On the Numerical Stability of Discretised Optimal Control Problems

Optimal Control Problems (OCP) consist in optimising 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.