arXiv subjects

Shishu Pal Singh

Publications and source records attributed to Shishu Pal Singh.

4 recordsLinked to original sources

arXiv2026

Finite element theta schemes for the viscous Burgers' equation with nonlinear Neumann boundary feedback control

In this article, we develop a fully discrete numerical scheme for the one-dimensional (1D) and two-dimensional (2D) viscous Burgers equations with nonlinear Neumann boundary feedback control. The temporal discretization employs a $θ$-scheme, while a conforming finite element method is used for the spatial approximation. The existence and uniqueness of the fully discrete solution are established. We further prove that the scheme is unconditionally exponentially stable for $θ\in [1/2, 1]$, thereby ensuring that the stabilization property of the continuous model is retained at the discrete level. In addition, optimal error estimates are obtained for both the state variable and the boundary control inputs in 1D and 2D frameworks. Finally, several numerical experiments are presented to validate our theoretical findings and to demonstrate the effectiveness of the proposed stabilization strategy under varying model parameters.

math.NA↗

arXiv2026

Global stabilization and finite element analysis of the viscous Burgers' equation with memory subject to Neumann boundary feedback control

This paper presents a global stabilization result of the viscous Burgers' equation with the memory term by applying Neumann boundary feedback control laws. We construct suitable feedback control inputs using the control Lyapunov functional and establish stabilization in the $L^{2}, H^{1},$ and $H^{2}$-norms. The existence and uniqueness of the solution are established through the Faedo-Galerkin method. Moreover, we show the global stabilization where the diffusion coefficient $ν$ is unknown. Then, we apply a $C^{0}$-conforming finite element method to the spatial variable while keeping the time variable continuous. Furthermore, we obtain global stabilization of the semi-discrete scheme and optimal error estimates for the state variable in the $L^{\infty}$, $L^{2}$, and $H^{1}$-norms, using the Ritz-Volterra projection. Additionally, error estimates for the feedback control laws are established. Lastly, we present some numerical simulations to demonstrate the theoretical findings.

math.OC↗

arXiv2025

Finite Difference Method for Global Stabilization of the Viscous Burgers' Equation with Nonlinear Neumann Boundary Feedback Control

This article focuses on a nonlinear Neumann boundary feedback control formulation for the viscous Burgers' equation and develops a class of finite difference schemes to achieve global stabilization. The proposed procedure, known as the $θ$-scheme with $θ\in [0,1]$, unifies explicit and implicit time discretizations and is suitable for handling the nonlinear boundary feedback control problem. Using the discrete energy method, we prove that the proposed difference scheme is conditionally stable for $0 \leq θ< \frac{1}{2}$ and unconditionally stable for $θ\geq \frac{1}{2}$. In addition, we establish the exponential stability of the fully discrete solution. The error analysis shows a first-order convergence rate of the state variable in the discrete $L^{2}$-, $H^{1}$-, and $L^{\infty}$-norms for $θ\geq \frac{1}{2}$, while preserving the exponential stability property. A first-order convergence rate for the boundary control inputs is also obtained. Numerical experiments are conducted to validate the theoretical findings and to demonstrate the effectiveness of the method for the inhomogeneous nonlinear Neumann boundary feedback control of the viscous Burgers' equation.

math.NA↗

arXiv2025

Finite Element Analysis for the Chafee-Infante Equation Using Distributed Feedback Control

In this paper, we propose a $ C^0 $-conforming finite element method for the Chafee-Infante equation with a finite-parameter feedback control. We establish error analysis for both the state variable and the control variable for the spatially discretized solution. Furthermore, we employ the backward Euler method for time discretization and discuss the stability analysis of the fully discrete scheme. Additionally, we develop error estimates for both the state variable and the control input in the fully discrete setting. Finally, we verify our theoretical conclusions using some numerical experiments.

math.NA↗