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

2019 ◽  
Vol 81 (2) ◽  
pp. 845-880 ◽  
Author(s):  
Sudeep Kundu ◽  
Amiya Kumar Pani
Keyword(s):  
Finite Element ◽  
Feedback Control ◽  
Error Analysis ◽  
Nonlinear Boundary ◽  
Global Stabilization ◽  
Control Laws ◽  
Boundary Feedback Control ◽  
Burgers Type Equations ◽  
Element Error ◽  
Nonlinear Boundary Feedback

scholarly journals Global Stabilization of Two Dimensional Viscous Burgers’ Equation by Nonlinear Neumann Boundary Feedback Control and Its Finite Element Analysis

2020 ◽  
Vol 84 (3) ◽  
Author(s):  
Sudeep Kundu ◽  
Amiya Kumar Pani
Keyword(s):  
Finite Element ◽  
Feedback Control ◽  
Burgers Equation ◽  
Neumann Boundary ◽  
Control Law ◽  
Global Stabilization ◽  
Two Dimensional ◽  
Spatial Direction ◽  
Boundary Feedback Control ◽  
Boundary Feedback

Abstract In this article, global stabilization results for the two dimensional viscous Burgers’ equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, applying $$C^0$$ C 0 -conforming finite element method in spatial direction, optimal error estimates in $$L^\infty (L^2)$$ L ∞ ( L 2 ) and in $$L^\infty (H^1)$$ L ∞ ( H 1 ) -norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, several numerical experiments are conducted to confirm our theoretical findings.

scholarly journals Yamamoto's principle and its applications to precise finite element error analysis

2003 ◽  
Vol 152 (1-2) ◽  
pp. 507-532 ◽  
Author(s):  
Takuya Tsuchiya ◽  
Kazuki Yoshida ◽  
Sae Ishioka
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Element Error

Stabilization of an Axially Moving String by Nonlinear Boundary Feedback

1999 ◽  
Vol 121 (1) ◽  
pp. 117-121 ◽  
Author(s):  
Rong-Fong Fung ◽  
Jinn-Wen Wu ◽  
Sheng-Luong Wu
Keyword(s):  
Nonlinear Boundary ◽  
Mechanical Energy ◽  
Control Laws ◽  
Axially Moving String ◽  
Boundary Feedback ◽  
Boundary Velocity ◽  
Total Mechanical Energy ◽  
Axially Moving ◽  
The Right ◽  
Nonlinear Boundary Feedback

In this paper, we consider the system modeled by an axially moving string and a mass-damper-spring (MDS) controller, applied at the right-hand side (RHS) boundary of the string. We are concerned with the nonlinear string and the effect of the control mechanism. We stabilize the system through a proposed boundary velocity feedback control law. Linear and nonlinear control laws through this controller are proposed. In this paper, we find that a linear boundary feedback caused the total mechanical energy of the system to decay an asymptotically, but it fails for an exponential decay. However, a nonlinear boundary feedback controller can stabilize the system exponentially. The asymptotic and exponential stability are verified.

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