 (2002), 509.
doi: 10.1016/S0362-546X(01)00843-4. |
[4] |
G. Wang and L. Wang, Maximum principle of state-constrained optimal control governed by fluid dynamic systems,, Nonlinear Anal., 52 (2003), 1911.
doi: 10.1016/S0362-546X(02)00282-1. |
[5] |
H. Liu, Optimal control problems with state constraint governed by Navier-Stokes equations,, Nonlinear Anal., 73 (2010), 3924.
doi: 10.1016/j.na.2010.08.026. |
[6] |
G. Wang, Optimal controls of 3-dimensional Navier-Stokes equations with state constraints,, SIAM J. Control Optim., 41 (2002), 583.
doi: 10.1137/S0363012901385769. |
[7] |
E. Casas, J.-P. Raymond and H. Zidani, Pontryagin's principle for local solutions of control problems with mixed control-state constraints,, SIAM J. Control Optim., 39 (2000), 1182.
|
[8] |
J.-P. Raymond and H. Zidani, Pontryagin's principles for state-constrained control problems governed by semilinear parabolic equations with unbounded controls,, SIAM J. Control Optim., 36 (1998), 1853. Google Scholar |
[9] |
F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey,, Math. Control Relat. Fields, 1 (2011), 267.
doi: 10.3934/mcrf.2011.1.267. |
[10] |
S. W. Hansen and O. Yu Imanuvilov, Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions,, Math. Control Relat. Fields, 1 (2011), 189.
doi: 10.3934/mcrf.2011.1.189. |
[11] |
V. Barbu, The time optimal control of Navier-Stokes equations,, Systems Control Lett., 30 (1997), 93.
doi: 10.1016/S0167-6911(96)00083-7. |
[12] |
L. Baudouin, E. Crépeau and J. Valein, Global Carleman estimate on a network for the wave equation and application to an inverse problem,, Math. Control Relat. Fields, 1 (2011), 307.
doi: 10.3934/mcrf.2011.1.307. |
[13] |
I. Lasiecka, R. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part I: $H^1(\Omega)$-estimates,, J. Inv. Ill-Posed Problems, 12 (2004), 43.
|
[14] |
M. Badra, Global Carleman inequalities for Stokes and penalized Stokes equations,, Math. Control Relat. Fields, 1 (2011), 149.
doi: 10.3934/mcrf.2011.1.149. |
[15] |
S. Ervedoza and E. Zuazua, Observability of heat processes by transmutation without geometric restrictions,, Math. Control Relat. Fields, 1 (2011), 177.
doi: 10.3934/mcrf.2011.1.177. |
[16] |
R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", CBMS-NSF Regional Conference Series in Applied Mathematics, (1983). Google Scholar |
[17] |
H. O. Fattorini and S. Sritharan, Necessary and sufficient conditions for optimal controls in viscous flow problems,, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 211.
doi: 10.1017/S0308210500028444. |
[18] |
E. Casas, J.-P. Raymond and H. Zidani, Optimal control problem governed by semilinear elliptic equations with integral control constraints and pointwise state constraints,, in, 126 (1998), 89.
|
[19] |
L. Cesari, "Optimization, Theory and Applications,", Springer-Verlag, (1983). Google Scholar |
[20] |
V. Barbu, Optimal control of Navier-Stokes equations with periodic inputs,, Nonlinear Anal., 31 (1998), 15.
doi: 10.1016/S0362-546X(96)00306-9. |
[21] |
X. J. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems,, SIAM J. Control Optim., 29 (1991), 895.
doi: 10.1137/0329049. |
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