On the numerical stability of discretised Optimal Control Problems
Ashutosh Bijalwan, Jose J Muñoz
TL;DR
This work investigates the numerical stability of time-discretised optimal control problems, focusing on MP and implicit Euler schemes applied to Hamiltonian boundary-value problems derived from indirect methods. It shows that stability and numerical oscillations depend on both the time-step $\Delta t$ and the control-cost weight $\alpha$, deriving explicit thresholds $\alpha_{th,MP}$ and $\alpha_{th,iE}$ for a linear OCP with dynamics $\dot{v}= -\tfrac{b}{m}v + \tfrac{u}{m} - a$ and cost $\mathcal{J}=\int_0^T \left[ \tfrac{1}{2}(v-v_t)^2 + \tfrac{\alpha}{2}u^2 \right] dt$. The analysis uses the discretised maps’ spectral properties to characterize oscillations and stability, and confirms the thresholds with numerical examples. Moreover, the authors show that similar oscillatory behavior persists in a nonlinear OCP, exemplified by an inverted elastic pendulum, where for small $\Delta t$ the conservative stability criterion scales as $\alpha_{th}=O(\Delta t^2)$, implying that smaller $\alpha$ requires correspondingly smaller time steps. Overall, the findings provide practical guidelines for choosing $\Delta t$ and $\alpha$ to avoid spurious numerical oscillations in discretised OCPs.
Abstract
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