# Feedback design for biological control by the sterile insect release technique exploiting monotone system theory

Amit Bhaya, Pierre-Alexandre Bliman

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## To cite this version:

Amit Bhaya, Pierre-Alexandre Bliman. Feedback design for biological control by the sterile insect release technique exploiting monotone system theory. European Journal of Control, In press, 86, pp.101292. 10.1016/j.ejcon.2025.101292. hal-05165091

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Submitted on 16 Jul 2025

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# Feedback design for biological control by the sterile insect release technique exploiting monotone system theory

Amit Bhaya*

Pierre-Alexandre Bliman†

July 12, 2025

## Abstract

The Sterile Insect Technique (SIT) is a promising control method against insect pests and insect vectors. It consists in releasing males previously sterilized in laboratory, in order to reduce or eliminate a specific wild population. We study the implementation of SIT-based elimination campaign of *Aedes* mosquitoes using feedback control. We provide state-feedback and output-feedback control laws and establish their convergence, as well as their robustness properties. In this design procedure, a pivotal role is played by the use of properties of monotone systems. Simple illustrative simulations are provided.

**Keywords.** Sterile Insect Technique; State-feedback design; Output-feedback design; Interval observers; Monotone system theory

## 1 Introduction

Many species of mosquitoes transmit established or potentially (re)emerging diseases, such as *Aedes aegypti* and *Aedes albopictus*, which are vectors of dengue, chikungunya and Zika. Other pests threaten agriculture, such as the Mediterranean fly (*Ceratitis capitata*) or the oriental fly (*Bactrocera dorsalis*), which oviposit under the skin surface of their host fruits and damage the harvest. Controlling these species is likely to become even more necessary in the future, including in the temperate zones of the world, due to global climate changes. Control through insecticides is now used with reluctance, due to impact on biodiversity and resistance-induced reduced efficiency. Alternative methods include biological control, which uses beneficial insects or pathogens that they transport to control unwanted insects, weeds, or diseases, and usually target specific species of interest, without harming other organisms.

The method we are interested in here is the Sterile Insect Technique (SIT) [19, 2], involving the release of very large numbers of males previously sterilized (usually by irradiation, or by an alternative technique like the genetic RIDL technique or the use of Wolbachia-infected males). The females inseminated by these males do not produce viable eggs. With sufficiently high number of sterile insects, most of the crosses are sterile, and sustained releases on an area-wide basis may then succeed in reducing the wild population, or even in driving it to extinction.

Designing practically successful release campaigns on a large scale for a limited cost is still a source of some important questions. In this paper, we are interested in the implementation of SIT on *Aedes* mosquitoes by feedback control.

SIT model and analysis results have been presented in [17, eq. (1)] and [25, eq. (1)] for *Aedes aegypti* mosquitoes, allowing the study of epidemic spread [15]. This species has the particularity of experiencing intraspecific competition, apparently through reduction of the oviposition rate in congested breeding

<footnote>*Department of Electrical Engineering, Federal University of Rio de Janeiro, PEE/COPPE/UFRJ, Rio de Janeiro, RJ 219945-970, Brazil, amit@nacad.ufrj.br</footnote>

<footnote>†Corresponding author – Sorbonne Université, Inria, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions UMR 7598, Equipe MUSCLEES, Paris, France, pierre-alexandre.bliman@inria.fr</footnote>
sites. Continuing in the same vein, a simpler, 3D model, including the Allee effect [18], was proposed in [11, eq. (2)]. As another example, intraspecific competition in the species *Anopheles*, vectors of malaria, occurs later in the aquatic stages of life, through mortality increase. Such a modelling option has been explored in [3, eq. (28)-(29)-(30)-(31)], subsequently simplified in [4, eq. (16)], resulting in a nearby 3D model. In the sequel we use a 3D model, simply obtained by ignoring the Allee effect in [11, eq. (2)]. This model, numbered (2) below, is similar to one of the models in [1] but does not make the limiting assumption of sex ratio 1:1 at birth and of identical male/female mortality rates, which reduce the system dimension. It has been used for feedback control in [6, 7, 13] and this allows comparison with the techniques proposed in this paper.

Notice that, from a control theory perspective, eliminating mosquito population is a problem of global stabilization of the extinction equilibrium. A key idea to achieve this goal in a biological control method such as SIT, is to design control strategies that ensure a sufficient proportion of sterile males within the total male population, in order to reduce the basic offspring number of the population to a subcritical value. This principle was used in [10] to design successful release campaign by periodic impulsive control, with the size of every release adjusted in accordance with the current estimate of the wild population size. It was also used in [8] as basis for feedback synthesis in slightly different context (replacement of wild mosquito population by *Wolbachia*-infected mosquitoes, whose vectorial capacity is reduced). These results were obtained in relatively simple situations, and understanding how to apply the technique to more complex models is a challenge we want to tackle here.

In [6], backstepping control [12] was used to build state feedback law stabilizing the extinction equilibrium of system (2), denoted $E_0$ in the sequel. The idea therein was to stabilize the sterile-to-wild male ratio at a given value, chosen sufficiently large in order to force elimination of the mosquito population. Static output feedback laws were also proposed in [6], based on measured output that is either the total number of males, or the number of wild males, but stabilization was not proved in general for the corresponding policies. Numerical simulations were provided and parametric robustness was tested numerically. The approach taken in [7] combined reinforcement learning with mathematical analysis to identify a candidate solution for an explicit stabilizing feedback control. The latter was based on measuring the total number of males and the total number of females in the field. However convergence of the controlled system was only suggested by numerical simulations. Lastly, in [13], backstepping was used to design stabilizing control law for a reduced 1D model and to test it numerically on the complete model (2).

