State Dependent Riccati for dynamic boundary control to optimize irrigation in Richards' Equation framework

TL;DR

This paper develops a boundary control method using State-Dependent Riccati equations to optimize water use and root water uptake in irrigation modeled by Richards' equation, validated through numerical simulations with noise.

Contribution

It introduces a novel feedback control approach for irrigation optimization based on Richards' equation, incorporating a cost functional balancing water input and root uptake.

Findings

Effective boundary control reduces water usage while maintaining root water uptake.

The feedback control approach remains robust under noisy conditions.

Numerical simulations validate the method's practical applicability.

Abstract

We present an approach for the optimization of irrigation in a Richards' equation framework. We introduce a proper cost functional, aimed at minimizing the amount of water provided by irrigation, at the same time maximizing the root water uptake, which is modeled by a sink term in the continuity equation. The control is acting on the boundary of the dynamics and due to the nature of the mathematical problem we use a State-Dependent Riccati approach which provides suboptimal control in feedback form, applied to the system of ODEs resulting from the Richards' equation semidiscretization in space. The problem is tested with existing hydraulic parameters, also considering proper root water uptake functions. The numerical simulations also consider the presence of noise in the model to further validate the use of a feedback control approach.