Fundamental Limits of Prescribed-Time Regulation: Gain Budgets, Accuracy Frontiers, and Arrival Costs
Prescribed-time controllers drive the state to the origin at a time $T$ chosen in advance, with gains that grow without bound as $t$ approaches $T$. This paper shows the growth is a necessity and computes the exact price of avoiding it. The starting point is a simple estimate: any trajectory with $\|\dot z(t)\|\le L(t)\|z(t)\|$ satisfies $\|z(t)\|\ge\|z(0)\|e^{-\int_0^t L}$. From this: a gain profile $L$ supports exact prescribed-time convergence, over the unrestricted class of closed-loop vector fields admitting it, if and only if its integral over $[0,T)$ is infinite; so every design must either use an envelope of infinite integral or violate the linear growth bound at the origin, as finite-time and fixed-time designs do. Under a finite gain budget $\mathcal B=\int_0^T L$ the best worst-case contraction over the class is exactly $e^{-\mathcal B}$, a floor valid for every plant, and a finite-budget trajectory converges to a definite point away from the origin. A comparison lemma then carries the theory beyond linear growth: for envelopes $\|\dot z\|\le L(t)\,ω(\|z\|)$, first arrival at the origin requires budget at least $\int_0^{\|z_0\|}du/ω(u)$; arrival is impossible whenever this integral diverges, and when it is finite it is the exact threshold, attained inside the class. The classical settling expressions of finite-time control reappear as hitting-time floors, and the known fixed-time condition is the two-ended convergence of the same integral. For linear time-varying loops the results refine to logarithmic-norm budgets, corollaries of a known two-sided estimate, with the terminal error of gain freezing described exactly.