Tikhonov-regularised projected gradient flow for equality-constrained bilinear quantum control

We study a projection-type gradient flow for equality-constrained maximisation of a smooth bilinear control objective on $\mathcal{H}=L^2(0,T;\mathbb{R})$, eliminating Lagrange multipliers through an $(M{+}1)\times(M{+}1)$ moving Gram matrix $\Gamma(s)_{\ell\ell'}=\int_0^T S(t)\,c_\ell(s,t)\,c_{\ell'}(s,t)\,\mathrm{d}t$. 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 $\Gamma$ with $\Gamma_{\varepsilon}:=\Gamma+\varepsilon^{2}I$ and prove: (i) an exact spectral identity giving $\kappa(\Gamma_{\varepsilon})=(\sigma_{\max}^{2}+\varepsilon^{2})/(\sigma_{\min}^{2}+\varepsilon^{2})$; (ii) objective monotonicity $\mathrm{d}J/\mathrm{d}s\ge 0$ for all $\varepsilon\ge 0$; (iii) constraint drift $|h_{m}-C_{m}|=\mathcal{O}(\varepsilon^{2})$ with a computable prefactor; (iv) convergence of the regularised trajectory to the unregularised one in $L^{2}(0,T)$ at rate $\mathcal{O}(\varepsilon^{2})$ under uniform invertibility of $\Gamma$; and (v) a discrete CFL criterion $\Delta s\,G\,\|\Gamma_{\varepsilon}^{-1}\|\le\alpha<2$ guaranteeing objective monotonicity of the forward-Euler scheme up to $\mathcal{O}(\Delta s^{2})$ local truncation error. The theory is validated on a three-level bilinear benchmark for all-optical Bell-state preparation, where $\kappa(\Gamma)\in[10^{9},10^{11}]$, the predicted $\varepsilon^{2}$ 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.