In this paper we propose several feedback control laws capable of achieving the extinction equilibrium for a population evolving according to system (2). A state-feedback control significantly simpler than [6] is first displayed. It is shown afterwards how to adapt it to the presence of model uncertainties. Last, an output-based control law is synthesized. To our knowledge, this constitutes the first contribution to robust control and output-based control laws for the implementation of Sterile Insect Technique. All these results are proved thanks to comparison principle and monotone system theory [23, 24]<fnref n="1" />, and one of the aims of the present paper is precisely to emphasize the theoretical and practical interest of this theory for the biological control of vectors and pests.

The model is presented in Section 2 and the control principle is described in Section 3. State-feedback synthesis is then achieved in Section 4. Taking advantage of the monotonicity properties [23, 24] of the life cycle model, we then show in Section 5 how to ensure robust stabilization against parametric and dynamic uncertainty. It is then shown in Section 6 how interval observers constitute a natural way to synthesize output-based feedback laws when only partial state measurement is available. Numerical simulations illustrate the findings in Section 7, before the Conclusion. A preliminary version of this paper was published in the Proceedings of the 2025 European Control Conference [9].

**Notations** For any $z \in \mathbb{R}$, define $z^+ := \max\{z; 0\}$. A vector $x \in \mathbb{R}^n$, $n \in \mathbb{N}$, is said *nonnegative* if all its components are nonnegative; *positive* if it is nonnegative and nonzero; *strictly positive* if all its components are positive. Defining the corresponding order relations, these properties are written respectively $x \ge 0_n$, resp. $x > 0_n$, resp. $x \gg 0_n$. Any two elements $x < y$ generate the *order interval* [23] $[x, y] := \{z : x \le z \le y\}$. A matrix is said to be *Metzler* if all its off-diagonal components are nonnegative. The *stability modulus* [20, p. 32] (or *spectral abscissa* [14]) of a matrix, is the greatest real part of its eigenvalues.

<footnote n="1">¹See especially Section 3. Competition between two agents/teams: An equation-free theory in [24]</footnote>
## 2 Model

The uncontrolled model that describes the dynamics of the wild mosquito population alone is first provided in (1). Its state variable incorporates three compartments, representing aquatic phase (E), males (M) and fertilized females (F). The model controlled through sterile male ($M_s$) releases is then presented in (2). Well-posedness and positiveness are treated in Theorem 1.

### • Uncontrolled system

$$ \dot{E} = \beta_E F \left(1 - \frac{E}{K}\right)^+ - (\nu_E + \delta_E)E, \quad (1a) $$

$$ \dot{M} = (1 - \nu)\nu_E E - \delta_M M, \quad (1b) $$

$$ \dot{F} = \nu\nu_E E - \delta_F F. \quad (1c) $$

The life of *Aedes* mosquitoes goes through different phases: eggs, larvae, pupae, and finally adults capable of reproducing. The last phase is aerial, but the three first ones are all aquatic, and subject to competition for space and food resources in the breeding sites. The state variable of the model has three components, *E*, *M* and *F*. The component *E* merges the whole aquatic phase in a single quantity, *M* represents the (fertile) males and *F* represents the adult females. In natural conditions, the females are inseminated immediately after hatching and produce offspring. This is why the birth term is proportional to *F*, but doesn't display the male population *M* — an apparently paradoxical modelling option, justified in the presence of a sufficiently large male population. Section 5.2 considers specifically the case of extreme male scarcity. The recruitment term in (1a) models egg breeding and the term in parentheses models 'skip oviposition' behavior: the females deposit comparatively fewer eggs in sites with already high occupation rate. In coherence with these explanations, the parameter $\beta_E$ represents the mean number of eggs laid by a female mosquito per time unit (typically per day); $\nu_E$ the hatching parameter (integrating survivorship and development during the whole aquatic phases); $\nu$ the sex-ratio in offspring; $K$ the egg carrying capacity; and $\delta_E, \delta_M, \delta_F$ the respective death rates of each of the three categories.

**Remark 1.** In the early forms of the model [17, 25, 15], the recruitment term in (1a) contained the factor $(1 - \frac{E}{K})$ in place of $(1 - \frac{E}{K})^+$. We feel that this slight modification, that enforces nonnegativity, is more faithful to the interpretation as a birth term. In any case, both systems possess the same positively invariant attractive set, where $E \le K$ (see Theorem 1 below).

### • Controlled system

$$ \dot{E} = \beta_E F \left(1 - \frac{E}{K}\right)^+ - (\nu_E + \delta_E)E, \quad (2a) $$

$$ \dot{M} = (1 - \nu)\nu_E E - \delta_M M, \quad (2b) $$

$$ \dot{F} = \nu\nu_E E \frac{M}{M + \gamma M_s} - \delta_F F, \quad (2c) $$

$$ \dot{M}_s = u - \delta_s M_s. \quad (2d) $$

The evolution of the compartments *E* and *M* obey the same equations in the controlled system. The latter contains the supplementary equation (2d), which accounts for the evolution of the population of sterilized males $M_s$. The latter, whose mortality rate is denoted $\delta_s$, are released as adults in the field with an instantaneous rate *u*: this time function is the control variable of the model. In the present paper, as well as in [6, 13, 7] we assume that this variable varies continuously in time. (In practice, the releases are made periodically, so it is more realistic to consider impulsive, periodic, releases [10].) The main subtlety of system (2) lies in (2c): here, due to the presence of the sterilized males, only a reduced proportion of mating gives rise to viable eggs. Now, the variable *F* represents only those (fertile) females that are inseminated by a fertile male, as the matings with the sterile males do not produce viable eggs. The parameter $\gamma$, usually smaller than 1, represents the relative mating competitiveness of the sterile
males. When $\gamma = 1$, the fraction $\frac{M}{M+M_s}$ is exactly the proportion of fertile males in the total male population.

