标题： 利用多项式平方和优化定位平面圆限制性三体问题的极值周期轨道 显示英文标题

标题： Locating Extremal Periodic Orbits for the Planar Circular Restricted Three Body Problem using Polynomial Sum-of-Squares Optimization

摘要： 随着人们对长期复杂空间任务设计兴趣的增加，寻找耗油量最少的轨道具有根本的重要性。 本文发展了现有的计算模型，用于通过和平方和（SOS）优化技术定位多项式动力系统中的不稳定周期轨道（UPOs），并提出了一种数值框架来收敛天体力学中平面圆限制性三体问题（PCR3BP）的UPOs。 这是通过开发多项式SOS优化技术，并将其扩展到具有非多项式和哈密顿动力学的系统中实现的。 首先，我们演示并利用了在大多项式次数下，收敛与可观测量相关的紧密界值对尺度因子变化的依赖性。 然后使用SOS优化来计算非负多项式，其最小化子水平集近似地定位相应的UPO的部分。 建议改进当前的非线性优化技术，以在相关子水平集中计算大量的点。 这些点为现有算法的UPO计算提供了良好的初始条件。 此类UPO的独特之处在于它们优化了感兴趣的输入可观测量的长时间平均值，该可观测量是状态变量的函数。 对于PCR3BP而言，这意味着这些空间轨道可以无需持续燃料消耗而无限期地随时间遍历。 作为航天任务设计的实际应用，我们通过最小化卫星与地球和月球之间距离平方的无穷时间平均值，收敛了地球-月球系统通信中继问题所需的最小传输功率的UPOs。 显示英文摘要

摘要： With an increasing interest in the design of long and complex space missions, the search for orbits that require the least amount of fuel is of fundamental interest. This paper develops existing computational models for locating Unstable Periodic Orbits (UPOs) in polynomial dynamical systems using Sum-of-Squares (SOS) optimization technique and proposes a numerical framework to converge UPOs for the Planar Circular Restricted Three-Body Problem (PCR3BP) in astrodynamics. This is done by developing the polynomial SOS optimization technique with extension to systems with non-polynomial and Hamiltonian dynamics. First, we demonstrate and exploit the dependency of convergence of tight bounds on an observable of interest with varying scaling factors for large polynomial degrees. SOS optimization is then used to compute nonnegative polynomials, the minimization sublevel sets of which, approximately localise parts of the corresponding UPO. Improvements in current non-linear optimization techniques are suggested to compute a large number of points inside the relevant sublevel sets. Such points provide good initial conditions for UPO computations with existing algorithms. The distinguishing feature of such UPOs is that they optimize the long-time average of an input observable of interest which is a function of state variables. For the PCR3BP this means that such orbits in space can be traversed indefinitely in time without continuous fuel expenditure. As practical applications to space mission designs, we converge UPOs that minimise transmitted power required by satellites for the Earth-Moon system in a communication relay problem by minimizing the infinite-time average of sum of squares of distances of a satellite from Earth and the Moon.