Linear Test Approach to Global Controllability of Higher-Order Nonlinear Dispersive Equations with Finite-Dimensional Control
Debanjit Mondal
TL;DR
The paper addresses approximate controllability for a class of higher-order nonlinear dispersive equations on the torus with finite-dimensional control. It develops a linear-test framework by constructing a reference trajectory from the inviscid Burgers equation, linearizing around it, and proving approximate controllability of the linearized problem via observable families and a finite-dimensional approximate right inverse, then lifting this to the nonlinear system. A key contribution is a uniform finite-dimensional control structure, with the control space’s dimension independent of the dispersion order and control time, along with a saturating-frequency framework that generalizes the result beyond a fixed Fourier mode set. The work advances controllability theory for dispersive PDEs under finite-dimensional forcing, providing a practical and structurally transparent approach that complements existing Lie-algebraic methods. It has potential implications for both theoretical analysis and applications in wave propagation where low-dimensional actuators are used.
Abstract
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.