All parameters in models (1) and (2) are assumed positive in the sequel, and $\nu < 1$. For Aedes mosquitoes, measurements have shown that the mortality rates are usually ordered as follows: $\delta_F \le \delta_M \le \delta_s$. While this is of course quite important in practice, we stress that the mathematical results in the paper are valid without these assumptions.

For simplicity, one considers control functions $u \in L_{\text{loc}}^1(0, +\infty)$ such that $u \ge 0$. We call solution of equation (2) any quadruple $(E, M, F, M_s)$ of functions in $C^1 \times C^1 \times C^1 \times W_{\text{loc}}^{1,1}$ fulfilling the equations almost everywhere.

**Theorem 1.** For any nonnegative initial condition $(E_0, M_0, F_0, M_{s,0})$, any nonnegative $u \in L_{\text{loc}}^1(0, +\infty)$, there exists a unique solution to (2). Moreover, the latter takes on nonnegative values and the current point $(E(t), M(t), F(t))$ converges to the forward-invariant order interval $[\mathbf{E}_0, \mathbf{E}^U]$, where

$$ \mathbf{E}_0 := (0, 0, 0), \quad \mathbf{E}^U := K \left( 1, \frac{(1-\nu)\nu_E}{\delta_M}, \frac{\nu\nu_E}{\delta_F} \right). \qquad (3) $$

Finally, $M(0) > 0$ implies $M(t) > 0, t > 0$; $E(0) > 0$ or $F(0) > 0$ implies $E(t), M(t), F(t) > 0, t > 0$; and $E(0) = F(0) = 0$ implies $E(t), F(t) = 0, t > 0$.

*Proof.* From (2a) one deduces that $-(\nu_E + \delta_E)E \le \dot{E} \le \beta_E F(1 - \frac{E}{K})$, yielding ultimate boundedness of $E$, more precisely

$$ 0 \le \liminf_{t \to +\infty} E(t) \le \limsup_{t \to +\infty} E(t) \le K, $$

for every trajectory. From (2b) one then deduces that $M$ is ultimately bounded, with

$$ 0 \le \liminf_{t \to +\infty} M(t) \le \limsup_{t \to +\infty} M(t) \le \frac{(1-\nu)\nu_E}{\delta_M} K. $$

On the other hand, equation (2c) implies $0 \le \dot{F} + \delta_F F \le \nu\nu_E E$, so that $F$ is ultimately bounded too, with

$$ 0 \le \liminf_{t \to +\infty} F(t) \le \limsup_{t \to +\infty} F(t) \le \frac{\nu\nu_E}{\delta_F} K, $$

for any initial condition and any $u$ as in the statement. This property implies that the solutions are defined on $[0, +\infty)$.

The fact that any component initially positive remains positive for any time instant comes from the inequalities

$$ \dot{E} \ge -(\nu_E + \delta_E)E, \quad \dot{M} \ge -\delta_M M, \quad \dot{F} \ge -\delta_F F. \qquad (4) $$

On the other hand, positivity of $E$ makes $M$ and $F$ positive, and positivity of $F$ renders $E$ positive. $\square$

# 3 Control principle

We describe the basic idea used for control synthesis. This requires precise analysis of the existence and stability of the equilibrium points of (1). This is first recalled in Section 3.1, after which the key result is displayed in Section 3.2 (Proposition 3).

## 3.1 Basic offspring number and analysis of the uncontrolled system (1)

In this section we recall the asymptotic behaviour of the uncontrolled system (1), highlighting the key role of the basic offspring number. Related analysis results and demonstrations have been provided in [3, 4, 6]. In order to introduce useful tools from monotone system theory, we provide a succinct proof below.

Ordering the state vector components as $(E, M, F)$, the Jacobian matrix of system (1) is

$$ \begin{pmatrix} -\frac{\beta_E}{K} F - (\nu_E + \delta_E) & 0 & \beta_E \left(1 - \frac{E}{K}\right) \\ (1 - \nu)\nu_E & -\delta_M & 0 \\ \nu\nu_E & 0 & -\delta_F \end{pmatrix}. $$
This matrix is a Metzler matrix at any $(E, M, F)$ in the forward-invariant set $[\mathbf{E}_0, \mathbf{E}^U]$ defined by (3), so that system (1) is cooperative [23] in $[\mathbf{E}_0, \mathbf{E}^U]$, see [23, Proposition 3.1.1].

The system of equations defining the equilibrium points of system (1) is

$$0 = \beta_E F \left(1 - \frac{E}{K}\right) - (\nu_E + \delta_E)E = (1 - \nu)\nu_E E - \delta_M M = \nu\nu_E E - \delta_F F, \quad (5)$$

and the extinction equilibrium $\mathbf{E}_0$ defined in (3) is always a solution.

On the other hand, due to the last two equations in (5), all components of an equilibrium have to respect given relative proportions, so that any possible positive equilibrium is indeed strictly positive, due to (1b) and (1c). From the last equation in (5), $F$ can be expressed in terms of $E$ and parameters, and substituting this in the first equation in (5) yields:

$$\beta_E \frac{\nu\nu_E}{\delta_F} \left(1 - \frac{E}{K}\right) = \nu_E + \delta_E,$$

which has zero or one positive solution, according to the sign of $\beta_E \nu\nu_E - (\nu_E + \delta_E)\delta_F$. The asymptotic behaviour is summarized by the following result. (A similar global stability result is given in [4, Theorem 1] for a related system.)

**Theorem 2.** Let

$$\mathcal{N} := \frac{\beta_E \nu \nu_E}{(\nu_E + \delta_E)\delta_F}. \qquad (6)$$

* If $\mathcal{N} \le 1$, then $\mathbf{E}_0$ is the unique equilibrium point of system (1), and it is GAS in $\mathbb{R}_+^3$.

