scholarly journals Approximate Controllability of a 3D Nonlinear Stochastic Wave Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Peng Gao
Keyword(s):  
Wave Equation ◽  
Galerkin Method ◽  
Approximate Controllability ◽  
Well Posedness ◽  
Maxwell System ◽  
Stochastic Wave Equation ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method ◽  
The Galerkin Method

We study the well-posedness of a 3D nonlinear stochastic wave equation which derives from the Maxwell system by the Galerkin method. Then we study the approximate controllability of this system by the Hilbert uniqueness method.

scholarly journals Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects

2018 ◽  
Vol 36 (4) ◽  
pp. 1199-1235 ◽  
Author(s):  
Umberto Biccari ◽  
Víctor Hernández-Santamaría
Keyword(s):  
Heat Equation ◽  
Approximate Controllability ◽  
Fractional Laplacian ◽  
Controllability Problem ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
One Dimensional ◽  
Fractional Heat Equation ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method

Abstract We analyse the controllability problem for a one-dimensional heat equation involving the fractional Laplacian $(-d_x^{\,2})^{s}$ on the interval $(-1,1)$. Using classical results and techniques, we show that, acting from an open subset $\omega \subset (-1,1)$, the problem is null-controllable for $s>1/2$ and that for $s\leqslant 1/2$ we only have approximate controllability. Moreover, we deal with the numerical computation of the control employing the penalized Hilbert Uniqueness Method and a finite element scheme for the approximation of the solution to the corresponding elliptic equation. We present several experiments confirming the expected controllability properties.

scholarly journals Exact Neumann boundary controllability for problems of transmission of the wave equation

1999 ◽  
Vol 41 (1) ◽  
pp. 125-139 ◽  
Author(s):  
WEIJIU LIU ◽  
GRAHAM H. WILLIAMS
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Exact Controllability ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Boundary Controllability ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method ◽  
Initial States

Using the Hilbert Uniqueness Method, we study the problem of exact controllability in Neumann boundary conditions for problems of transmission of the wave equation. We prove that this system is exactly controllable for all initial states in L2(Ω)×(H1(Ω))′.

scholarly journals Convergence of Non-autonomous Attractors for Subquintic Weakly Damped Wave Equation

2021 ◽  
Author(s):  
Jakub Banaśkiewicz ◽  
Piotr Kalita
Keyword(s):  
Wave Equation ◽  
Galerkin Method ◽  
Growth Condition ◽  
Global Attractor ◽  
Time Dependent ◽  
Strichartz Estimates ◽  
Damped Wave Equation ◽  
Independent Function ◽  
The Galerkin Method

AbstractWe study the non-autonomous weakly damped wave equation with subquintic growth condition on the nonlinearity. Our main focus is the class of Shatah–Struwe solutions, which satisfy the Strichartz estimates and coincide with the class of solutions obtained by the Galerkin method. For this class we show the existence and smoothness of pullback, uniform, and cocycle attractors and the relations between them. We also prove that these non-autonomous attractors converge upper-semicontinuously to the global attractor for the limit autonomous problem if the time-dependent nonlinearity tends to a time independent function in an appropriate way.

scholarly journals Controllability for a Wave Equation with Moving Boundary

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Lizhi Cui ◽  
Libo Song
Keyword(s):  
Wave Equation ◽  
Dirichlet Boundary ◽  
Moving Boundary ◽  
Energy Estimates ◽  
Boundary Controllability ◽  
One Dimensional ◽  
Elastic String ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method ◽  
Dimensional Wave

We investigate the controllability for a one-dimensional wave equation in domains with moving boundary. This model characterizes small vibrations of a stretched elastic string when one of the two endpoints varies. When the speed of the moving endpoint is less than1-1/e, by Hilbert uniqueness method, sidewise energy estimates method, and multiplier method, we get partial Dirichlet boundary controllability. Moreover, we will give a sharper estimate on controllability time that only depends on the speed of the moving endpoint.

CARLEMAN ESTIMATE AND UNIQUE CONTINUATION PROPERTY FOR THE LINEAR STOCHASTIC KORTEWEG–DE VRIES EQUATION

2014 ◽  
Vol 90 (2) ◽  
pp. 283-294 ◽  
Author(s):  
PENG GAO
Keyword(s):  
Galerkin Method ◽  
Uniqueness Theorem ◽  
Vries Equation ◽  
Unique Continuation ◽  
Carleman Estimate ◽  
Well Posedness ◽  
Unique Continuation Property ◽  
De Vries ◽  
Continuation Property ◽  
The Galerkin Method

AbstractIn this paper, we obtain the well posedness of the linear stochastic Korteweg–de Vries equation by the Galerkin method, and then establish the Carleman estimate, leading to the unique continuation property (UCP) for the linear stochastic Korteweg–de Vries equation. This UCP cannot be obtained from the classical Holmgren uniqueness theorem.

Ritz Legendre Multiwavelet Method for the Damped Generalized Regularized Long-Wave Equation

2011 ◽  
Vol 7 (1) ◽  
Author(s):  
S. A. Yousefi ◽  
Z. Barikbin
Keyword(s):  
Wave Equation ◽  
Galerkin Method ◽  
Numerical Method ◽  
Variable Coefficient ◽  
Algebraic Equations ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
The Galerkin Method

In this paper, a numerical method is proposed to approximate the solution of the nonlinear damped generalized regularized long-wave (DGRLW) equation with a variable coefficient. The method is based upon Ritz Legendre multiwavelet approximations. The properties of Legendre multiwavelet are first presented. These properties together with the Galerkin method are then utilized to reduce the nonlinear DGRLW equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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