Quadratic Returns for a Moving Water Tank: An Explicit Right Inverse and a Sylvester Representation

Coron's return method established local nonlinear controllability of a moving water tank in sufficiently large time. We study the quadratic response at uniform equilibrium, where the acceleration input in the linearized model cannot excite the Fourier modes of the fluid with even indices. At three one-way wave crossings (T=3), we construct a nonzero smooth acceleration, whose first two final-state coefficients vanish. A planar curve reduces this cancellation to a scalar identity. At this reference acceleration, two integrations give an explicit control correction that preserves the linear return and produces any given leading change in the quadratic final state. The correction maps periodic $H^r$ targets with zero mean to $H^r$ controls for every nonnegative integer $r$. A contraction argument also gives local exact solvability of the quadratic final-state equation. Finally, a time-dependent Sylvester equation provides an alternative representation of the coupling responsible for the second-order correction. The associated change of state coordinates involves one spatial derivative, while the control correction preserves regularity in appropriate Sobolev spaces. These results concern the first two variation equations and not yet the fully nonlinear model.