Sensitivity of ODE Solutions and Quantities of Interest with Respect to Component Functions in the Dynamics

TL;DR

This paper investigates how solutions of ODEs and related quantities of interest are affected by changes in component functions within the system, extending classical sensitivity analysis to include these functions.

Contribution

It introduces a new sensitivity analysis framework for component functions in ODEs using the Implicit Function Theorem, providing tighter error bounds than classical methods.

Findings

Sensitivity-based bounds are often tighter than Gronwall-type bounds.

The method applies to Zermelo's problem and hypersonic vehicle trajectory simulations.

Continuous Fréchet differentiability of solutions w.r.t. component functions is established.

Abstract

This work analyzes the sensitivities of the solution of a system of ordinary differential equations (ODEs) and a corresponding quantity of interest (QoI) to perturbations in a state-dependent component function that appears in the governing ODEs. This extends existing ODE sensitivity results, which consider the sensitivity of the ODE solution with respect to state-independent parameters. It is shown that with Carath\'eodory-type assumptions on the ODEs, the Implicit Function Theorem can be applied to establish continuous Fr\'echet differentiability of the ODE solution with respect to the component function. These sensitivities are used to develop new estimates for the change in the ODE solution or QoI when the component function is perturbed. In applications, this new sensitivity-based bound on the ODE solution or QoI error is often much tighter than classical Gronwall-type error bounds. The sensitivity-based error bounds are applied to Zermelo's problem and to a trajectory simulation for a hypersonic vehicle.