arXiv subjects

Sudeep Kundu

Publications and source records attributed to Sudeep Kundu.

12 recordsLinked to original sources

arXiv2026

Penalty-Based Feedback Control and Finite Element Analysis for the Stabilization of Nonlinear Reaction-Diffusion Equations

In this work, first we employ a penalization technique to analyze a Dirichlet boundary feedback control problem pertaining to reaction-diffusion equation. We establish the stabilization result of the equivalent Robin problem in the $H^{2}$-norm with respect to the penalty parameter. Furthermore, we prove that the solution of the penalized control problem converges to the corresponding solution of the Dirichlet boundary feedback control problem as the penalty parameter $ε$ approaches zero. A $C^{0}$-conforming finite element method is applied to this problem for the spatial variable while keeping the time variable continuous. We discuss the stabilization of the semi-discrete scheme for the penalized control problem and present an error analysis of its solution. Finally, we validate our theoretical findings through numerical experiments including showing that penalized solution converges to the original solution.

math.NA↗

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↗

arXiv2020

Global Stabilization of Two Dimensional Viscous Burgers' Equation by Nonlinear Neumann Boundary Feedback Control and its Finite Element Analysis

In this article, global stabilization results for the two dimensional (2D) 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 $C^0$-conforming finite element method in spatial direction, optimal error estimates in $L^\infty(L^2)$ and in $L^\infty(H^1)$- 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.

math.NA↗

arXiv2020

Policy iteration for Hamilton-Jacobi-Bellman equations with control constraints

Policy iteration is a widely used technique to solve the Hamilton Jacobi Bellman (HJB) equation, which arises from nonlinear optimal feedback control theory. Its convergence analysis has attracted much attention in the unconstrained case. Here we analyze the case with control constraints both for the HJB equations which arise in deterministic and in stochastic control cases. The linear equations in each iteration step are solved by an implicit upwind scheme. Numerical examples are conducted to solve the HJB equation with control constraints and comparisons are shown with the unconstrained cases.

math.OC↗

arXiv2019

Global Stabilization of 2D Forced Viscous Burgers' Equation Around Nonconstant Steady State Solution by Nonlinear Neumann Boundary Feedback Control:Theory and Finite Element Analysis

Global stabilization of viscous Burgers' equation around constant steady state solution has been discussed in the literature. The main objective of this paper is to show global stabilization results for the 2D forced viscous Burgers' equation around a nonconstant steady state solution using nonlinear Neumann boundary feedback control law, under some smallness condition on that steady state solution. On discretizing in space using $C^0$ piecewise linear elements keeping time variable continuous, a semidiscrete scheme is obtained. Moreover, global stabilization results for the semidiscrete solution and optimal error estimates for the state variable in $L^\infty(L^2)$ and $L^\infty(H^1)$-norms are derived. Further, optimal convergence result is established for the boundary feedback control law. All our results in this paper preserve exponential stabilization property. Finally, some numerical experiments are documented to confirm our theoretical findings.

math.OC↗

arXiv2019

Robust feedback control of nonlinear PDEs by numerical approximation of high-dimensional Hamilton-Jacobi-Isaacs equations

We propose an approach for the synthesis of robust and optimal feedback controllers for nonlinear PDEs. Our approach considers the approximation of infinite-dimensional control systems by a pseudospectral collocation method, leading to high-dimensional nonlinear dynamics. For the reduced-order model, we construct a robust feedback control based on the $\cH_{\infty}$ control method, which requires the solution of an associated high-dimensional Hamilton-Jacobi-Isaacs nonlinear PDE. The dimensionality of the Isaacs PDE is tackled by means of a separable representation of the control system, and a polynomial approximation ansatz for the corresponding value function. Our method proves to be effective for the robust stabilization of nonlinear dynamics up to dimension $d\approx 12$. We assess the robustness and optimality features of our design over a class of nonlinear parabolic PDEs, including nonlinear advection and reaction terms. The proposed design yields a feedback controller achieving optimal stabilization and disturbance rejection properties, along with providing a modelling framework for the robust control of PDEs under parametric uncertainties.

math.OC↗

arXiv2018

Stabilization of Kelvin-Voigt viscoelastic Fuid Fow model

In this article, stabilization result for the viscoelastic fluid flow problem governed by Kelvin-Voigt model, that is, convergence of the unsteady solution to a steady state solution is proved under the assumption that linearized self-adjoint steady state eigenvalue problem has a minimal positive eigenvalue. Both power and exponential convergence results are derived under various conditions on the forcing function. It is shown that results are valid uniformly in the time relaxation or some times called regularization parameter $κ$ as $κ\to 0$, which in turn, establishes results for the Navier-Stokes system.

math.NA↗

arXiv2018

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

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 $L^\infty(L^2)$, $L^\infty(H^1)$ and $L^\infty(L^\infty)$-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.

math.NA↗

arXiv2015

On Kirchhoff's Model of Parabolic Type

In this paper, existence of a strong global solution for all finite time is derived for the Kirchhoff's model of parabolic type. Based on exponential weight function, some new regularity results which reflect the exponential decay property are obtained for the exact solution. For the related dynamics, existence of a global attractor is shown to hold for the problem, when the non- homogeneous forcing function is either independent of time or in $L^{\infty}(L^2)$. With finite element Galerkin method applied in spatial direction keeping time variable continuous, a semidiscrete scheme is analyzed and it is, further, established that the semi-discrete system has a global discrete attractor. Optimal error estimates in $L^{\infty}(H^1_0)$-norm are derived which are valid uniformly in time. Further, based on a Backward Euler method, a completely discrete scheme is developed and error estimates are derived. It is further observed that in case $f = 0$ or $f =O(e^{-γ_0 t})$ with $γ_0 > 0,$ the discrete solutions and also error estimates decay exponentially. Finally, some numerical experiments are discussed which confirm our theoretical findings.

math.NA↗