# Contribution of Viability Theory and Constrained Optimal Control to Fertigation

Mahugnon G Dadjo, Alain Rapaport, Denis Efimov, Jérôme Harmand,
Rosane Ushirobira

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Mahugnon G Dadjo, Alain Rapaport, Denis Efimov, Jérôme Harmand, Rosane Ushirobira. Contribution of Viability Theory and Constrained Optimal Control to Fertigation. REUSE EUROMED 2024: Euro-Mediterranean Conference on Wastewater Reuse, Oct 2024, Montpellier, France. hal-05527470

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# Contribution of Viability Theory and Constrained Optimal Control to Fertigation

Mahugnon G. Dadjo¹,²,³, Alain Rapaport¹, Denis Efimov²,
Jérôme Harmand³, Rosane Ushirobira²

¹MISTEA, Univ. Montpellier, Institut Agro, Montpellier, France

²Inria, Univ. Lille, CNRS, 59000 Lille, France

³LBE-INRAE, Univ. Montpellier, Narbonne, France

## 1 Introduction

The population growth increases food demand in most regions of the world, and particulary in most European countries. One of 21st-century agriculture's challenges is to increase agricultural production to meet food needs [6]. Crops growth depends notably on water. In the prevailing context of climate change, water resources are subjected to mounting pressures, not only in southern nations but also across Europe. The adaptation of societies to these global transformations necessitates a comprehensive overhaul of resource management strategies, coupled with the implementation of soberness measures. Agriculture, as the most water-intensive human activity, accounts for nearly 40% of water withdrawals in Europe. Consequently, the optimizing irrigation with treated wastewater emerges as a paramount objective for the future of agriculture. Over the past several decades, agronomists have developed models to enhance our understanding of crop growth dynamics, predict crop yield and optimize agricultural production. Crops need nutrients that can be present in soil or provided as fertilizers. The focus here on fertigation, where nutrients are delivered with the irrigated water. Initially concentrated on water supply, these models have progressively integrated nitrogen dynamics [4]. The application of such models to optimize the water and nutrient supply to crops is essential for reducing the use of synthetic inputs and minimizing water consumption in agriculture, ensuring good crop production, particularly within the framework of water reuse, where the goal is to diminish or eliminate reliance on chemical fertilizers. This paper presents an approach grounded in viability theory and optimal control to optimize fertigation practices. To implement this approach, a model is required.

## 2 Model description

The crop model describes the interaction between soil moisture $S(t)$, total soil nitrogen content $N(t)$, and biomass production $B(t)$. The dynamics are as follows:

* $\dot{S} = k_1(-\varphi(t)K_S(S) - (1-\varphi(t))K_R(S) + k_2u)$ is the soil water balance equation and is modeled as a balance between gains from irrigation flow rate $u$ and losses mainly due to soil evaporation $(1-\varphi(t))K_R(S)$ and crop transpiration $\varphi(t)K_S(S)$. The function represents the vegetal cover at time $t$ (related to the LAI, Leaf Area Indice). The function $K_S$ (see Fig.a) is used to capture the plant stomatal response to soil moisture condition, the function $K_R$ (see Fig.b) reduce evaporation according to soil humidity and the function.

* $\dot{N} = -k_3\varphi(t)K_S(S)f(\frac{N}{S}) + k_4C_N^{in}u$ is the soil nitrogen balance where $k_3\varphi(t)K_S(S)f(\frac{N}{S})$ is the nitrogen uptake and $k_4C_N^{in}u$ is the nitrogen input from irrigation water and the function $f$ (see Fig.c) limits the nitrogen uptake above a certain critical concentration $\eta_C$.

* $\dot{B} = \varphi(t)K_S(S)f(\frac{N}{S})$ is biomass production. The model assumes that biomass production is proportional to nitrogen uptake. A graph of $K_S$, $K_R$ and $f$ is shown below where $S_w$ is the wilting point, $S^*$ is the point of incipient stomatal closure and $S_h$ is the hygroscopic point.

