Differentiability in perturbation parameter of measure solutions to perturbed transport equation

TL;DR

This paper proves that solutions to a perturbed transport equation are differentiable with respect to the perturbation parameter in a specific measure space, facilitating applications in optimal control.

Contribution

It establishes the differentiability of measure solutions to a perturbed transport equation with respect to the perturbation parameter in a suitable Banach space.

Findings

Solutions are differentiable in the measure space with respect to perturbations.

Differentiability is established in a Banach space predual to Hölder space.

Results are relevant for optimal control applications.

Abstract

We consider a linear perturbation in the velocity field of the transport equation. We investigate solutions in the space of bounded Radon measures and show that they are differentiable with respect to the perturbation parameter in a proper Banach space, which is predual to the H\"older space $\mathcal{C}^{1+\alpha}(\mathbb{R}^d)$. This result on differentiability is necessary for application in optimal control theory, which we also discuss.