Optimal Control Problems consist on the optimisation of an objective functional subjected to a set of Ordinary Differential Equations. In this work, we consider the effects on the stability of the numerical solution when this optimisation is discretised in time. In particular, we analyse a OCP with a quadratic functional and linear ODE, discretised with Mid-point and implicit Euler. We show that the numerical stability and the presence of numerical oscillations depends not only on the time-step size, but also on the parameters of the objective functional, which measures the amount of control input. Finally, we also show with an illustrative example that these results also carry over non-linear optimal control problems

翻译：最优控制问题旨在优化受常微分方程约束的目标泛函。本文研究了时间离散化处理对该优化问题数值解稳定性的影响。具体而言，我们分析了采用中点格式和隐式欧拉格式离散化的二次型泛函与线性常微分方程构成的最优控制问题。研究表明，数值稳定性及数值振荡现象不仅取决于时间步长，还取决于衡量控制输入量的目标泛函参数。最后，我们通过一个算例证明，上述结论同样适用于非线性最优控制问题。