scholarly journals Robustness of global attractors: Abstract framework and application to dissipative wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sergey Dashkovskiy ◽  
Oleksiy Kapustyan
Keyword(s):  
Large Class ◽  
Global Attractor ◽  
Wave Equations ◽  
Global Attractors ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Local Input ◽  
Abstract Framework

<p style='text-indent:20px;'>We establish local input-to-state stability and the asymptotic gain property for a class of infinite-dimensional systems with respect to the global attractor of the respective undisturbed system. We apply our results to a large class of dissipative wave equations with nontrivial global attractors.</p>

Stability of Global Attractors of Impulsive Infinite-Dimensional Systems

2018 ◽  
Vol 70 (1) ◽  
pp. 30-41 ◽  
Author(s):  
O. V. Kapustyan ◽  
M. O. Perestyuk ◽  
I. V. Romanyuk
Keyword(s):  
Global Attractors ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems

The existence of global attractors for a class of nonlinear dissipative evolution equations

2005 ◽  
Vol 135 (5) ◽  
pp. 887-913 ◽  
Author(s):  
Eduardo Arbieto Alarcon ◽  
Rafael José Iorio
Keyword(s):  
Cauchy Problem ◽  
Global Attractor ◽  
Hamiltonian Systems ◽  
Evolution Equations ◽  
Global Attractors ◽  
Infinite Dimensional ◽  
The Cauchy Problem ◽  
Well Posed

In this work we consider the Cauchy problem associated with dissipative perturbations of infinite-dimensional Hamiltonian systems. we describe abstract conditions under which the problem is locally and globally well posed. Moreover, we establish the existence of global attractor. Finally, we present several applications of the theory.

scholarly journals Strong Global Attractors for 3D Wave Equations with Weakly Damping

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Fengjuan Meng
Keyword(s):  
Wave Equation ◽  
Global Attractor ◽  
Wave Equations ◽  
Global Attractors ◽  
Energy Space ◽  
Damped Wave Equation

We consider the existence of the global attractorA1for the 3D weakly damped wave equation. We prove thatA1is compact in(H2(Ω)∩H01(Ω))×H01(Ω)and attracts all bounded subsets of(H2(Ω)∩H01(Ω))×H01(Ω)with respect to the norm of(H2(Ω)∩H01(Ω))×H01(Ω). Furthermore, this attractor coincides with the global attractor in the weak energy spaceH01(Ω)×L2(Ω).

scholarly journals On Global Attractors for a Class of Reaction-Diffusion Equations on Unbounded Domains with Some Strongly Nonlinear Weighted Term

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Jin Zhang ◽  
Chengkui Zhong
Keyword(s):  
Global Attractor ◽  
Equilibrium Points ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Index Theory ◽  
Global Attractors ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Infinite Dimensional ◽  
Strongly Nonlinear

We consider the existence and properties of the global attractor for a class of reaction-diffusion equation∂u/∂t-Δu-u+κ(x)|u|p-2u+f(u)=0,  in  Rn×R+;  u(x,0)=u0(x),  in  Rn. Under some suitable assumptions, we first prove that the problem has a global attractorAinL2(Rn). Then, by using theZ2-index theory, we verify thatAis an infinite dimensional set and it contains infinite distinct pairs of equilibrium points.

Extremum Seeking Feedback With Wave Partial Differential Equation Compensation

2020 ◽  
Vol 143 (4) ◽  
Author(s):  
Tiago Roux Oliveira ◽  
Miroslav Krstic
Keyword(s):  
Wave Equations ◽  
Small Neighborhood ◽  
Extremum Seeking ◽  
Distributed Parameter ◽  
Infinite Dimensions ◽  
Partial Differential ◽  
Actuator Dynamics ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Practical Convergence

Abstract This paper addresses the compensation of wave actuator dynamics in scalar extremum seeking (ES) for static maps. Infinite-dimensional systems described by partial differential equations (PDEs) of wave type have not been considered so far in the literature of ES. A distributed-parameter-based control law using back-stepping approach and Neumann actuation is initially proposed. Local exponential stability as well as practical convergence to an arbitrarily small neighborhood of the unknown extremum point is guaranteed by employing Lyapunov–Krasovskii functionals and averaging theory in infinite dimensions. Thereafter, the extension for wave equations with Dirichlet actuation, antistable wave PDEs as well as the design for the delay-wave PDE cascade are also discussed. Numerical simulations illustrate the theoretical results.

scholarly journals The global attractors and dimensions estimation for the Kirchho type wave equations with nonlinear strongly damped terms

2016 ◽  
Vol 12 (3) ◽  
pp. 6087-6102
Author(s):  
Chengfei Ai ◽  
Huixian Zhu ◽  
Guoguang Lin
Keyword(s):  
Global Attractor ◽  
Wave Equations ◽  
Global Attractors ◽  
Time Behavior ◽  
Priori Estimate ◽  
Type Wave ◽  
Long Time ◽  
Uniqueness Of The Solution ◽  
The Galerkin Method ◽  
Strongly Damped

This paper studies the long time behavior of the solution to the initial boundaryvalue problems for a class of strongly damped Kirchho type wave equations:utt "1ut + j ut jp1 ut + j u jq1 u (kruk2)u = f(x):Firstly, we prove the existence and uniqueness of the solution by priori estimate and the Galerkin method. Then we obtain to the existence of the global attractor. Finally, we consider that the estimation of the upper bounds of Hausdor and fractal dimensionsfor the global attractor is obtained.

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