Exact discrete-adjoint optimization of trap timing and placement in a Stieltjes-time reaction-diffusion model: A Galicia case study

Seasonal population models must combine diffusion, inactive periods and abrupt biological transfers while remaining differentiable with respect to localized interventions. We formulate a finite-dimensional timing-and-placement control problem for a Stieltjes-time reaction--diffusion model and derive the exact adjoint of the fully discrete residual. Continuous finite elements and an implicit Stieltjes--Euler scheme represent propagation, compartment replacement, resets and historical phase averages. For the resulting box-constrained penalized problem, we prove well-posedness and differentiability of the discrete control-to-state map, existence of a minimizer, first-order stationarity, and equivalence between direct-sensitivity and adjoint gradients. Benchmark tests show machine-precision agreement (below $3\times10^{-15}$) and adjoint speed-ups from about $7.5$ to $30.9$ as the number of traps increases from one to four. The method is applied to \textit{Vespa velutina} trap campaigns on a realistic Galicia mesh. Smoothed activation windows, domain-normalized moving kernels and the complete adjoint are independently verified. The final two-cluster control remains admissible and has a scaled gradient infinity norm of $2.97\times10^{-5}$ with 120 temporal subdivisions per month. Fixed-control evaluations at 60, 120 and 240 subdivisions yield a refinement-increment ratio of $0.4984$. At the finest level, the joint control outperforms time-only and reference controls, reducing the diagnostic objective by $0.49209$. This reduction depends on uncalibrated trap-to-mortality intensities and should not be interpreted as field capture efficacy. The framework is fully reproducible, without claiming global optimality or a calibrated field-management prescription.