Moreover, we say that crops suffer water stress when the function is not maximum i.e. $S < S^*$ and they suffer nitrogen stress when the function is not maximum i.e $\frac{N}{S} < \eta_C$. We assume that other essential resources for crop growth, such as phosphorus and minerals, are not limiting factors. The distinguishing
<figure><img src="image_2.png" /><figcaption>Figure 1: The water stress function (blue), the evaporation reduction function (green) as a function of soil moisture *S* and the nitrogen uptake limitation function (red).</figcaption></figure>

feature of this model, in contrast to those found in existing literature, lies in its explicit consideration of nitrogen supply through irrigation, a process known as "fertigation". Depending on the nitrogen concentration in the irrigation water, a dilution effect may occur, reducing the available nitrogen for the crop, thereby exacerbating nitrogen stress while simultaneously lowering costs, even as water stress is avoided. The next section presents the irrigation decision-making based in viability theory and optimal control.

# 3 Results and Discussions

## Viability approach

The objective of this study is to leverage the model to identify the optimal balance between water and nitrogen inputs necessary for achieving robust growth performance. Fertigation is generally employed when soil nitrogen content is inadequate to ensure maximum growth; this deficiency can be offset by the nitrogen present in the irrigation water. Both the initial nitrogen concentration in the soil and the nitrogen concentration in the irrigation water play pivotal roles. To determine an optimal strategy, it is essential to possess knowledge of the future needs of the crops. According to the model, biomass production is maximized when crops neither suffer from water stress nor nitrogen stress. Therefore, the future needs should avoid both water and nitrogen stress. Geometrically, this entails ensuring that the future needs of crops remain within the sky blue set E. The mathematical theory of viability helps ensure that future needs remain within the limits defined by these two constraints. Thus, the set E, defined

<figure><img src="image_3.png" /><figcaption>Figure 2: Illustration of the viability set. The boundary of the hydric stress is depicted in blue and of the nitrogen one in red.</figcaption></figure>

by the two constraints (water and nitrogen in soil), is considered viable if, starting from an initial soil moisture level and nitrogen quantity that respect the boundaries of E at the beginning of the season, there exists a strategy (not necessarily optimal) that ensures current and futures needs do not exceed any of the imposed limits. If the moisture level and nitrogen availability at the beginning of the season are such that irrigation prevents both water and nitrogen stress at any point during the season, then maximal biomass production is guaranteed. In other words, it means that neither water nor nitrogen is limiting during the season. Depending on whether the operational parameters($C_N^{in}$ and $u_{max}$) of the fertigation exceed certain thresholds, there are several irrigation strategies that can ensure this optimal condition at harvest time. Our viability analysis, utilizing the model, enables precise delineation of the permissible ranges for nitrogen concentration in the irrigation water and the maximum irrigation rate necessary to uphold the viability property. The viability thresholds identified by the model delineate the minimum input requirements to achieve maximum biomass at harvest time while avoiding unnecessary nitrogen accumulation in the soil. A preliminary analysis indicates that, given an initial humidity level and nitrogen quantity that prevent both water stress and nitrogen stress at the beginning of the season, the optimal irrigation strategy minimizes the terminal nitrogen while maximizing biomass production. Subsequently, we explore control strategies aimed at maximizing biomass (i.e. maintaining conditions
within the viability domain, (see Figure 2) while minimizing total water consumption. This constitutes an optimal control problem under a viability constraint, a completely new approach that has not been used for such a problem before. We identified a variety of markedly different control strategies, which are illustrated below, depending on threshold values of the initial nitrogen concentrations $N_0^b > N_0^\dagger > N_0^\sharp > \eta_C$ (the existence of such values have been proved mathematically in [1]):

## Strategy 1

When the initial quantity of nitrogen is high ($N_0 > N_0^b$), the optimal strategy consists of no irrigation until the humidity threshold $S^*$ is met, and then keeping the humidity level exactly at this threshold up to the final time.

<figure><img src="image_4.png" /><figcaption>Figure 3: Optimal strategy for $N_0 = N_0^b$</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Optimal strategy for $N_0 > N_0^b$</figcaption></figure>

## Strategy 2

For lower value of initial nitrogen $N_0^\dagger \le N_0 \le N_0^b$, the lack of irrigation causes crops to suffer from nitrogen stress before experiencing water stress, making the preceding strategy is suboptimal. The optimal solution consists in keeping the system at the edge of this constraint until it reaches the humidity threshold, and then keeping the humidity level exactly at this threshold.

