Measure solutions to perturbed structured population models - differentiability with respect to perturbation parameter
Abstract
This paper is devoted to study measure solutions μth to perturbed nonlinear structured population models where t denotes time and h controls the size of perturbation. We address differentiability of the map h ↦ μth. After showing that this type of results cannot be expected in the space of bounded Radon measures M (R+) equipped with the flat metric, we move to the slightly bigger spaces Z =M (R+) ‾ (C 1 + α) *. We prove that when α >1/2, the map h ↦ μth 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.
- Publication:
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Journal of Differential Equations
- Pub Date:
- April 2020
- DOI:
- 10.1016/j.jde.2019.10.024
- arXiv:
- arXiv:1812.01747
- Bibcode:
- 2020JDE...268.4119S
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- 42 pages + 13 pages of Appendix