* If $\mathcal{N} > 1$, then, apart from $\mathbf{E}_0$, system (1) also admits the positive equilibrium point

$$\mathbf{E}^* := K \left(1 - \frac{1}{\mathcal{N}}\right) \left(1, \frac{(1 - \nu)\nu_E}{\delta_M}, \frac{\nu\nu_E}{\delta_F}\right), \quad (7)$$

and the latter is GAS in $\mathbb{R}_+^3 \setminus \{(0, M, 0) : M \ge 0\}$.

The constant $\mathcal{N}$ is called the basic offspring number. It characterizes the viability of the considered population in absence of control. Notice that $\mathbf{E}^* = (1 - \frac{1}{\mathcal{N}})\mathbf{E}^U$. One may show without difficulty that the GAS equilibrium points are indeed exponentially stable when $\mathcal{N} \ne 1$.

*Proof of Theorem 2.* As a preliminary, notice that (1a)-(1c) evolves independently of the variable $M$. The Jacobian matrix of the subsystem describing the evolution of $(E, F)$ is

$$\begin{pmatrix} -\frac{\beta_E}{K}F - (\nu_E + \delta_E) & \beta_E \left(1 - \frac{E}{K}\right) \\ \nu\nu_E & -\delta_F \end{pmatrix} \qquad (8)$$

and is irreducible for any $(E, F) \in [0, K) \times [0, \frac{\nu\nu_E}{\delta_F}K]$. Therefore, the $(E, F)$-subsystem is strongly monotone [23, p. 3] in the forward-invariant, attractive, set $[0, K) \times [0, \frac{\nu\nu_E}{\delta_F}K)$, due to [23, Theorem 4.1.1]. According to the value of $\mathcal{N}$, (1a)-(1c) admits one or two equilibrium points, which are the projections of $\mathbf{E}_0$ and $\mathbf{E}^*$ on their 1st and 3rd components.

* Assume $\mathcal{N} \le 1$. There is no equilibrium point of the $(E, F)$-subsystem other than the projection $(0, 0)$ of $\mathbf{E}_0$. Notice that the right-hand side of the equation is negative at the point which is the projection of $\mathbf{E}^U$. Therefore the trajectory originating from this point is decreasing [23, Proposition 3.2.1]. Being bounded from below, it converges, and the limit has to be $(0, 0)$. As the projection of $\mathbf{E}^U$ is an upper bound of the positively-invariant set under consideration, the trajectory originating from this point dominates every other trajectory in this set. The latter thus converge and the origin is GAS. Under these conditions, the $M$ component in (1b) also vanishes, and the equilibrium $\mathbf{E}_0$ of (1) is GAS.

* Assume $\mathcal{N} > 1$. System (1) then admits only the two distinct equilibrium points $\mathbf{E}_0$ and $\mathbf{E}^* \gg \mathbf{E}_0$. To show that $\mathbf{E}^*$ attracts every trajectory from $\mathbb{R}_+^3 \setminus \{(0, M, 0) : M \ge 0\}$, consider the $(E, F)$-subsystem. It possesses two equilibrium points $(0, 0)$ and $K(1 - \frac{1}{\mathcal{N}})\left(1, \frac{\nu\nu_E}{\delta_F}\right)$, deduced from $\mathbf{E}_0$ and $\mathbf{E}^*$ by projection on the 1st and 3rd components. In absence of a third equilibrium point in the interval between these two points, [23, Theorem 2.2.2] ensures that one of the two equilibrium points attracts
the trajectories issuing from any initial condition except for the other equilibrium, due to the fact that the flow is strictly-order preserving [23, p. 2]. The Jacobian matrix (8) in (0,0) admits the characteristic polynomial $s^2 + (\nu_E + \delta_E + \delta_F)s + (\nu_F + \delta_E)\delta_F - \nu\nu_E\beta_E = s^2 + (\nu_E + \delta_E + \delta_F)s + (\nu_E + \delta_E)\delta_F(1 - \mathcal{N})$. As $\mathcal{N} > 1$, the origin is unstable, and one deduces that every trajectory is in fact attracted to the strictly positive equilibrium point of the $(E, F)$-subsystem. The third component $M$ evolves as an output of the previous subsystem, and one finally obtains the GAS property for $\mathbf{E}^*$. This concludes the proof of Theorem 2. $\square$

**Remark 2.** Notice that the complete system (1) is not strongly order preserving [23], otherwise $\mathbf{E}^*$ would be GAS in $\mathbb{R}_+^3 \setminus \{\mathbf{E}_0\}$ instead of $\mathbb{R}_+^3 \setminus \{(0, M, 0) : M \ge 0\}$.

## 3.2 Principle of the proposed method: control of the apparent offspring number

From now on and in the rest of the paper, we assume that the wild population is viable, that is

$$ \mathcal{N} > 1, $$

and for sake of simplicity denote the state of system (1) as

$$ x := (E \ M \ F)^T. \qquad (9) $$

We study in this section a general principle of control synthesis, given in Proposition 3. The latter is general enough and provides useful insights, for state or output feedback synthesis.

From the point of view of the wild population, the presence of the sterile insects modifies solely the birth term in (1a), through the multiplication by the ratio $\frac{M(t)}{M(t)+\gamma M_s(t)}$. This is the idea exploited in the present paper, which serves as a general principle for control synthesis in what follows, see also [10, 6]. Before stating the main result of this section, we introduce an important notion.

**Definition 1.** We call a nonzero trajectory of (2) any trajectory generated by a nonnegative control $u \in L_{\text{loc}}^1(0, +\infty)$ and an initial condition $(x_0, M_{s,0}) \in \mathbb{R}_+^4$ such that $x_0 \neq \mathbf{E}_0$. We define the apparent offspring number of a nonzero trajectory as the corresponding function

$$ \mathcal{N}_{\text{app}}(t) := \mathcal{N} \frac{M(t)}{M(t) + \gamma M_s(t)}. \qquad (10) $$

Due to Theorem 1, the apparent offspring number of a nonzero trajectory is well defined for any $t > 0$.

