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

TL;DR

This work develops a rigorous sensitivity framework for ODEs with state-dependent component functions by casting the IVP in a Carathéodory setting and applying the Implicit Function Theorem to prove continuous Fréchet differentiability of the solution with respect to the component function $\mathbf{g}$. It introduces Fréchet differentiability of the governing Nemytskii operator and derives sensitivity equations for $\mathbf{x}$ and QoIs, along with adjoint-formulations. Using these differentiability results, the authors derive new sensitivity-based error bounds for both the ODE solution and QoI when $\mathbf{g}$ is perturbed, which often outperform classical Grönwall-type estimates. They validate the approach on hypersonic-vehicle trajectory simulations and Zermelo’s problem, demonstrating tight, computable bounds that facilitate surrogate-model refinement and uncertainty quantification in dynamical systems.

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.

Figures (11)

• Figure 1: Perturbed ($\epsilon = 0.01$) and true trajectories for hypersonic ODE.
• Figure 2: Left: Sensitivity-based trajectory error estimate (blue) and trajectory error (orange) in good agreement, and trajectory error bound (green) yields a much tighter bound on the trajectory error than the Gronwall-type error bound (red, capped at $10^{10}$). Right: Approximate logarithmic Lipschitz constant along perturbed trajectory for hypersonic ODE.
• Figure 3: Left: Strong agreement between the $L^2$ trajectory error estimates (blue), trajectory errors (orange), and error bounds (green) for hypersonic ODE for a range of perturbations $\epsilon$. Right: Strong agreement between the QoI error estimates (blue), QoI errors (orange), and error bounds (green) for hypersonic ODE for a range of perturbations $\epsilon$.
• Figure SM.1: Solutions to the true and perturbed Zermelo ODE for $\epsilon = 0.1$.
• Figure SM.2: Left: Sensitivity-based trajectory error estimate (blue) and trajectory error (orange) in good agreement, and trajectory error bound (green) yields a much tighter bound on the trajectory error than the Gronwall-type error bound (red). Right: Approximate logarithmic Lipschitz constant along perturbed trajectory for Zermelo ODE.

Theorems & Definitions (29)

• Theorem 2.2
• Proof 1
• Theorem 2.4
• Theorem 2.5
• Corollary 2.6
• Proof 2
• Lemma 2.7
• Proof 3
• Theorem 2.8: Implicit Function Theorem
• Theorem 2.9