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Evolution Equations and Control Theory (EECT)
 

Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation
Pages: 281 - 296, Issue 3, September 2015

doi:10.3934/eect.2015.4.281      Abstract        References        Full text (450.4K)           Related Articles

Peng Gao - School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China (email)

1 J. P. Aubin, L'analyse Non Linéaire et ses Motivations Économiques, Masson, Paris, 1984.       
2 O. Bodart, M. Gonzalez-Burgos and R. Pérez-García, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Comm. Partial Diff. Eq., 29 (2004), 1017-1050.       
3 E. Cerpa and A. Mercado, Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation, J. Differential Equations, 250 (2011), 2024-2044.       
4 L. H. Chen and H. C. Chang, Nonlinear waves on liquid film surfaces-II. Bifurcation analyses of the long-wave equation, Chem. Eng. Sci., 41 (1986), 2477-2486.
5 M. Chen, Null controllability with constraints on the state for the linear Korteweg-de Vries equation, Archiv der Mathematik., 104 (2015), 189-199.       
6 P. Collet, J. P. Eckmann, H. Epstein and J. Stubbe, A global attracting set for the Kuramoto-Sivashinsky equation, Comm. Math. Phys., 152 (1993), 203-214.       
7 C. Foias, B. Nicolaenko, G. R. Sell and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl., 67 (1988), 197-226.       
8 P. Gao, Insensitizing controls for the Cahn-Hilliard type equation, Electron. J. Qual. Theory Differ. Equ, 35 (2014), 1-22.       
9 P. Gao, A new global Carleman estimate for the one-dimensional Kuramoto-Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem, Nonlinear Anal., 117 (2015), 133-147.       
10 A. Gonzalez and A. Castellanos, Nonlinear electrohydrodynamic waves on films falling down an inclined plane, Phys. Rev. E., 53 (1996), 3573-3578.
11 J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Comm. Pure Appl. Math., 47 (1994), 293-306.       
12 P. G. Meléndez, Lipschitz stability in an inverse problem for the main coefficient of a Kuramoto-Sivashinsky type equation, J. Math. Anal. Appl., 408 (2013), 275-290.       
13 A. P. Hooper and R. Grimshaw, Nonlinear instability at the interface between two viscous fluids, Phys. Fluids, 28 (1985), 37-245.
14 M. S. Jolly, I. G. Kevrekidis and E. S. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Phys. D, 44 (1990), 38-60.       
15 Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems, Theor. Phys., 54 (1975), 687-699.
16 Y. Kuramoto, Diffusion-induced chaos in reaction systems, Suppl. Prog. Theor. Phys, 64 (1978), 346-367.
17 Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369.
18 C. Louis-Rose, A null controllability problem with a finite number of constraints on the normal derivative for the semilinear heat equation, Electron. J. Qual. Theory Differ. Equ., 95 (2012), 1-34.       
19 J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971.       
20 J. L. Lions, Sentinelles Pour Les Systèmes Distribués à Données Incomplètes, Masson, Paris, 1992.       
21 R. E. Laquey, S. M. Mahajan, P. H. Rutherford and W. M. Tang, Nonlinear saturation of the trapped-ion mode, Phys. Rev. Lett., 34 (1975), 391-394.
22 G. M. Mophou, Null controllability with constraints on the state for nonlinear heat equations, Forum Math., 23 (2011), 285-319.       
23 G. M. Mophou and O. Nakoulima, Null controllability with constraints on the state for the semilinear heat equation, J. Optim. Theory Appl., 143 (2009), 539-565.       
24 O. Nakoulima, Optimal control for distributed systems subject to null-controllability. Application to discriminating sentinels, ESAIM Control Optim. Calc. Var., 13 (2007), 623-638.       
25 B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Diff. Eq., 14 (1989), 245-297.       
26 S. Somdouda and G. M. Mophou, Null controllability with constraints on the state for the age-dependent linear population dynamics problem, Adv. Differ. Equ. Control Process., 10 (2012), 113-130.       
27 J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.       
28 G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-I Derivation of basic equations, Acta Astronaut., 4 (1977), 1177-1206.       
29 R. Temam and X. Wang, Estimates on the lowest dimension of inertial manifolds for the Kuramoto-Sivashinsky equation in the general case, Differential Integral Equations, 7 (1994), 1095-1108.       
30 Z. C. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation, Taiwanese J. Math., 16 (2012), 1991-2017.       

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