The stabilization of wave equations with moving boundary

In this paper, we consider the stabilization of wave equations with moving boundary. First, we show the solution behaviour of wave equation with Neumann boundary conditions, that is, the energy of wave equation with mixed boundary conditions may decrease, increase or conserve depending on the different range of parameter. Second, we prove the wellposedness and stabilization for the wave equation with time delay and moving boundary.

[1]  M. Fowler,et al.  Function Spaces , 2022 .

[2]  Zi-Qing Tian,et al.  Stabilization of one-dimensional wave equation by non-collocated boundary feedback , 2016, Eur. J. Control.

[3]  L. Libelo,et al.  Solution of the Wave Equation for Moving Boundaries , 1968 .

[4]  Claude Bardos,et al.  Control and Stabilization for the Wave Equation, Part III: Domain with Moving Boundary , 1981 .

[5]  Serge Nicaise,et al.  Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks , 2006, SIAM J. Control. Optim..

[6]  P. Duclos,et al.  STABILITY AND INSTABILITY OF THE WAVE EQUATION SOLUTIONS IN A PULSATING DOMAIN , 1998 .

[7]  Vilmos Komornik,et al.  Rapid boundary stabilization of the wave equation , 1991 .

[8]  Zhen-Hu Ning,et al.  Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback , 2010 .

[9]  Martin Gugat,et al.  Optimal boundary feedback stabilization of a string with moving boundary , 2007, IMA J. Math. Control. Inf..

[10]  Abdelmouhcene Sengouga.  Observability of the 1-D Wave Equation with Mixed Boundary Conditions in a Non-cylindrical Domain , 2018 .

[11]  Stabilization of wave dynamics by moving boundary , 2018 .

[12]  P. K. C. Wang,et al.  Stabilization and control of distributed systems with time-dependent spatial domains , 1990 .

[13]  Xiaoyu Fu.  Stabilization of hyperbolic equations with mixed boundary conditions , 2015 .

[14]  An Example of Pure Stability for the Wave Equation with Moving Boundary , 1998 .

[15]  Kaïs Ammari,et al.  Stabilization of the wave equation with moving boundary , 2018, Eur. J. Control.

[17]  Bernhard H. Haak,et al.  Exact Observability of a 1-Dimensional Wave Equation on a Noncylindrical Domain , 2019, SIAM J. Control. Optim..

[18]  Hang Gao,et al.  Exact controllability for a one-dimensional wave equation in non-cylindrical domains☆ , 2013 .

[19]  J. Lagnese,et al.  An example of the effect of time delays in boundary feedback stabilization of wave equations , 1985, 1985 24th IEEE Conference on Decision and Control.

[20]  E. Zuazua,et al.  The rate at which energy decays in a string damped at one end , 1995 .

[21]  Genqi Xu,et al.  Stabilization of wave systems with input delay in the boundary control , 2006 .

[22]  V. Komornik.  Exact Controllability and Stabilization: The Multiplier Method , 1995 .