Strong Legendre Condition

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

A second-order necessary condition in optimal control requiring the Hessian of the maximized Hamiltonian with respect to the control to be uniformly negative definite along the reference extremal. An extremal satisfying this condition is called regular and admits a field of extremals, which is prerequisite for proving local optimality. The logarithmic barrier formulation naturally satisfies this condition.

Application Value

The Strong Legendre Condition plays a significant role in cislunar space mission design, analysis, and control. In orbital design, it can be leveraged for transfer trajectory optimization; in navigation and control, it improves mission execution precision and reliability; in system analysis, it facilitates deeper understanding of complex multi-body dynamical behavior, guiding mission planning and risk assessment.

References

  • Caillau et al. 2012 - Minimum fuel control of the planar circular restricted three-body problem
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Contributors: ouyangjiahong, Ou Yang Jiahong