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Debanjit Mondal

Publications and source records attributed to Debanjit Mondal.

4 recordsLinked to original sources
arXiv2025

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.

math.OC↗
arXiv2025

Linear Test Approach to Global Controllability of Higher-Order Nonlinear Dispersive Equations with Finite-Dimensional Control

We investigate a class of higher-order nonlinear dispersive equations posed on the circle, subject to additive forcing by a finite-dimensional control. Our main objective is to establish approximate controllability by using the controllability of the inviscid Burgers system, linearized around a suitably constructed trajectory. In contrast to earlier approaches based on Lie algebraic techniques, our method offers a more concise proof and sheds new light on the structure of the control. Although the approach necessitates a higher-dimensional control space, both the structure and dimension of the control remain uniform with respect to the order of the dispersive equation and the control time.

math.AP↗
arXiv2024

Global Controllability of the Kawahara Equation at Any Time

In this article, we prove that the nonlinear Kawahara equation on the periodic domain \(\mathbb{T}\) (the unit circle in the plane) is globally approximately controllable in \(H^s(\mathbb{T})\) for \(s \in \mathbb{N}\), at any time \(T > 0\), using a two-dimensional control force. The proof is based on the Agrachev-Sarychev approach in geometric control theory.

math.AP↗