Bilinear controllability for the linear KdV-Schr{ö}dinger equation
This result extends controllability theory to a coupled KdV-Schrödinger system, but is incremental as it adapts an existing saturation method to a new equation.
The paper proves small-time global approximate controllability in L2(T) for a linear KdV-Schrödinger equation on the torus using bilinear controls spanning a finite number of Fourier modes, achieving controllability between any pair of states with the same norm.
We study the controllability of a linear KdV-Schr{ö}dinger equation on the one-dimensional torus via purely imaginary bilinear controls. Considering controls spanning a suitable finite number of Fourier modes, we prove small-time global approximate controllability in L2(T). The result holds between any pair of states with the same norm and is obtained via the saturation method by following the idea introduced in [Poz24]. We first establish small-time controllability for phase multiplications, and then generate transport operators associated with diffeomorphisms of the torus. Finally, we combine these results to recover global approximate controllability. Note that the controllability property holds independently of the Schr{ö}dinger component of the dynamics, which may in particular be taken to vanish.