scholarly journals Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise

2020 ◽  
Vol 25 (7) ◽  
pp. 2461-2493 ◽  
Author(s):  
Renhai Wang ◽  
◽  
Bixiang Wang ◽  
Keyword(s):  
Wave Equations ◽  
Infinite Dimensional ◽  
Random Dynamics ◽  
Lattice Wave ◽  
Nonlinear Noise

scholarly journals Existence and approximation of attractors for nonlinear coupled lattice wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lianbing She ◽  
Mirelson M. Freitas ◽  
Mauricio S. Vinhote ◽  
Renhai Wang
Keyword(s):  
Asymptotic Behavior ◽  
Global Attractor ◽  
Wave Equations ◽  
Upper Semicontinuity ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Asymptotic Compactness ◽  
Finite Dimensional ◽  
Lattice Wave ◽  
Infinite Lattices

<p style='text-indent:20px;'>This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in <inline-formula><tex-math id="M1">\begin{document}$ E: = \ell^2\times\ell^2\times\ell^2\times\ell^2 $\end{document}</tex-math></inline-formula>. We then prove that the solution semigroup has a unique global attractor in <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula>. We finally prove that this attractor can be approximated in terms of upper semicontinuity of <inline-formula><tex-math id="M3">\begin{document}$ E $\end{document}</tex-math></inline-formula> by a finite-dimensional global attractor of a <inline-formula><tex-math id="M4">\begin{document}$ 2(2n+1) $\end{document}</tex-math></inline-formula>-dimensional truncation system as <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices.</p>

scholarly journals Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations

2016 ◽  
Vol 168 (1-2) ◽  
pp. 717-757 ◽  
Author(s):  
M. Soledad Aronna ◽  
J. Frédéric Bonnans ◽  
Axel Kröner
Keyword(s):  
Optimal Control ◽  
Wave Equations ◽  
Bilinear Systems ◽  
Infinite Dimensional

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>

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 DIFFRACTION OF BLOCH WAVE PACKETS FOR MAXWELL'S EQUATIONS

2013 ◽  
Vol 15 (06) ◽  
pp. 1350040 ◽  
Author(s):  
GRÉGOIRE ALLAIRE ◽  
MARIAPIA PALOMBARO ◽  
JEFFREY RAUCH
Keyword(s):  
Maxwell’S Equations ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Plane Waves ◽  
Approximate Solutions ◽  
Bloch Wave ◽  
System Structure ◽  
Periodic Medium ◽  
Scalar Wave ◽  
Infinite Dimensional

We study, for times of order 1/h, solutions of Maxwell's equations in an [Formula: see text] modulation of an h-periodic medium. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order h. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation. A weak ray average hypothesis guarantees stability. Compared to earlier work on scalar wave equations, the generator is no longer elliptic. Coercivity holds only on the complement of an infinite-dimensional kernel. The system structure requires many innovations.

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