**Proposition 3.** Consider a nonzero trajectory of (2) such that

$$ \exists \theta \in (0, 1), \exists T \ge 0, \forall t \ge T, \quad \mathcal{N}_{\text{app}}(t) \le \theta. \qquad (11) $$

Then there exist $v \gg 0_3$, $c > 0$, $\lambda > 0$ depending only on $\theta$, $T$ and the parameters of the system, such that $v^\top x(t) \le c v^\top x_0 e^{-\lambda t}$ for any $t \ge 0$.

**Proof of Proposition 3.** Consider a nonzero trajectory $(E, M, F, M_s)$ of (2) that fulfils (11). One has

$$ \nu\nu_E \frac{M(t)}{M(t) + \gamma M_s(t)} = \nu\nu_E \frac{\mathcal{N}_{\text{app}}(t)}{\mathcal{N}}, $$

so this solution of (2) fulfils, for any $t \ge T$, the following system of differential inequalities:

$$ \dot{E} \le \beta_E F - (\nu_E + \delta_E)E, $$

$$ \dot{M} = (1 - \nu)\nu_E E - \delta_M M, $$

$$ \dot{F} = \nu\nu_E \frac{\mathcal{N}_{\text{app}}(t)}{\mathcal{N}} E - \delta_F F. $$

Define the vector $v \gg 0_3$ by

$$ v^\top := \left( (1 - \nu) \frac{\nu_E}{\nu_E + \delta_E} \quad \frac{1 - \theta}{3} \quad \frac{1 + \theta}{2} \frac{1 - \nu}{\theta} \quad \nu \mathcal{N} \right). $$
One has

$$
\begin{align*}
v^\top \begin{pmatrix} -(\nu_E + \delta_E) & 0 & \beta_E \\ (1-\nu)\nu_E & -\delta_M & 0 \\ \nu\nu_E \frac{\mathcal{N}_{\text{app}}(t)}{\mathcal{N}} & 0 & -\delta_F \end{pmatrix} &= \left( (1-\nu)\nu_E \left(-1+\frac{1-\theta}{3}+\frac{1}{2}\frac{1+\theta}{\theta}\mathcal{N}_{\text{app}}(t)\right) - \frac{1-\theta}{3}\delta_M \frac{(1-\nu)\nu_E\beta_E}{\nu_E+\delta_E} \left(1-\frac{1}{2}\frac{1+\theta}{\theta}\right) \right) \\
&= \left( -\frac{1}{6}(1-\nu)\nu_E \left(6+2(\theta-1)-3\frac{1+\theta}{\theta}\mathcal{N}_{\text{app}}(t)\right) - \frac{1-\theta}{3}\delta_M - \frac{1}{2}\frac{(1-\nu)\nu_E\beta_E}{\nu_E+\delta_E}\frac{1-\theta}{\theta} \right).
\end{align*}
$$

The last two components of the vector are negative. On the other hand, when $t \ge T$, then $\mathcal{N}_{\text{app}}(t) \le \theta$, so that

$$6 + 2(\theta - 1) - 3\frac{1+\theta}{\theta}\mathcal{N}_{\text{app}}(t) \geq 6 + 2(\theta - 1) - 3(1+\theta) = 1 - \theta > 0.$$

Therefore, there exists $\lambda > 0$ such that, for any $t \ge T$,

$$v^\top \begin{pmatrix} -(\nu_E + \delta_E) & 0 & \beta_E \\ (1-\nu)\nu_E & -\delta_M & 0 \\ \nu\nu_E \frac{\mathcal{N}_{\text{app}}(t)}{\mathcal{N}} & 0 & -\delta_F \end{pmatrix} \le -\lambda v^\top,$$

and thus

$$\frac{dv^\top x(t)}{dt} \le -\lambda v^\top x(t).$$

On the other hand, using the fact that $\mathcal{N}_{\text{app}}(t) \le \mathcal{N}$, there exists a real number $\mu \ge 0$ such that, for any $0 \le t \le T$,

$$v^\top \begin{pmatrix} -(\nu_E + \delta_E) & 0 & \beta_E \\ (1-\nu)\nu_E & -\delta_M & 0 \\ \nu\nu_E \frac{\mathcal{N}_{\text{app}}(t)}{\mathcal{N}} & 0 & -\delta_F \end{pmatrix} \le \mu v^\top.$$

By use of Gronwall's lemma, one thus has $v^\top x(t) \le v^\top x(0)e^{\mu t} \le v^\top x(0)e^{\mu T}$ for any $0 \le t \le T$; while $v^\top x(t) \le v^\top x(T)e^{-\lambda(t-T)} \le cv^\top x(0)e^{(\lambda+\mu)T}e^{-\lambda t}$ for any $t \ge T$, with $c := e^{(\lambda+\mu)T}$. This is the claimed property, and the proof of Proposition 3 is complete. $\square$

**Remark 3.** Same arguments than in the proof of Theorem 4 allow to show that, if

$$\exists \theta > 1, \exists T \ge 0, \forall t \ge T, \quad \mathcal{N}_{\text{app}}(t) \ge \theta,$$

instead of (11), then the corresponding set of differential inequalities is fulfilled

$$\dot{E} = \beta_E F \left(1 - \frac{E}{K}\right)^+ - (\nu_E + \delta_E)E,$$

$$\dot{M} = (1 - \nu)\nu_E E - \delta_M M,$$

$$\dot{F} \ge \nu \nu_E E \frac{\theta}{\mathcal{N}} - \delta_F F.$$

This allows to show that all nonzero trajectories are bounded ultimately from below by the positive equilibrium of the corresponding system of differential equations. The latter has basic offspring number equal to $\theta > 1$, and the corresponding case in Theorem 2 applies.

