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 $\mathrm{\parallel }\dot{z}(t)\mathrm{\parallel }\le L(t)\mathrm{\parallel }z(t)\mathrm{\parallel }$ satisfies $\mathrm{\parallel }z(t)\mathrm{\parallel }\ge \mathrm{\parallel }z(0)\mathrm{\parallel }{e}^{-{\int }_0^{t}L}$. 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 $\mathrm{\parallel }\dot{z}\mathrm{\parallel }\le L(t)\omega (\mathrm{\parallel }z\mathrm{\parallel })$, first arrival at the origin requires budget at least ${\int }_0^{\mathrm{\parallel }{z}_0\mathrm{\parallel }}du\mathrm{/}\omega (u)$; 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.