Optimal dynamical stabilization

The paper formulates the same control-affine problem ẍ + u(t)x = 0 with u ∈ [u−, u+] and cost ∫ u dt, applies the Pontryagin Maximum Principle to obtain the threshold law u = u− if x2 < λ, u = u+ if x2 > λ, and asserts that the unique global optimum is a symmetric bang–bang policy with exactly two switchings; it also derives, in the large-T limit, the transcendental condition √|u−| = √u+ tan(√u+ T+∞/2) and explains the quantized family of admissible durations T+ i∞ via a Schrödinger/Sturm–Liouville reduction. These statements align with the candidate solution’s PMP derivation, two-switch structure, Riccati-uniqueness argument, and the Sturm–Liouville quantization T+ i∞ = 2( arctan(√|u−|/√u+) + iπ )/√u+. The paper’s appendices summarize the normality/no-singular-arc facts and two-commutation result, while the model supplies some missing proof details (e.g., a clean Riccati monotonicity showing uniqueness). Thus, both are correct and proceed by essentially the same method, with the model giving a more explicit proof sketch of steps the paper cites to a companion reference. See the paper’s PMP threshold and two-switch structure and large-T formula and quantized set discussions .