A representation of a fundamental solution group for a class of wave equations is constructed by exploiting connections between stationary action and optimal control. By using a Yosida approximation of the associated generator, an approximation of the group of interest is represented for sufficiently short time horizons via an idempotent convolution kernel that describes all possible solutions of a corresponding short time horizon optimal control problem. It is shown that this representation of the approximate group can be extended to longer horizons via a concatenation of such short horizon optimal control problems, provided that the associated initial and terminal conditions employed in concatenating trajectories are determined via a stationarity rather than an optimality based condition. The construction is illustrated by its application to the approximate solution of a two point boundary value problem.

通过利用平稳作用和最优控制之间的联系，构造了一类波动方程的基本解组的表示。通过使用相关发电机的Yosida逼近，通过幂等卷积核代表了足够短的时间范围内感兴趣组的逼近，该幂等的卷积核描述了相应的短时间范围最优控制问题的所有可能解。结果表明，只要通过平稳性而非基于最优性的条件来确定级联轨迹中采用的相关初始条件和最终条件，就可以通过这种短视野最优控制问题的级联将近似组的表示扩展到更长的视野。 。