Null controllability of a linearised coupled Korteweg–de Vries equation via a direct matrix backstepping approach

This paper establishes the null controllability of a linear coupled Korteweg–de Vries (KdV) system posed on a finite interval. Although the system is diagonalisable, we develop a novel direct matrix backstepping approach that avoids preliminary diagonalisation and treats the coupled dynamics as a whole. The method is based on a matrix Volterra transformation that maps the original system directly to an exponentially stable target system, leading to a coupled system of third-order kernel equations whose structure reflects the interaction between components. We establish sharp exponential estimates for the kernel matrix, which enable the implementation of a piecewise control strategy achieving null controllability without resorting to duality methods or observability inequalities. As a result, we obtain explicit piecewise continuous boundary controls that are bounded in L∞(0,T), in contrast with the H1/3(0,T) regularity typically required in observability-based approaches. The proposed approach preserves the intrinsic coupling structure and provides a constructive framework that can be extended to non-diagonalisable coupled dispersive systems.