arXiv subjects

Subrata Majumdar

Publications and source records attributed to Subrata Majumdar.

9 recordsLinked to original sources

arXiv2026

Small-time global controllability of a class of bilinear fourth-order parabolic equations

In this work, we address the small-time global controllability properties of a class of fourth-order nonlinear parabolic equations driven by bilinear controls posed on the one-dimensional torus. The controls depend only on time and act through a prescribed family of spatial profiles. Our first result establishes the small-time global approximate controllability of the system using three scalar controls, between states that share the same sign. This property is obtained by adapting the geometric control approach to the fourth-order setting, using a finite family of frequency-localized controls. We then study the small-time global exact controllability to non-zero constant states for the concerned system. This second result is achieved by analyzing the null controllability of an appropriate linearized fourth-order system and by deducing the controllability of the nonlinear model through a fixed-point argument together with the small-time global approximate control property. To the best of our knowledge, this work provides the first contribution toward the study of controllability properties of fourth-order parabolic equations by means of bilinear controls.

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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.

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arXiv2026

Global controllability of the Cahn-Hilliard equation

This paper investigates the global controllability properties of the Cahn--Hilliard equation posed on the $d$-dimensional flat torus $\mathbb{T}^d$. We first establish small-time global approximate controllability of the system by means of controls acting on finitely many Fourier modes, relying on techniques inspired by geometric control theory. We then prove null controllability of the linearized equation using a spatially localized control supported on an arbitrary measurable subset of positive Lebesgue measure, based on quantitative propagation of smallness estimates for the free dynamics. For dimensions $d \in \{1,2,3\}$, we further derive local null controllability for the full nonlinear system via a fixed-point argument. By combining these results, we establish global null controllability of the Cahn--Hilliard equation. This work provides the first result on global controllability for this equation, achieved through a two-stage strategy in which the control is first localized in Fourier space and subsequently restricted to a set of positive measure.

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arXiv2025

On the controllability of the Kuramoto-Sivashinsky equation on multi-dimensional cylindrical domains

In this article, we investigate null controllability of the Kuramoto-Sivashinsky (KS) equation on a cylindrical domain $Ω=Ω_x\times Ω_y$ in $\mathbb R^N$, where $Ω_x=(0,a),$ $a>0$ and $Ω_y$ is a smooth domain in $\mathbb R^{N-1}$. We first study the controllability of this system by a control acting on $\{0\}\times ω$, $ω\subset Ω_y$, through the boundary term associated with the Laplacian component. The null controllability of the linearized system is proved using a combination of two techniques: the method of moments and Lebeau-Robbiano strategy. We provide a necessary and sufficient condition for the null controllability of this system along with an explicit control cost estimate. Furthermore, we show that there exists minimal time $T_0(x_0)>0$ such that the system is null controllable for all time $T > T_0(x_0)$ by means of an interior control exerted on $γ= \{x_0\} \times ω\subset Ω$, where $x_0/a\in (0,1)\setminus \mathbb{Q}$ and it is not controllable if $T 1$, then we prove the controllability for any time $T>0.$ Finally, for the case of $N=2 \text{ or } 3$, we show the local null controllability of the main nonlinear system by employing the source term method followed by the Banach fixed point theorem.

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arXiv2025

Local null-controllability of a system coupling Kuramoto-Sivashinsky-KdV and elliptic equations

This paper deals with the null-controllability of a system of {\em mixed parabolic-elliptic pdes} at any given time $T>0$. More precisely, we consider the \textit{Kuramoto-Sivashinsky--Korteweg-de Vries equation} coupled with a second order elliptic equation posed in the interval $(0,1)$. We first show that the linearized system is globally null-controllable by means of a localized interior control acting on either the KS-KdV or the elliptic equation. Using the \textit{Carleman approach}, we provide the existence of a control with the explicit cost $Ce^{C/T}$ with some constant $C>0$ independent in $T$. Then, applying the source term method followed by the \textit{Banach fixed point theorem}, we conclude the small-time local null-controllability result of the nonlinear systems.

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arXiv2025

Boundary null controllability of a class of 2-d degenerate parabolic PDEs

This article deals with the boundary null controllability of some degenerate parabolic equations posed on a square domain, presenting the first study of boundary controllability for such equations in multidimensional settings. The proof combines two classical techniques: the method of moments and the Lebeau-Robbiano strategy. A key novelty of this work lies in the analysis of boundary control localized on a subset of the boundary where the degeneracy occurs. Furthermore, we establish the Kalman rank condition as a full characterization of boundary controllability for coupled degenerate systems. The results are extended to $N$-dimensional domains, and potential extensions and open problems are discussed to motivate further research in this area.

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arXiv2025

Controllability and Stabilizability of the Linearized Compressible Navier-Stokes System with Maxwell's Law

In this paper, we study the control properties of the linearized compressible Navier-Stokes system with Maxwell's law around a constant steady state $(ρ_s, u_s, 0), ρ_s>0, u_s>0$ in the interval $(0, 2π)$ with periodic boundary data. We explore the exact controllability of the coupled system by means of a localized interior control acting in any of the equations when time is large enough. We also study the boundary exact controllability of the linearized system using a single control force when the time is sufficiently large. In both cases, we prove the exact controllability of the system in the space $L^2(0,2π)\times L^2(0, 2π)\times L^2(0, 2π)$. We establish the exact controllability results by proving an observability inequality with the help of an Ingham-type inequality. Moreover, we prove that the system is exactly controllable at any time if the control acts everywhere in the domain in any of the equations. Next, we prove the small time lack of controllability of the concerned system. Further, using a Gramian-based approach demonstrated by Urquiza, we prove the exponential stabilizability of the corresponding closed-loop system with an arbitrary prescribed decay rate using boundary feedback control law.

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arXiv2024

Event-triggered boundary control of the linearized FitzHugh-Nagumo equation

In this paper, we address the exponential stabilization of the linearized FitzHugh-Nagumo system using an event-triggered boundary control strategy. Employing the backstepping method, we derive a feedback control law that updates based on specific triggering rules while ensuring the exponential stability of the closed-loop system. We establish the well-posedness of the system and analyze its input-to-state stability in relation to the deviations introduced by the event-triggered control. Numerical simulations demonstrate the effectiveness of this approach, showing that it stabilizes the system with fewer control updates compared to continuous feedback strategies while maintaining similar stabilization performance.

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arXiv2022

Null controllability of the linear Stabilized Kuramoto-Sivashinsky system using moment method

This paper deals with the null controllability of a coupled parabolic system, which is Kuramoto-Sivashinsky-Korteweg-de Vries equation coupled with heat equation through first order derivative. More precisely, we prove the null controllability of the system with a single localized bilinear interior control acting on either of the components of the coupled system, and with a single periodic boundary control acting through zeroth order derivatives of either of the components. We employ the well-known moment method to study the controllability of the concerned system.

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