Sharp Critical Minimax Laws and No-Learning Thresholds in Continuous-Time Adaptive Control

We study episodic continuous-time control with an unknown vector control gain, scalar state, quadratic action cost, and smooth convex terminal cost. In the scalar Gaussian experiment, let Δ(H) denote the minimax improvement over zero control and set δ= sqrt2H 2-1. We prove the critical law Δ(H) δ 4 sqrt (1/δ) (δ 0), with a matching lower and upper bound. The lower bound follows from a uniform deficit--energy inequality valid for fully adaptive controls with unbounded amplitudes, while a moving soft-threshold feedback attains the rate. For local parameters θ=N-1/4h, h H, we show that the normalized minimax regret over N episodes is within O(N-1/2) of a fixed-horizon Gaussian sequential control problem, without an additional dimension factor. The terminal task enters the limit only through c g= Var(g(Z)). The Gaussian problem exhibits an exact no-learning phase transition: C d T(H)=TH 2/2 H 4T d 2/2, yielding the asymptotic task boundary c gH 4=d 2/2.