Moscow, 1961: Pontryagin — blinded at fourteen — proves that to reach a target in minimum time, the optimal control does not ease; it slams to the bounds. This panel is runnable: it integrates the bang–bang law for the double integrator and measures the minimum time by trajectory, not by a looked-up constant.
source L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Moscow, 1961; Interscience trans. 1962) — the maximum principle. Room: THE NUMBER. Rendered, not quoted.
Lev Semyonovich Pontryagin (1908–1988) lost his sight at 14 in a stove explosion; his mother read the mathematics aloud and he worked it in his head, becoming one of the great Soviet topologists. Turning to control, he and his students Boltyanskii, Gamkrelidze and Mishchenko published The Mathematical Theory of Optimal Processes in 1961.
Its central theorem — the maximum principle — founded modern optimal control. He sits in THE NUMBER beside the St Petersburg–Moscow analysts Lyapunov, Chebyshev and Markov.
Form the Hamiltonian H = p₁x₂ + p₂u along the flow, with costate p solving ṗ₁=0, ṗ₂=−p₁. To be time-optimal the control must maximise H over u at every instant. Since H is linear in u, the maximiser is forced to a corner:
u* = sign(p₂) — i.e. u = ±1, never in between. p₂(t)=p₂(0)−p₁t is linear, so it changes sign at most once: the optimal control is bang–bang with a single switch. Smoothness is not merely unhelpful; it is excluded.
The maximum principle is the necessary condition of optimal control, dual to Bellman’s dynamic programming (the sufficient, value-function side, USA, same years). Together they are the two doors into the field.
Downstream: aerospace guidance, minimum-fuel and minimum-time trajectories, model-predictive control, and the continuous-time backbone of modern reinforcement learning (the Hamilton–Jacobi–Bellman equation, the co-state as the gradient the policy climbs).
The double integrator: ẋ₁=x₂ (position), ẋ₂=u (velocity), actuator bound |u|≤1. Drive a state to the origin (0,0) in minimum time. Pick a start; the engine builds the bang–bang law and integrates it.
|u|≤1 is the actuator bound — the real limit of the machine. r=(1,0) is the canonical anchor: minimum time T=2.
The switching curve. The two arcs that flow into the origin form x₁ = −½ x₂|x₂|. Above it drive u=−1; below it drive u=+1; ride it into the origin. One crossing = one switch.
switch at t = — · switch state — · switches: —
| t | x₁ | x₂ | u |
|---|
Every row is integrated (Euler, dt=1e-4), not tabulated. For a rest start at position p the theory gives T = 2√p; the engine reaches it by trajectory to <1e-3.
| control | bound | reaches origin | time | legal |
|---|
The bang–bang law is compared to a sub-maximal control (|u|=0.5): it is strictly slower — it does not reach the origin by the bang–bang time. No |u|≤1 control beats the bang–bang minimum.
The engine makes it concrete: the sub-maximal control (|u|=0.5) never overtakes the bang–bang law — it arrives late or not at all within the same clock.
“Smooth control is always best.” Cut. Time-optimal control is bang–bang — it slams to ±1 and switches; the smooth path is strictly slower.
“You need to solve the differential equation to find the optimum.” Cut. The maximum principle hands you the structure (sign of the costate) before any trajectory is integrated.
“More switches must mean a faster path.” Cut. For the double integrator the costate is linear — at most one switch is optimal; extra switches only waste time.
The red move: let the control exceed the bound, |u|=2. Now the machine reaches the origin faster than T=2 — but that time is a lie: it was bought by cheating the limit, not by better control.
Break the bound and the engine posts a time below the |u|≤1 minimum. The witness compares it to the legal minimum, sees the bound violated, and turns red. Beating T=2 means cheating |u|≤1, not finding a valid control.