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Carleman Estimates and null controllability of coupled degenerate systems
Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term
1. | Laboratoire Jacques-Louis Lions, U.M.R C.N.R.S. 7598, Université Pierre et Marie Curie, Boite courrier 187, 75252 Paris Cedex 05 |
2. | Université de Carthage, Institut Préparatoire aux Etudes Scientifiques et Techniques, B.P. 51, 2070 La Marsa, Tunisia |
References:
[1] |
F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert space, SIAM J. Control Optim., 38 (2000), 1102-1119.
doi: 10.1137/S0363012998335802. |
[2] |
F. Alvarez and H. Attouch, Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria, ESAIM, Control Optim. Calc. Var., 6 (2001), 539-552.
doi: 10.1051/cocv:2001100. |
[3] |
H. Attouch, X. Goudou and P. Redont, The heavy ball with friction method, I. The continuous dynamical system: Global exploration of the local minima of a real-valued function asymptotic by analysis of a dissipative dynamical system, Commun. Contemp. Math., 2 (2000), 1-34.
doi: 10.1142/S0219199700000025. |
[4] |
A. Cabot and P. Frankel, Asymptotics for some semilinear hyperbolic equations with non-autonomous damping, J. Differential Equations, 252 (2012), 294-322.
doi: 10.1016/j.jde.2011.09.012. |
[5] |
A. Cabot, H. Engler and S. Gadat, On the long time behavior of second order differential equations with asymptotically small dissipation, Trans. Amer. Math. Soc., 361 (2009), 5983-6017.
doi: 10.1090/S0002-9947-09-04785-0. |
[6] |
L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dyn. Differ. Equations, 20 (2008), 643-652.
doi: 10.1007/s10884-007-9099-5. |
[7] |
D. D'Acunto and K. Kurdyka, Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials, Ann. Polon. Math., 87 (2005), 51-61.
doi: 10.4064/ap87-0-5. |
[8] |
A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Masson, Paris, 1991, xii+132 pp. |
[9] |
A. Haraux, Positively homogeneous functions and the Łojasiewicz gradient inequality, Ann. Polon. Math., 87 (2005), 165-174.
doi: 10.4064/ap87-0-13. |
[10] |
A. Haraux, Some applications of the Łojasiewicz gradient inequality, Comm. Pure and Applied Analysis, 11 (2012), 2417-2427.
doi: 10.3934/cpaa.2012.11.2417. |
[11] |
A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differential Equations, 144 (1998), 313-320.
doi: 10.1006/jdeq.1997.3393. |
[12] |
A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9 (1999), 95-124.
doi: 10.1007/s005260050133. |
[13] |
A. Haraux and M. A. Jendoubi, Decay estimates of the solutions to some evolution problems with an analytic nonlinearity, Asymptotic Analysis, 26 (2001), 21-36. |
[14] |
A. Haraux, P. Martinez and J. Vancostenoble, Asymptotic stability for intermittently controlled second-order evolution equations, Asymptotic Analysis, 43 (2006), 2089–-2108. .
doi: 10.1137/S0363012903436569. |
[15] |
A. Haraux and T. S. Pham, On the Łojasiewicz exponents of quasi-homogeneous functions, Universitatis Iagellonicae Acta Mathematica, 49 (2011), 45-57. |
[16] |
S. Łojasiewicz, Ensembles semi-analytiques, I. H. E. S. Notes, (1965). |
show all references
References:
[1] |
F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert space, SIAM J. Control Optim., 38 (2000), 1102-1119.
doi: 10.1137/S0363012998335802. |
[2] |
F. Alvarez and H. Attouch, Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria, ESAIM, Control Optim. Calc. Var., 6 (2001), 539-552.
doi: 10.1051/cocv:2001100. |
[3] |
H. Attouch, X. Goudou and P. Redont, The heavy ball with friction method, I. The continuous dynamical system: Global exploration of the local minima of a real-valued function asymptotic by analysis of a dissipative dynamical system, Commun. Contemp. Math., 2 (2000), 1-34.
doi: 10.1142/S0219199700000025. |
[4] |
A. Cabot and P. Frankel, Asymptotics for some semilinear hyperbolic equations with non-autonomous damping, J. Differential Equations, 252 (2012), 294-322.
doi: 10.1016/j.jde.2011.09.012. |
[5] |
A. Cabot, H. Engler and S. Gadat, On the long time behavior of second order differential equations with asymptotically small dissipation, Trans. Amer. Math. Soc., 361 (2009), 5983-6017.
doi: 10.1090/S0002-9947-09-04785-0. |
[6] |
L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dyn. Differ. Equations, 20 (2008), 643-652.
doi: 10.1007/s10884-007-9099-5. |
[7] |
D. D'Acunto and K. Kurdyka, Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials, Ann. Polon. Math., 87 (2005), 51-61.
doi: 10.4064/ap87-0-5. |
[8] |
A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Masson, Paris, 1991, xii+132 pp. |
[9] |
A. Haraux, Positively homogeneous functions and the Łojasiewicz gradient inequality, Ann. Polon. Math., 87 (2005), 165-174.
doi: 10.4064/ap87-0-13. |
[10] |
A. Haraux, Some applications of the Łojasiewicz gradient inequality, Comm. Pure and Applied Analysis, 11 (2012), 2417-2427.
doi: 10.3934/cpaa.2012.11.2417. |
[11] |
A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differential Equations, 144 (1998), 313-320.
doi: 10.1006/jdeq.1997.3393. |
[12] |
A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9 (1999), 95-124.
doi: 10.1007/s005260050133. |
[13] |
A. Haraux and M. A. Jendoubi, Decay estimates of the solutions to some evolution problems with an analytic nonlinearity, Asymptotic Analysis, 26 (2001), 21-36. |
[14] |
A. Haraux, P. Martinez and J. Vancostenoble, Asymptotic stability for intermittently controlled second-order evolution equations, Asymptotic Analysis, 43 (2006), 2089–-2108. .
doi: 10.1137/S0363012903436569. |
[15] |
A. Haraux and T. S. Pham, On the Łojasiewicz exponents of quasi-homogeneous functions, Universitatis Iagellonicae Acta Mathematica, 49 (2011), 45-57. |
[16] |
S. Łojasiewicz, Ensembles semi-analytiques, I. H. E. S. Notes, (1965). |
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