# Measure solutions to perturbed structured population models - differentiability with respect to perturbation parameter

## TL;DR

This paper investigates the differentiability of measure solutions to perturbed structured population models with respect to perturbation size, establishing differentiability in a specific function space for certain regularity conditions.

## Contribution

It demonstrates that measure solutions are differentiable with respect to perturbations in a space larger than Radon measures, under certain smoothness conditions, advancing the analysis of such models.

## Key findings

- Differentiability fails in the space of bounded Radon measures with flat metric.
- Differentiability holds in the space Z for ter rac{1}{2} regularity.
- The results support the use of space Z for optimal control applications.

## Abstract

This paper is devoted to study measure solutions $\mu_t^h$ to perturbed nonlinear structured population models where $t$ denotes time and $h$ controlls the size of perturbation. We address differentiability of the map $h \mapsto \mu_t^h$. After showing that this type of results cannot be expected in the space of bounded Radon measures $\mathcal{M}(\mathbb{R}^+)$ equipped with the flat metric, we move to the slightly bigger spaces $Z = \overline{\mathcal{M}(\mathbb{R}^+)}^{(C^{1+\alpha})^*}$. We prove that when $\alpha > \frac{1}{2}$, the map $h \mapsto \mu_t^h$ is differentiable in $Z$. The proof exploits approximation scheme of a nonlinear problem from previous studies and is based on the iteration of an implicit integral equations obtained from study of the linear equation. The result shows that space $Z$ is a promising setting for optimal control of phenomena governed by such type of models.