arXiv:2610.12380v1 Announce Type: new Abstract: High-altitude platform stations (HAPS) are a promising component of non-terrestrial networks (NTNs) for wide-area connectivity. However, serving multiple users over large user footprints requires efficient power-control mechanisms that account for the unique propagation characteristics and strict power constraints of HAPS. This paper develops a tractable analytical and optimization framework for downlink power allocation in HAPS systems under statistical channel state information, 3GPP NTN path loss, and Rician fading. A closed-form expression for the signal-to-interference-plus-noise ratio (SINR) with mean-channel maximal-ratio transmission (MRT) precoding is derived, capturing geometry-dependent inter-user interference coupling. Based on this, two power-control schemes are proposed: quality-of-service (QoS)-constrained power minimization and max-min SINR fairness. The QoS solution is obtained in closed form or through a low-complexity fixed-point iteration, while the max-min solution is obtained through bisection. Numerical results demonstrate up to 6 dB reduction in required transmit power at moderate-to-high target rates and improved worst-user SINR compared to the uniform power, inverse path-loss, and geometry-aware heuristic allocations. The proposed QoS-optimal scheme also achieves a QoS feasibility probability of approximately 0.95 at R_req = 2.5 bits/s/Hz and P_max = 35 dBm, while the uniform power, inverse path-loss, and geometry-aware heuristic allocations achieve less than 0.4. Moreover, the proposed schemes achieve performance comparable to the linear programming (LP)-based benchmarks at a fraction of the computational cost, reducing the average computation time by approximately 99.97% when K = 20.