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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 $$
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