On conditions for asymptotic stability of dissipative infinite-dimensional systems with intermittent damping

Journal of Differential Equations - Tập 252 - Trang 5569-5593 - 2012
Falk M. Hante1, Mario Sigalotti2, Marius Tucsnak3,4
1Mathematics Center of Heidelberg (MATCH), and Interdisciplinary Center for Scientific Computing (IWR), Im Neuenheimer Feld 368, 69120 Heidelberg, Germany
2INRIA Saclay–Île-de-France, Team GECO, and CMAP, UMR 7641, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
3Institut Élie Cartan (IECN), UMR 7502, BP 239, Vandœuvre-lès-Nancy 54506, France
4CORIDA, INRIA Nancy–Grand Est., France

Tài liệu tham khảo

Aeyels, 1998, A new asymptotic stability criterion for nonlinear time-variant differential equations, IEEE Trans. Automat. Control, 43, 968, 10.1109/9.701102 Anderson, 1986 Ball, 1977, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 63, 370 Ball, 1982, Controllability for distributed bilinear systems, SIAM J. Control Optim., 20, 575, 10.1137/0320042 Ball, 1979, Feedback stabilization of distributed semilinear control systems, Appl. Math. Optim., 5, 169, 10.1007/BF01442552 Benavides Guzmán, 2003, Energy decay estimates for the damped plate equation with a local degenerated dissipation, Systems Control Lett., 48, 191, 10.1016/S0167-6911(02)00264-5 Chaillet, 2008, Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases, Math. Control Signals Systems, 20, 135, 10.1007/s00498-008-0024-1 Chitour, 2010, On the stabilization of persistently excited linear systems, SIAM J. Control Optim., 48, 4032, 10.1137/080737812 Fattorini, 2005, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems, vol. 201 Fragnelli, 2008, Stability of solutions for some classes of nonlinear damped wave equations, SIAM J. Control Optim., 47, 2520, 10.1137/070689735 Guo, 2005, Regularity of a Schrödinger equation with Dirichlet control and colocated observation, Systems Control Lett., 54, 1135, 10.1016/j.sysconle.2005.04.008 Hante, 2011, Converse Lyapunov theorems for switched systems in Banach and Hilbert spaces, SIAM J. Control Optim., 49, 752, 10.1137/100801561 Haraux, 1989, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Port. Math., 46, 245 Haraux, 2005, Asymptotic stability for intermittently controlled second-order evolution equations, SIAM J. Control Optim., 43, 2089, 10.1137/S0363012903436569 Hatvani, 1996, Integral conditions on the asymptotic stability for the damped linear oscillator with small damping, Proc. Amer. Math. Soc., 124, 415, 10.1090/S0002-9939-96-03266-2 Hatvani, 1995, A necessary and sufficient condition for the asymptotic stability of the damped oscillator, J. Differential Equations, 119, 209, 10.1006/jdeq.1995.1087 Hörmander, 2003, The Analysis of Linear Partial Differential Operators. I: Distribution Theory and Fourier Analysis, 10.1007/978-3-642-61497-2 Martinez, 1999, Decay of solutions of the wave equation with a local highly degenerate dissipation, Asymptot. Anal., 19, 1 Martinez, 2002, Stabilization of the wave equation by on–off and positive–negative feedbacks, ESAIM Control Optim. Calc. Var., 7, 335, 10.1051/cocv:2002015 Mizel, 1997, An abstract bang-bang principle and time-optimal boundary control of the heat equation, SIAM J. Control Optim., 35, 1204, 10.1137/S0363012996265470 Morgan, 1977, On the stability of nonautonomous differential equations x˙=(a+b(t))x with skew-symmetric matrix b(t), SIAM J. Control Optim., 15, 163, 10.1137/0315013 Nakao, 1996, Decay of solutions of the wave equation with a local degenerate dissipation, Israel J. Math., 95, 25, 10.1007/BF02761033 Pazy, 1983, Semigroups of Linear Operators and Applications to Partial Differential Equations, 10.1007/978-1-4612-5561-1 K.D. Phung, G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, preprint, 2011. Pucci, 1996, Asymptotic stability for nonautonomous dissipative wave systems, Comm. Pure Appl. Math., 49, 177, 10.1002/(SICI)1097-0312(199602)49:2<177::AID-CPA3>3.0.CO;2-B Reifler, 1994, Unique continuation of some dispersive waves, Comm. Partial Differential Equations, 19, 1203, 10.1080/03605309408821051 Seidman, 1986, The coefficient map for certain exponential sums, Indag. Math. (N.S.), 48, 463, 10.1016/1385-7258(86)90031-4 Seidman, 1988, How violent are fast controls?, Math. Control Signals Systems, 1, 89, 10.1007/BF02551238 Slemrod, 1974, A note on complete controllability and stabilizability for linear control systems in Hilbert space, SIAM J. Control, 12, 500, 10.1137/0312038 Smith, 1961, Asymptotic stability of x″+a(t)x′+x=0, Quart. J. Math. Oxford Ser. (2), 12, 123, 10.1093/qmath/12.1.123 Tcheugoué Tébou, 1998, On the decay estimates for the wave equation with a local degenerate or nondegenerate dissipation, Port. Math., 55, 293 Tenenbaum, 2007, New blow-up rates for fast controls of Schrödinger and heat equations, J. Differential Equations, 243, 70, 10.1016/j.jde.2007.06.019 Tucsnak, 2009, Observation and Control for Operator Semigroups, 10.1007/978-3-7643-8994-9 Wang, 2008, L∞-null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optim., 47, 1701, 10.1137/060678191 Weiss, 1994, Transfer functions of regular linear systems, Part I: Characterizations of regularity, Trans. Amer. Math. Soc., 342, 827 Zuazua, 2003, Remarks on the controllability of the Schrödinger equation, vol. 33, 193 Zygmund, 2002, Trigonometric Series, vols. I, II