**Remark 4.** The rate of convergence of the population to zero under a control that verifies (11) may be guaranteed (that is, estimated from below) using the stability modulus of the matrix appearing in the comparison system in the proof of Proposition 3. On the other hand, the fact that (4) holds, whatever the control $u$, imposes hard limitations on the convergence rate. However, this is usually not restrictive in practice. Notice that these bounds are effective: when $\mathcal{N}_{\text{app}}$ converges arbitrarily fast to 0, the exponential convergence to $\mathbf{E}_0$ occurs at rate $\min\{\nu_E + \delta_E, \delta_M, \delta_F\}$.
Ensuring condition (11) by adequate feedback for every nonzero trajectory is thus sufficient to stabilize the origin of the wild insect equations. We will apply this idea in Section 4 (resp. Section 6) to the synthesis of stabilizing state feedback (resp. output feedback) control laws through continuous-time releases. It is intuitively reasonable to expect that (11) may be fulfilled provided the released quantities are ‘large enough’. However, we also aim at constructing control laws that vanish when reaching the control goal. We will see in the sequel that it is even possible to complete stabilization with a finite total amount of released insects, that is with the property

$$ \int_{0}^{+\infty} u(t) \cdot dt < +\infty. \qquad (12) $$

Before going further, notice that (11) is equivalent to

$$ \exists \theta \in (0, 1), \exists T \ge 0, \forall t \ge T, \quad \frac{M_s(t)}{M(t)} \ge \alpha(\theta) := \frac{1}{\gamma} \left( \frac{\mathcal{N}}{\theta} - 1 \right). $$

The map $\theta \mapsto \alpha(\theta)$ is decreasing and maps $(0, 1)$ into $(\frac{1}{\gamma}(\mathcal{N} - 1), +\infty)$, and finally (11) is equivalent to

$$ \exists \alpha > \alpha_{\text{crit}} := \frac{1}{\gamma}(\mathcal{N} - 1), \exists T \ge 0, \forall t \ge T, \quad \frac{M_s(t)}{M(t)} \ge \alpha. \qquad (13) $$

This form is used in the sequel as the cornerstone for control synthesis.

# 4 State feedback synthesis

In this section we apply the idea developed in Section 3 to obtain state feedback controls. We aim to choose $u$ in such a way as to achieve condition (13) (or equivalently (11)) by use of permanent releases, and then benefit from the application of Proposition 3. For this, we first recall in Section 4.1 an idea for control design introduced in [8], refined and exploited in the sequel. We then propose a key control law (Theorem 4) in Section 4.2 and show its stabilizing effect.

## 4.1 Design of an adapted control law

Let $\alpha > \alpha_{\text{crit}} = \frac{1}{\gamma}(\mathcal{N} - 1)$ and $\omega > 0$. If $u$ is chosen in such a way that

$$ \frac{d(M_s - \alpha M)}{dt} + \omega(M_s(t) - \alpha M(t)) \ge 0, \quad t \ge 0, \qquad (14) $$

that is $\dot{M}_s \ge \alpha \dot{M} - \omega(M_s(t) - \alpha M(t))$, then by integration

$$ M_s(t) - (M_s(0) - \alpha M(0))e^{-\omega t} \ge \alpha M(t), \qquad (15) $$

for any $t \ge 0$, and thus

$$ \liminf_{t \to +\infty} (M_s(t) - \alpha M(t)) \ge 0, \quad t \ge 0. $$

However, this does not imply that $M_s(t) \ge \alpha M(t)$ on $[T, +\infty)$ for large enough $T > 0$. In fact, (15) implies

$$ \frac{M_s(t)}{M(t)} \ge \alpha + \frac{\alpha M(0) - M_s(0)}{M(t)} e^{-\omega t}. $$

But the second term of the right-hand side here is not guaranteed to be positive as $M(t)$ goes to zero. The numerator of the fraction is typically positive at the beginning of a campaign, when $M_s(0) = 0$. However it could possibly be negative, for example, after a halt and a new beginning, and the fraction itself could take negative arbitrarily large values as $M(t) \to 0$. In such conditions, there is no evidence that (13) would be guaranteed, not even for some value in $(\alpha_{\text{crit}}, \alpha)$ instead of $\alpha$.

As an alternative to (14), consider the condition

$$ \frac{d}{dt} \left( \frac{M_s(t)}{M(t)} - \alpha \right) + \omega \left( \frac{M_s(t)}{M(t)} - \alpha \right) \ge 0, \quad t \ge 0. \qquad (16) $$
When (16) is verified, one obtains

$$ \frac{M_s(t)}{M(t)} \geq \alpha + \left( \frac{M_s(0)}{M(0)} - \alpha \right) e^{-\omega t}, \quad t \geq 0, $$

which indeed ensures the desired property (13) with a constant $\alpha$ replaced by any value in $(\alpha_{crit}, \alpha)$, contrary to (15). Inequality (16) is expressed equivalently as

$$ \dot{M}_s \geq \frac{M_s(t)}{M(t)} \dot{M}(t) + \omega(\alpha M(t) - M_s(t)), \quad t \geq 0. $$

From (2d), this imposes the following unilateral constraint on the value of the control input:

$$ u(t) \geq \frac{M_s(t)}{M(t)} \dot{M}(t) + \omega \alpha M(t) + (\delta_s - \omega) M_s(t), \quad t \geq 0, \qquad (17) $$

while, for its part, (14) yields

$$ u(t) \geq \alpha \dot{M}(t) + \omega \alpha M(t) + (\delta_s - \omega) M_s(t), \quad t \geq 0. \qquad (18) $$

This solution is also not fully satisfying, as the first term on the right-hand side of (17) could possibly take on large or unbounded values when the denominator vanishes, requiring large or unbounded values for the control $u$. This has to be taken into account, especially due to the fact that $M(t)$ and $\dot{M}(t)$ are assumed to be measured.

