Sensitivity of Optimal Control Solutions and Quantities of Interest with Respect to Component Functions

TL;DR

This paper develops a theoretical framework to analyze how optimal control solutions and quantities of interest change with perturbations in component functions within the system dynamics and objectives, extending existing sensitivity analysis methods.

Contribution

It introduces a novel sensitivity analysis approach for optimal control problems with component functions, using the Implicit Function Theorem under Carathéodory assumptions.

Findings

Established continuous Fréchet differentiability of OCP solutions w.r.t. component functions.

Derived new estimates for QoI changes due to component function perturbations.

Demonstrated applicability on hypersonic vehicle trajectory optimization.

Abstract

This work establishes sensitivities of the solution of an optimal control problem (OCP) and a corresponding quantity of interest (QoI) to perturbations in a state/control-dependent component function that appears in the governing ODEs and the objective function. This extends existing OCP sensitivity results, which consider the sensitivity of the OCP solution with respect to state/control-independent parameters. It is shown that with Carath\'eodory-type assumptions, the Implicit Function Theorem can be applied to establish continuous Fr\'echet differentiability of the OCP solution with respect to the component function. These sensitivities are used to develop new estimates for the change in the QoI when the component function is perturbed. Applicability of the theoretical results is demonstrated on a trajectory optimization problem for a hypersonic vehicle.