Distributed Feedback Control of the Benjamin-Bona-Mahony-Burgers Equation by a Reduced-Order Model

Piao, Guang-Ri; Lee, Hyung-Chun

doi:10.4208/eajam.210214.061214a

Cited by 4 publications

(1 citation statement)

References 25 publications

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“…For stabilization of the BBM-B equation, the authors in [10] have shown global stabilization results corresponding to µ = 1 with zero Dirichlet boundary condition at one end and Neumann boundary control on the other end. Using a reduced order model, distributed feedback control for the BBM-B equation is discussed in [21]. Also, quadratic B-spline finite element method followed by linear quadratic regulator theory to design feedback control, is used to stabilize in [22] without any convergence analysis.…”

Section: Introductionmentioning

confidence: 99%

Global Stabilization of BBM–Burgers’ Type Equations by Nonlinear Boundary Feedback Control Laws: Theory and Finite Element Error Analysis

Kundu

Pani

2019

J Sci Comput

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In this article, global stabilization results for the Benjamin-Bona-Mahony-Burgers' (BBM-B) type equations are obtained using nonlinear Neumann boundary feedback control laws. Based on the C 0 -conforming finite element method, global stabilization results for the semidiscrete solution are also discussed. Optimal error estimates in L ∞ (L 2 ), L ∞ (H 1 ) and L ∞ (L ∞ )-norms for the state variable are derived, which preserve exponential stabilization property. Moreover, for the first time in the literature, superconvergence results for the boundary feedback control laws are established. Finally, several numerical experiments are conducted to confirm our theoretical findings.

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Section: Introductionmentioning

confidence: 99%

Global Stabilization of BBM–Burgers’ Type Equations by Nonlinear Boundary Feedback Control Laws: Theory and Finite Element Error Analysis

Kundu

Pani

2019

J Sci Comput

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A New Highly Accurate Numerical Scheme for Benjamin–Bona–Mahony–Burgers Equation Describing Small Amplitude Long Wave Propagation

2023

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Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh

Gao

et al. 2017

Discrete Dynamics in Nature and Society

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A finite element model is proposed for the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation with a high-order dissipative term; the scheme is based on adaptive moving meshes. The model can be applied to the equations with spatial-time mixed derivatives and high-order derivative terms. In this scheme, new variables are needed to make the equation become a coupled system, and then the linear finite element method is used to discretize the spatial derivative and the fifth-order Radau IIA method is used to discretize the time derivative. The simulations of 1D and 2D BBM-Burgers equations with high-order dissipative terms are presented in numerical examples. The numerical results show that the method keeps a second-order convergence in space and provides a smaller error than that based on the fixed mesh, which demonstrates the effectiveness and feasibility of the finite element method based on the moving mesh. We also study the effect of the dissipative terms with different coefficients in the equation; by numerical simulations, we find that the dissipative term plays a more important role than in dissipation.

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Distributed Feedback Control of the Benjamin-Bona-Mahony-Burgers Equation by a Reduced-Order Model

Cited by 4 publications

References 25 publications

Global Stabilization of BBM–Burgers’ Type Equations by Nonlinear Boundary Feedback Control Laws: Theory and Finite Element Error Analysis

Global Stabilization of BBM–Burgers’ Type Equations by Nonlinear Boundary Feedback Control Laws: Theory and Finite Element Error Analysis

A New Highly Accurate Numerical Scheme for Benjamin–Bona–Mahony–Burgers Equation Describing Small Amplitude Long Wave Propagation

Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh

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