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

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 Carathéodory-type assumptions on the ODEs, the Implicit Function Theorem can be applied to establish continuous Fréchet 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.

27 pages, 11 figures, added supplemental materials with additional numerical examples