arXiv subjects

Debanjana Mitra

Publications and source records attributed to Debanjana Mitra.

5 recordsLinked to original sources

arXiv2026

Local controllability of the Cahn-Hilliard-Burgers' equation around certain steady states

In this article we study the local controllability of the one-dimensional Cahn-Hilliard-Navier-Stokes equation, that is Cahn-Hilliard-Burgers' equation, around a certain steady state using a localized interior control acting only in the concentration equation. To do it, we first linearize the nonlinear equation around the steady state. The linearized system turns out to be a system coupled between second order and fourth order parabolic equations and the control acts in the fourth order parabolic equation. The null controllability of the linearized system is obtained by a duality argument proving an observability inequality. To prove the observability inequality, a new Carleman inequality for the coupled system is derived. Next, using the source term method, it is shown that the null controllability of the linearized system with non-homogeneous terms persists provided the non-homogeneous terms satisfy certain estimates in a suitable weighted space. Finally, using a Banach fixed point theorem in a suitable weighted space, the local controllability of the nonlinear system is obtained.

math.OC↗

arXiv2025

LQR based $ω-$stabilization of a heat equation with memory

We consider a heat equation with memory which is defined on a bounded domain in $\mathbb{R}^d$ and is driven by $m$ control inputs acting on the interior of the domain. Our objective is to numerically construct a state feedback controller for this equation such that, for each initial state, the solution of the closed-loop system decays exponentially to zero with a decay rate larger than a given rate $ω>0$, i.e. we want to solve the $ω$-stabilization problem for the heat equation with memory. We first show that the spectrum of the state operator $A$ associated with this equation has an accumulation point at $-ω_0<0$. Given a $ω\in(0,ω_0)$, we show that the $ω$-stabilization problem for the heat equation with memory is solvable provided certain verifiable conditions on the control operator $B$ associated with this equation hold. We then consider an appropriate LQR problem for the heat equation with memory. For each $n\in\mathbb{N}$, we construct finite-dimensional approximations $A_n$ and $B_n$ of $A$ and $B$, respectively, and then show that by solving a corresponding approximation of the LQR problem a feedback operator $K_n$ can be computed such that all the eigenvalues of $A_n + B_n K_n$ have real part less than $-ω$. We prove that $K_n$ for $n$ sufficiently large solves the $ω$-stabilization problem for the heat equation with memory. A crucial and nontrivial step in our proof is establishing the uniform (in $n$) stabilizability of the pair $(A_n+ωI, B_n)$. We have validated our theoretical results numerically using two examples: an 1D example on a unit interval and a 2D example on a square domain.

eess.SY↗

arXiv2023

Feedback stabilization of parabolic coupled system and its numerical study

In the first part of this article, we study feedback stabilization of a parabolic coupled system by using localized interior controls. The system is feedback stabilizable with exponential decay $-ω<0$ for any $ω>0$. A stabilizing control is found in feedback form by solving a suitable algebraic Riccati equation. In the second part, a conforming finite element method is employed to approximate the continuous system by a finite dimensional discrete system. The approximated system is also feedback stabilizable (uniformly) with exponential decay $-ω+ε$, for any $ε>0$ and the feedback control is obtained by solving a discrete algebraic Riccati equation. The error estimate of stabilized solutions as well as stabilizing feedback controls are obtained. We validate the theoretical results by numerical implementations.

math.AP↗

arXiv2022

Some Controllability Results for Linearized Compressible Navier-Stokes System with Maxwell's Law

In this article, we study the control aspects of the one-dimensional compressible Navier-Stokes equations with Maxwell's law linearized around a constant steady state with zero velocity. We consider the linearized system with Dirichlet boundary conditions and with interior controls. We prove that the system is not null controllable at any time using localized controls in density and stress equations and even everywhere control in the velocity equation. However, we show that the system is null controllable at any time if the control acting in the density or stress equation is everywhere in the domain. This is the best possible null controllability result obtained for this system. Next, we show that the system is approximately controllable at large time using localized controls. Thus, our results give a complete understanding of the system in this direction.

math.AP↗

arXiv2018

Null Controllability of the Incompressible Stokes Equations in a 2-D Channel Using Normal Boundary Control

In this paper, we consider the Stokes equations in a two-dimen- sional channel with periodic conditions in the direction of the channel. We establish null controllability of this system using a boundary control which acts on the normal component of the velocity only. We show null controllability of the system, subject to a constraint of zero average, by proving an observability inequality with the help of a Muntz-Szasz Theorem.

math.AP↗