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We investigate a class of higher-order nonlinear dispersive equations posedon the circle, subject to additive forcing by a finite-dimensional control. Ourmain objective is to establish approximate controllability by using thecontrollability of the inviscid Burgers system, linearized around a suitablyconstructed trajectory. In contrast to earlier approaches based on Liealgebraic techniques, our method offers a more concise proof and sheds newlight on the structure of the control. Although the approach necessitates ahigher-dimensional control space, both the structure and dimension of thecontrol remain uniform with respect to the order of the dispersive equation andthe control time.

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