论文标题
周期轨道的最佳稳定
Optimal Stabilization of Periodic Orbits
论文作者
论文摘要
在这一贡献中，通过不变的歧管理论和符号几何形状研究了周期性轨道的最佳稳定问题。最佳点稳定案例的稳定歧管理论被推广到周期性轨道稳定的情况下，其中正常双曲线不变的歧管（NHHIM）扮演双曲线平衡的作用。 根据周期性的riccati微分方程，得出了相关的哈密顿系统的NHIM的充分条件。结果表明，如果线性定期系统满足稳定性和可检测性，则最佳轨道稳定问题具有解决方案。沿周期轨道采用了移动的正交坐标系，这是轨道稳定和线性化参数的自然框架。 示例说明的例子包括弹簧质量振荡器系统的最佳控制问题，该系统应在一定的能级下稳定，以及卫星的轨道转移问题，卫星构成了典型的轨道力学控制问题。
In this contribution, the optimal stabilization problem of periodic orbits is studied via invariant manifold theory and symplectic geometry. The stable manifold theory for the optimal point stabilization case is generalized to the case of periodic orbit stabilization, where a normally hyperbolic invariant manifold (NHIM) plays the role of a hyperbolic equilibrium. A sufficient condition for the existence of an NHIM of an associated Hamiltonian system is derived in terms of a periodic Riccati differential equation. It is shown that the problem of optimal orbit stabilization has a solution if a linearized periodic system satisfies stabilizability and detectability. A moving orthogonal coordinate system is employed along the periodic orbit which is a natural framework for orbital stabilization and linearization argument. Examples illustrated include an optimal control problem for a spring-mass oscillator system, which should be stabilized at a certain energy level, and an orbit transfer problem for a satellite, which constitutes a typical control problem of orbital mechanics.