<figure><img src="image_6.png" /><figcaption>Figure 5: Optimal solution for $N_0 = N_0^\dagger$</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6: Optimal solution for $N_0 \in (N_0^\dagger, N_0^b)$</figcaption></figure>

While these two first strategies are quite intuitive and can be practically interpreted, our analysis reveals two other strategies that are very different and non-intuitive.

## Strategy 3

A surprising feature occurs when keeping the system at the edge of the nitrogen stress does not allow reaching the humidity threshold (because the time horizon as been reached before), for small initial nitrogen content. When the initial nitrogen is between $\eta_C$ and $N_0^\sharp$, the optimal strategy consists of irrigating from the beginning to maintain the humidity level at its maximal level up to a precise time, from which stopping the irrigation conducts the system to reach the nitrogen stress exactly at the terminal time.

<figure><img src="image_8.png" /><figcaption>Figure 7: Optimal solution for $N_0 = \eta_C$</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 8: Optimal solution for $N_0 = N_0^\sharp$</figcaption></figure>
## Strategy 4

Finally, another non-intuitive feature occurs when the initial nitrogen level is between $N_0^\sharp$ and $N_0^\dagger$. The optimal strategy involves continuous irrigation following specific profiles while avoiding both nitrogen and water stress.

This research represents an initial step toward developing a decision support tool for irrigation management under water scarcity, particularly with the integration of reuse strategies (see [1]). Note that the optimal irrigation strategies can be quite different for $N_0 < N_0^\sharp$ or $N_0 > N_0^\dagger$, the minimal water consumption is always decreasing with initial $N_0$, down to the constant level for which the system receives enough nitrogen to never face a nitrogen stress, that is when $N_0 \ge N_0^\flat$.

<figure><img src="image_10.png" /><figcaption>Figure 9: Optimal solution for $N_0 = \eta_C$</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 10: Optimal solution for $N_0 \in (N_0^\sharp, N_0^\dagger)$ with the singular control</figcaption></figure>

# 4 Conclusion

Water scarcity, affecting approximately 40% of European regions, is driven by climate change, drought, and certain human activities. This severe issue is precipitating an unprecedented food crisis, with agricultural output declining by over 30%. In this scenario, utilizing wastewater for crop irrigation emerges as a potential solution, though it requires substantial investment due to stringent European regulations on treated wastewater reuse, despite high treatment costs. This paper proposed an optimized fertigation strategy using a crop model, viability analysis, and optimal control. It suggests tailoring water and nutrient supplies to maximize biomass production based on initial nitrogen availability. However, the model does not account for excess nitrogen that plants do not uptake, which could contaminate soil and groundwater. Therefore, future research should focus on minimizing nitrogen leaching due to surplus irrigation water.

# References

[1] Gildas Mahugnon Dadjo, Alain Rapaport, Jérôme Harmand, Rosane Ushirobira, Denis Efimov. Water-minimizing strategies under viability constraint for a crop fertigation model. 2023. Preprint. Available at: https://hal.inrae.fr/hal-04267286.

[2] Antoine Haddon, Alain Rapaport, Sébastien Roux, Jérôme Harmand. Model Based Optimization of Fertilization with Treated Wastewater Reuse. Advances in Water Resources, 2023, 181 (104561). https://doi.org/10.1016/j.advwatres.2023.104561.

[3] Kenza Boumaza, Nesrine Kalboussi, Alain Rapaport, Sébastien Roux, Carole Sinfort. Optimal control of a crop irrigation model under water scarcity. Optimal Control Applications and Methods, 2021, 42 (6), pp.1612-1631. https://doi.org/10.1002/oca.2749.

[4] N. Pelak, R. Revelli, and A. Porporato. A dynamical systems framework for crop models: Toward optimal fertilization and irrigation strategies under climatic variability. Ecological Modelling, 365:80–92, 2017. https://doi.org/10.1016/j.ecolmodel.2017.10.003

[5] Aubin J.P. Viability Theory, Birkauser, 2009.

[6] https://www.fao.org/fileadmin/templates/wsfs/docs/Issues_papers/HLEF2050_Global_Agriculture.pdf