AI 中文总结
研究火星动力下降优化问题，结合精确凝聚、强凸正则化等方法简化问题，经固定预算外推比例积分投影梯度迭代求解，大幅减少迭代次数，实现微秒级快速求解凸核。
AI 中文摘要
以燃料为主导的动力下降可表示为凸规划，但常规的全状态上图公式仍包含许多状态变量、燃料上图变量和动力学等式。此外，纯燃料目标没有强凸曲率。本文结合了三种结构简化方法。一是通过维度一致的低权重能量项使控制解唯一；二是终端状态灵敏度递归精确消除每个中间状态，经可逆行归一化将有300个原始变量和174个等式乘子的30节点基线问题转化为有90个控制变量和6个终端乘子的问题；三是燃料范数和推力球的共享径向结构给出由组收缩后跟幅度裁剪组成的精确闭式近端算子。通过在固定大小的C17数组中实现的固定预算外推比例积分投影梯度迭代求解凝聚问题。在火星动力下降案例中，能量权重为0.02且相对参考解容差为\(10^{-3}\)时，迭代次数从纯燃料全状态上图基线的2744次降至93次，而燃料指标仅增加0.033%。在英特尔i7 - 10875H上，平均端到端求解时间为68.2微秒，P99为128.1微秒。由于此测试案例中的推力集已是凸球，贡献在于凸核的快速求解而非新的无损凸化定理。
英文摘要
Fuel-dominant powered descent can be written as a convex program, but the usual full-state epigraph formulation still carries many state variables, fuel epigraph variables, and dynamics equalities. In addition, the pure-fuel objective provides no strong-convexity curvature. This paper combines three structural reductions. First, a dimensionally consistent low-weight energy term makes the control solution unique. Second, a terminal-state sensitivity recursion eliminates every intermediate state exactly; invertible row normalization turns the 30-node baseline with 300 primal variables and 174 equality multipliers into a problem with 90 control variables and six terminal multipliers. Third, the shared radial structure of the fuel norm and thrust ball gives an exact closed-form proximal operator consisting of group shrinkage followed by magnitude clipping. The condensed problem is solved with a fixed-budget extrapolated proportional--integral projected-gradient iteration implemented in fixed-size C17 arrays. In a Mars powered-descent case with energy weight 0.02 and relative reference-solution tolerance $10^{-3}$, the iteration count decreases from 2744 for the pure-fuel full-state epigraph baseline to 93, while the fuel metric increases by only 0.033\%. The mean end-to-end solve time is \SI{68.2}{\micro\second}, and P99 is \SI{128.1}{\micro\second}, on an Intel i7-10875H. Because the thrust set in this test case is already a convex ball, the contribution is fast solution of the convex core rather than a new lossless-convexification theorem.
