Connections between the principle of least action and optimal control are explored with a view to describing the trajectories of energy conserving systems, subject to temporal boundary conditions, as solutions of corresponding systems of characteristics equations on arbitrary time horizons. Motivated by the relaxation of least action to stationary action for longer time horizons, due to loss of convexity of the action functional, a corresponding relaxation of optimal control problems to stationary control problems is considered. In characterizing the attendant stationary controls, corresponding to generalized velocity trajectories, an auxiliary stationary control problem is posed with respect to the characteristic system of interest. Using this auxiliary problem, it is shown that the controls rendering the action functional stationary on arbitrary time horizons have a state feedback representation, via a verification theorem, that is consistent with the optimal control on short time horizons. An example is provided to illustrate application via a simple mass-spring system.

探索最小作用量原理和最优控制之间的联系，以将受时间边界条件约束的能量守恒系统的轨迹描述为任意时间范围上的相应特征方程系统的解。由于动作泛函的凸性损失，在较长时间范围内将最小动作松弛为静止动作的动机，考虑将最优控制问题相应地松弛为静止控制问题。在表征伴随的平稳控制时，对应于广义速度轨迹，关于感兴趣的特征系统提出了一个辅助平稳控制问题。使用这个辅助问题，结果表明，在任意时间范围内呈现动作泛函平稳的控制具有状态反馈表示，通过验证定理，这与短时间范围内的最优控制一致。提供了一个例子来说明通过一个简单的质量弹簧系统的应用。