To circumvent this difficulty, we slightly adapt condition (17) and (18), in a way convenient for our purpose. This adaptation, presented in (19) in the next section, amounts to (18) when $\dot{M} \geq 0$, but to a stronger condition when $\dot{M} < 0$. Also it is more stringent than (17) when $\dot{M} > 0$ and $\frac{M_s}{M} < \alpha$, and otherwise weaker.

## 4.2 Stabilizing state feedback control law

**Theorem 4.** Let $\alpha > \alpha_{crit} = \frac{1}{\gamma}(\mathcal{N} - 1)$ and $\omega > 0$. Assume

$$ u(t) \geq \alpha \dot{M}^{+} + \omega \alpha M(t) + (\delta_s - \omega) M_s(t), \quad t \geq 0. \qquad (19) $$

Then,

$$ \forall \theta \in \left( \frac{\mathcal{N}}{1 + \alpha\gamma}, 1 \right), \quad \forall t \geq \frac{1}{\omega} \left( \ln \alpha\gamma - \ln \left( 1 - \frac{\mathcal{N}}{\theta} + \alpha\gamma \right) \right) > 0, \quad \mathcal{N}_{\text{app}}(t) \leq \theta, \qquad (20) $$

and the extinction equilibrium $\mathbf{E}_0$ of system (2) is globally exponentially stable in the sense that:

$$ \exists v \gg 0_3, \exists c > 0, \exists \lambda > 0, \forall x(0) \geq 0_3, \forall t \geq 0, \quad v^T x(t) \leq c v^T x(0) e^{-\lambda t}, \qquad (21) $$

for every trajectory of (2) with control $u$ fulfilling (19).

Moreover, if $\omega \geq \delta_s$ and there exists $C \geq 1$ such that

$$ u(t) \leq C \left( \alpha \dot{M}^{+} + \omega \alpha M(t) + (\delta_s - \omega) M_s(t) \right)^{+}, \quad t \geq 0, \qquad (22) $$

then $M_s$ and $u$ vanish exponentially and (12) holds. The same holds if $\omega < \delta_s$ and

$$ C < \frac{\delta_s}{\delta_s - \omega}. \qquad (23) $$

Theorem 4 states that any release strategy satisfying condition (19) ensures exponential stabilization of the equilibrium $\mathbf{E}_0$. This condition depends upon the two design parameters $\alpha$ (which defines the target 'safety value' of $\frac{M_s}{M}$) and $\omega$ (which assesses the convergence rate towards this value). Thanks to the use of monotone system theory, it is expressed as an inequality defining minimal release rate, and possesses inherent robustness properties (see also Section 5).
By (2b), one has $\alpha \dot{M}^+ + \omega \alpha M = \max\{\alpha(1 - \nu)\nu_E E + \alpha(\omega - \delta_M)M; \omega \alpha M\}$. As the release rate $u$ takes on only nonnegative values, one may express (19) equivalently as

$$u(t) \geq \max\{\alpha(1 - \nu)\nu_E E + \alpha(\omega - \delta_M)M + (\delta_s - \omega)M_s; \omega\alpha M + (\delta_s - \omega)M_s; 0\}, \quad t \geq 0, \quad (24)$$

and (22) may be transformed similarly. This form highlights the fact that the expression on the right-hand side of (24) is piecewise linear in the state variable.

Inequality (22) ensures stabilization with a vanishing control $u$. It is worth noting that, for any $\omega > 0$, applying (19) with an equality ensures stabilization with finite total release (i.e. (12)), since this amounts to taking (22) with $C = 1$ in (22).

**Proof of Theorem 4.** The idea of the proof is of course to show that Proposition 3 applies, more precisely that (13) holds, for values of $\theta$ and $T$ independent of the trajectory. We will show that, for any nonzero trajectory,

$$z(t) \leq z(0)e^{-\omega t}, \quad z(t) := \left( \alpha - \frac{M_s(t)}{M(t)} \right)^{+}. \quad (25)$$

Before this, let us show that this condition is sufficient to conclude the proof. As a matter of fact, when (25) holds for any $t > 0$, then

$$\frac{M_s(t)}{M(t)} \geq \alpha - z(t) \geq \alpha - z(0)e^{-\omega t} \geq \alpha(1 - e^{-\omega t}).$$

In such a case, for any $\theta$ and $t$ as in (20), one has

$$\frac{M_s(t)}{M(t)} \geq \alpha(1 - e^{-\omega t}) \geq \alpha - \frac{1 - \frac{\mathcal{N}}{\theta} + \alpha\gamma}{\gamma} = \frac{\frac{\mathcal{N}}{\theta} - 1}{\gamma},$$

so that

$$\mathcal{N}_{\text{app}}(t) = \mathcal{N} \frac{1}{1 + \gamma \frac{M_s(t)}{M(t)}} \leq \theta.$$

As the values of $v \gg 0_3$, $c > 0$, $\lambda > 0$ in Proposition 3 depend only upon $\theta$, $T$ and the parameters of the system, common values may be taken for all trajectories, yielding the exponential convergence property (21).

