Abstract
We study a projection-type gradient flow for equality-constrained maximisation of a smooth bilinear control objective on H=L²(0,T;R), eliminating Lagrange multipliers through an (M+1)×(M+1) moving Gram matrix Γ(s)_ℓℓ'=∫₀^T S(t)c_ℓ(s,t)c_ℓ'(s,t)dt. The flow generates monotonic ascent in continuous time but becomes unstable on discretisation; existing implementations rely on heuristic step-size safeguards lacking rigorous justification. We close this gap by replacing Γ with Γ_ε:=Γ+ε²I and prove: (i) an exact spectral identity giving κ(Γ_ε)=(σ_max²+ε²)/(σ_min²+ε²); (ii) objective monotonicity dJ/ds≥ 0 for all ε≥ 0; (iii) constraint drift |h_m-C_m|=O(ε²) with a computable prefactor; (iv) convergence of the regularised trajectory to the unregularised one in L²(0,T) at rate O(ε²) under uniform invertibility of Γ; and (v) a discrete CFL criterion Δ sGΓ_ε⁻¹≤α<2 guaranteeing objective monotonicity of the forward-Euler scheme up to O(Δ s²) local truncation error. The theory is validated on a three-level bilinear benchmark for all-optical Bell-state preparation, where κ(Γ)∈[10⁹,10¹¹], the predicted ε² rate is confirmed over eight decades, and moderate regularisation eliminates step rejections and reduces constraint drift by more than an order of magnitude at unchanged final fidelity.