# Weighted Irrigation Plans

## TL;DR

This paper models weighted irrigation networks with a system of ODEs, proving existence, uniqueness, and lower semicontinuity of the cost functional, which ensures the existence of optimal plans considering branch weights.

## Contribution

It introduces a novel weighted model for irrigation networks, establishing key mathematical properties and existence results for optimal weighted irrigation plans.

## Key findings

- Proved existence and uniqueness of solutions for the weighted irrigation model.
- Established lower semicontinuity of the cost functional under pointwise convergence.
- Confirmed the existence of optimal weighted irrigation plans.

## Abstract

We model an irrigation network where lower branches must be thicker in order to support the weight of the higher ones. This leads to a countable family of ODEs, one for each branch, that must be solved by backward induction. Having introduced conditions that guarantee the existence and uniqueness of solutions, our main result establishes the lower semicontinuity of the corresponding cost functional, w.r.t. pointwise convergence of the irrigation plans. In turn, this yields the existence of an optimal irrigation plan, in the presence of these additional weights.