Constant rotation is optimal: an exact finite-time contraction bound for anisotropic damping under bounded rotational control

We consider a planar linear control system with anisotropic damping, given by two damping rates d1 > d2 >= 0, and a bounded rotational control u(t) with |u| <= beta. For the worst-case contraction of the propagator at a fixed final time T we show: if beta is at most half the difference of the two damping rates, then for every T > 0 the constant controls u = +beta and u = -beta are optimal, and for beta > 0 they are the only optimal controls. The optimal value is given in closed form. In particular, time-dependent control cannot improve on the best constant control at any finite time. The resulting asymptotic decay rate is not new in substance and also follows from known estimates for cooperative linear systems. The new content is the exact finite-time statement. The proof is an elementary comparison argument in hyperbolic coordinates. The system describes the mean of a two-dimensional Ornstein-Uhlenbeck process with bounded rotational drift and, equivalently, a two-rate Bloch system for a qubit in a plane under a bounded resonant field. Status: checked symbolically, numerically and by several independent AI-based reviews, but not by a human referee. Novelty is not established.