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 $Δ(H)\asymp δ^4\sqrt{\log(1/δ)} \qquad (δ\downarrow0),$ 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 $θ=N^{-1/4}h$, $|h|\le H$, 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 $c_g=\operatorname{Var}(g(Z))$. The Gaussian problem exhibits an exact no-learning phase transition: $C_d^T(H)=TH^2/2 \iff H^4T\le d^2/2,$ yielding the asymptotic task boundary $c_gH^4=d^2/2$.