Uniform stabilization of 1-d wave equation with anti-damping and delayed control

Journal of the Franklin Institute - Tập 357 - Trang 12473-12494 - 2020
Li Zhang1, Gen Qi Xu1, Hao Chen2
1School of Mathematics, Tianjin University, Tianjin 300350, PR China
2Department of Mechatronical Engineering, Beijing Institute of Technology, Beijing 100081, PR China

Tài liệu tham khảo

Rolewicz, 1970, On controllability of systems of strings, Stud. Math., 36, 105, 10.4064/sm-36-2-105-110 Xu, 2007, Stabilization of string system with linear boundary feedback, Nonlinear Anal., 1, 383 Su, 2017, Boundary stabilization of wave equation with velocity recirculation, IEEE Trans. Autom. Control, 9, 4760, 10.1109/TAC.2017.2688128 Hassine, 2017, Rapid exponential stabilization of a 1-d transmission wave equation with in-domain anti-damping, Asian J. Control, 19, 1, 10.1002/asjc.1509 Xu, 2019, Saturated boundary feedback stabilization of a linear wave equation, SIAM J. Control Optim., 57, 290, 10.1137/15M1034350 Suh, 1980, Use of time-delay actions in the controller design, IEEE Trans. Autom. Control, 25, 600, 10.1109/TAC.1980.1102347 Abdallah, 1993, Delayed-positive feedback can stabilize oscillatory systems, 3106 Zeng, 2020, New insights on stability of sampled-data systems with time-delay, Appl. Math. Comput., 374 Datko, 1988, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26, 697, 10.1137/0326040 Datko, 1986, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24, 152, 10.1137/0324007 Datko, 1991, Two questions concerning the boundary control of certain elastic systems, J. Differ. Equ., 92, 27, 10.1016/0022-0396(91)90062-E Datko, 1997, Two examples of ill-posedness with respect to time delays revisited, IEEE Trans. Autom. Control, 42, 511, 10.1109/9.566660 Zeng, 2019, A generalized free-matrix-based integral inequality for stability analysis of time-varying delay systems, Appl. Math. Comput., 354, 1, 10.1016/j.amc.2019.02.009 Zeng, 2015, New results on stability analysis for systems with discrete distributed delay, Automatica, 60, 189, 10.1016/j.automatica.2015.07.017 Zhang, 2019, Overview of recent advances in stability of linear systems with time-varying delays, IET Control Theory Appl., 13, 1, 10.1049/iet-cta.2018.5188 Prieur, 2019, Feedback stabilization of a 1-d linear reaction-diffusion equation with delay boundary control, IEEE Trans. Autom. Control, 64, 1415, 10.1109/TAC.2018.2849560 Fridman, 2009, Exponential stability of linear distributed parameter systems with time-varying delays, Automatica, 45, 194, 10.1016/j.automatica.2008.06.006 Nicaise, 2009, Stability of the heat and wave equations with boundary time-varying delays, Discret. Contin. Dyn. Syst., 2, 559 Nicaise, 2007, Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks, Netw. Heterogeneous Media, 2, 425, 10.3934/nhm.2007.2.425 Nicaise, 2006, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45, 1561, 10.1137/060648891 Krstic, 2009, Compensating actuator and sensor dynamics governed by diffusion PDEs, Syst. Control Lett., 58, 372, 10.1016/j.sysconle.2009.01.006 Kang, 2017, Boundary control of delayed ODE-heat cascade under actuator saturation, Automatica, 83, 252, 10.1016/j.automatica.2017.06.014 Ahmed-Ali, 2015, Observer design for a class of nonlinear ODE-PDE cascade systems, Syst. Control Lett., 83, 19, 10.1016/j.sysconle.2015.06.003 Zhou, 2015, Stabilization of a second order ODE-heat system coupling at intermediate point, Automatica, 60, 57, 10.1016/j.automatica.2015.06.039 Xu, 2006, Stabilization of wave systems with input delay in the boundary control, ESAIM, 12, 770 Shang, 2012, Stabilization of an Euler-Bernoulli beam with input delay in the boundary control, Syst. Control Lett., 61, 1069, 10.1016/j.sysconle.2012.07.012 Shang, 2015, Dynamic feedback control and exponential stabilization of a compound system, J. Math. Anal. Appl., 422, 858, 10.1016/j.jmaa.2014.09.013 Wang, 2013, Exponential stabilization of 1-d wave equation with input delay, WSEAS Trans. Math., 12, 1001 Xu, 2013, Stabilization of Timoshenko beam system with delay in the boundary control, Int. J. Control, 86, 1165, 10.1080/00207179.2013.787494 Liu, 2013, Exponential stabilization for Timoshenko beam with distributed delay in the boundary control, Abstr. Appl. Anal., 10.1155/2013/726794 Liu, 2015, Exponential stabilization for Timoshenko beam with different delays in the boundary control, IMA J. Math. Control Inf., 10.1093/imamci/dnv036 Han, 2013, Output-based stabilization of Euler-Bernoulli beam with time-delay in boundary input, IMA J. Math. Control Inf., 31, 533, 10.1093/imamci/dnt030 Shang, 2016, Output-based stabilization for a one-dimensional wave equation with distributed input delay in the boundary control, IMA J. Math. Control Inf., 33, 95, 10.1093/imamci/dnu030 Cox, 1994, The rate at which energy decays in a damped string, Commun. Partial Differ. Equ., 19, 213, 10.1080/03605309408821015 Shubov, 1997, Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string, Trans. Am. Math. Soc, 349, 4481, 10.1090/S0002-9947-97-02044-8 Benhassi, 2009, Feedback stabilization of a class of evolution equations with delay, J. Evol. Equ., 9, 103, 10.1007/s00028-009-0004-z Guo, 2012, Exponential stabilization of variable coefficient wave equations in a generic tree with small time-delays in the nodal feedbacks, J. Math. Anal. Appl., 395, 727, 10.1016/j.jmaa.2012.05.079 Shang, 2012, Stability analysis of Euler-Bernoulli beam with input delay in the boundary control, Asian J. Control, 14, 186, 10.1002/asjc.279 Smyshlyaev, 2010, Boundary stabilization of a 1-d wave equation with in-domain antidamping, SIAM J. Control Optim., 48, 4014, 10.1137/080742646 Liu, 2018, Integral-type feedback controller and application to the stabilization of heat equation with boundary input delay, WSEAS Trans. Math., 17, 311 Liu, 2019, Solvability of the nonlocal initial value problem and application to design of controller for heat-equation with delay, J. Math. Study, 52, 127, 10.4208/jms.v52n2.19.02 Pazy, 1983 Miyadera, 1966, On perturbation theory for semi-groups of operators, Tohoku Math. J., 18, 299, 10.2748/tmj/1178243419 Voigt, 1977, On the perturbation theory for strongly continuous semigroups, Math. Ann., 229, 163, 10.1007/BF01351602 Engel, 2000 O’Halloran, 1987, Feedback equivalence of constant linear system, Syst. Control Lett., 8, 241, 10.1016/0167-6911(87)90033-8 Gardner, 1990, Feedback equivalence for general control systems, Syst. Control Lett., 15, 15, 10.1016/0167-6911(90)90039-W