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29 results on '"Mitra, Debanjana"'

1. Feedback stabilization of parabolic coupled system and its numerical study

Author

Akram, Wasim, Mitra, Debanjana, Nataraj, Neela, and Ramaswamy, Mythily

Subjects
Mathematics - Analysis of PDEs, Mathematics - Numerical Analysis, 35K20, 93D15, 65M12, 65M22, 65M60
Abstract

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 $-\omega<0$ for any $\omega>0$. 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 $-\omega+\epsilon$, for any $\epsilon>0$ 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.

Published
2023

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

Author

Ahamed, Sakil and Mitra, Debanjana

Subjects
Mathematics - Analysis of PDEs, 35Q30, 93B05, 93B07, 35Q35
Abstract

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.

Published
2022

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

Author

Chowdhury, Shirshendu, Mitra, Debanjana, and Renardy, Michael

Subjects
Mathematics - Analysis of PDEs, Primary: 93C20, 93B05, Secondary: 76D05
Abstract

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., Comment: Accepted in Evolution Equations and Control Theory, 2018

Published
2018

7. Feedback stabilization of a parabolic coupled system and its numerical study.

Author

Akram, Wasim, Mitra, Debanjana, Nataraj, Neela, and Ramaswamy, Mythily

Subjects
RICCATI equation, ALGEBRAIC equations, FINITE element method, DISCRETE systems
Abstract

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 $ -\omega<0 $ for any $ \omega>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 $ -\omega+\epsilon $, for any $ \epsilon>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. [ABSTRACT FROM AUTHOR]

Published
2024
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16. Lack of complete stabilizability of some coupled ode-parabolic systems

Author

Maity, Debayan, Mitra, Debanjana, Ramaswamy, Mythily, Center for Applicable Mathematics [Bangalore] (TIFR-CAM), Tata Institute for Fundamental Research (TIFR), Department of Mathematics, Indian Institute of Technology Bombay, Powai, Maharashtra 400076, India, International Centre for Theoretical Sciences [TIFR] (ICTS-TIFR), and Maity, Debayan

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]
Abstract

In this article, we consider several coupled ode-parabolic systems. These systems are known to be not null controllable at any time by localized interior controls. We show that these systems are not exponentially stabilizable with arbitrary decay rates. And, consequently, we recover the known results that they are not null controllable any time.

Published
2021

17. Controllability of one dimensional linear coupled transport-parabolic system with variable coefficients

Author

Ahamed, Sakil, Maity, Debayan, Mitra, Debanjana, Department of Mathematics, Indian Institute of Technology Bombay, Powai, Maharashtra 400076, India, Center for Applicable Mathematics [Bangalore] (TIFR-CAM), and Tata Institute for Fundamental Research (TIFR)

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Abstract

In this article, we study the null controllability of linear coupled transport-parabolic systems with variable coefficients in one space dimension. We consider coupled systems with coupling of order zero, one and two. The systems are considered with homogeneous Dirichlet boundary conditions and with localized interior controls acting on both transport and parabolic equations. We show that coupled systems are not null controllable at small time. This time depends on the transport velocity and the support of the controls. When the transport velocity is identically zero, the systems are not null controllable at any time. To achieve these results, we construct highly localized solutions, known as Gaussian beams, corresponding to the adjoint systems, and using them, we show that the corresponding observability inqualities fail. However, these systems are null controllable at any time by controls acting everywhere in the parabolic equation, under suitable assumptions on the initial data and the coefficients.

Published
2020

18. LACK OF COMPLETE STABILIZABILITY OF SOME COUPLED ODE-PARABOLIC SYSTEMS.

Author

MAITY, DEBAYAN, MITRA, DEBANJANA, and RAMASWAMY, MYTHILY

Subjects
PARABOLA, VISCOELASTICITY, NAVIER-Stokes equations, MATHEMATICAL models, MATHEMATICAL analysis
Abstract

In this article, we consider several coupled ode-parabolic systems. These systems are known to be not null controllable at any time by localized interior controls. We show that, these systems are not exponentially stabilizable with arbitrary decay rate. And consequently, we recover the known results that they are not null controllable at any time. [ABSTRACT FROM AUTHOR]

Published
2022

22. LOCAL STABILIZATION OF COMPRESSIBLE NAVIER-STOKES EQUATIONS IN ONE DIMENSION AROUND NON-ZERO VELOCITY

Author

Mitra, Debanjana, Ramaswamy, Mythily, Raymond, Jean-Pierre, Virginia Tech [Blacksburg], Center for Applicable Mathematics [Bangalore] (TIFR-CAM), Tata Institute for Fundamental Research (TIFR), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Tata Institute of Fundamental Research [Bombay] (TIFR), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Compressible Navier-Stokes equations, feedback control, Applied Mathematics, 93C20, 76N25, AMS: 93C20, 93D15, 76N25, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 93D15, local stabilization, Analysis, localized interior control
Abstract

In this paper, we study the local stabilization of one dimensional compressible Navier-Stokes equations around a constant steady solution $(\rho_s, u_s)$, where $\rho_s>0, u_s\neq 0$. In the case of periodic boundary conditions, we determine a distributed control acting only in the velocity equation, able to stabilize the system, locally around $(\rho_s, u_s)$, with an arbitrary exponential decay rate. In the case of Dirichlet boundary conditions, we determine boundary controls for the velocity and for the density at the inflow boundary, able to stabilize the system, locally around $(\rho_s, u_s)$, with an arbitrary exponential decay rate.

Published
2017

28. Interior local null controllability of one-dimensional compressible flow near a constant steady state.

Author

Mitra, Debanjana, Ramaswamy, Mythily, and Renardy, Michael

Subjects
*CONTROLLABILITY in systems engineering, *COMPRESSIBLE flow, *PHYSICAL constants, *DIMENSIONAL analysis, *STEADY state conduction, *MATHEMATICAL proofs
Abstract

We consider the one-dimensional compressible Navier-Stokes equations with periodic boundary conditions, with initial conditions in a small neighborhood of a state of uniform density and uniform nonzero velocity. We prove that, with a control given only by a body force localized in a subinterval, we can steer the system to uniform density and velocity. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Published
2017
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