On the numerical stability of discretised Optimal Control Problems

TL;DR

This paper investigates how discretization methods and objective functional parameters affect the numerical stability and oscillations in discretized optimal control problems, including nonlinear cases.

Contribution

It provides a detailed analysis of stability dependencies on discretization schemes and control parameters, extending findings from linear to nonlinear optimal control problems.

Findings

Numerical stability depends on time-step size and control parameters.

Mid-point and implicit Euler methods exhibit different stability behaviors.

Results extend from linear to nonlinear optimal control problems.

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