Let us now demonstrate that the hypotheses of the statement ensure condition (25). When $0 < z(t) = \alpha - \frac{M_s(t)}{M(t)}$, then $\frac{M_s(t)}{M(t)} < \alpha$ and one has

$$\begin{align*}
\dot{z} &= -\frac{1}{M}\dot{M}_s + \frac{M_s}{M^2}\dot{M} \\
&= -\frac{1}{M}(u - \delta_s M_s) + \frac{M_s}{M^2}\dot{M} \\
&\leq -\frac{1}{M}\left(\alpha\dot{M}^+ + \omega\alpha M + (\delta_s - \omega)M_s - \delta_s M_s\right) + \frac{M_s}{M^2}\dot{M} \\
&= -\omega\left(\alpha - \frac{M_s}{M}\right) - \frac{1}{M}\left(\alpha\dot{M}^+ - \frac{M_s}{M}\dot{M}\right) \\
&= -\omega\left(\alpha - \frac{M_s}{M}\right) - \frac{1}{M}\left(\alpha - \frac{M_s}{M}\right)\dot{M}^+ - \frac{M_s}{M^2}\left(\dot{M}^+ - \dot{M}\right) \\
&\leq -\omega\left(\alpha - \frac{M_s}{M}\right) \quad (\text{as } z > 0) \\
&= -\omega z.
\end{align*}$$

On the other hand, when $\frac{M_s(t)}{M(t)} \geq \alpha$, then $z(t) = 0$ and $\frac{dz}{dt} = 0$. Overall, $\dot{z} \leq -\omega z$ for almost every $t \geq 0$. This yields the exponential decrease formula (25), and then the convergence property (21).
Assume now that (22) also holds. Due to (2d), the evolution of $M_s$ then fulfils

$$ \begin{aligned} \dot{M}_s &\le C (\alpha \dot{M}^+ + \omega \alpha M(t) + (\delta_s - \omega) M_s(t))^+ - \delta_s M_s(t) \\ &= \max\{C\alpha \dot{M}^+ + C\omega\alpha M(t) + ((C-1)\delta_s - C\omega)M_s(t); -\delta_s M_s(t)\}. \end{aligned} $$

The coefficient $(C-1)\delta_s - C\omega$ is negative if and only if $\delta_s \le \omega$, or if $\delta_s > \omega$ and (23) holds. In such case, one then has

$$ \dot{M}_s + \min\{C\omega - (C-1)\delta_s; \delta_s\} M_s \le C(\alpha \dot{M}^+ + \omega \alpha M(t)). $$

The coefficient of $M_s$ in this differential inequality is positive, and the right-hand side converges exponentially to zero. Applying Gronwall's lemma shows that $M_s$ converges exponentially to zero and $u$ as well, so that (12) holds. This achieves the proof of Theorem 4. □

# 5 Robustness issues

We discuss in this section how the previous stabilization result may be extended in presence of uncertainty. This flexibility comes as a major benefit of the monotonicity-based argument. We present in Section 5.1 a result concerning uncertainty on the parameters of system (2); and in Section 5.2 an important extension, in case where an Allee effect is added to the model (dynamic uncertainty).

## 5.1 Parametric uncertainties

Due to Theorem 4, (19) may be considered as the desired control goal. This condition has two remarkable properties. It aims at ensuring permanently a value of the ratio $M_s/M$ sufficient to reduce the wild population until it gets extinct. As such, it depends only upon the dynamics (2d) of $M_s$ and upon the value of $M$ and $\dot{M}$. It takes the form (24) when (2b) holds, but other dynamics of $M$ could be handled in the same way. Second, condition (19) does not impose a given supply rate, but only a minimal value for the latter. Both this inequality form and the previously mentioned genericity of the approach ensure intrinsic robustness properties.

We propose a robustness result here as an illustration, which ensures the same results as Theorem 4, based on upper and lower $(U$ and $L)$ estimates of the parameters and of the state variables.

**Theorem 5.** Let $\gamma^L \le \gamma$, $\mathcal{N} \le \mathcal{N}^U$, $\nu^L \le \nu$, $\nu_E^U \ge \nu_E$, $\delta_M^L \le \delta_M$, $\delta_s \le \delta_s^U$, and $E^U(t) \ge E(t)$, $M^U(t) \ge M(t)$, $M_s^L(t) \le M_s(t)$ for any $t \ge 0$. Let $\alpha > \frac{1}{\gamma^L}(\mathcal{N}^U - 1)$ and

$$ \omega \ge \max\{\delta_M^L, \delta_s^U\}. \qquad (26) $$

Then the exponential convergence property (21) holds for every trajectory of system (2), provided that

$$ u(t) \ge \max \{ \alpha(1 - \nu^L)\nu_E^U E^U + \alpha(\omega - \delta_M^L)M^U + (\delta_s^U - \omega)M_s^L; \omega\alpha M^U + (\delta_s^U - \omega)M_s^L; 0 \}, \quad t \ge 0. \quad (27) $$

Theorem 5 shows how to ensure robustness of the feedback assessed in Theorem 4 against parametric uncertainty, but also against state estimation error. Condition (27) is similar to condition (24), that is (19).

A natural attempt to ensure the property (12) of finite control integral consists in adapting hypothesis (22) to the present setting. Assuming that

$$ u(t) \le C' \max \{ \alpha(1 - \nu^L)\nu_E^U E^U + \alpha(\omega - \delta_M^L)M^U + (\delta_s^U - \omega)M_s^L; \omega\alpha M^U + (\delta_s^U - \omega)M_s^L; 0 \}, \quad t \ge 0, $$

for some $C' \ge 1$, is not always sufficient to obtain such a result, as the gap between the components $E^U$, $M^U$, $M_s^L$ of the state estimate and the actual values $E$, $M$, $M_s$ may be large. However, if e.g.

$$ E^U(t) \le c_E E(t), \quad M^U(t) \le c_M M(t), \quad M_s^L(t) \ge c_s M_s(t), \quad t \ge 0, $$

for given constants $c_E$, $c_M$, $c_s \ge 1$, one may state, using arguments identical to the one developed for the proof of Theorem 4, a sufficient condition in the spirit of (22).
Proof of Theorem 5. Under condition (26), one has

$$ (\omega - \delta_M^L)M^U - (\omega - \delta_M)M = (\omega - \delta_M^L)(M^U - M) + (\delta_M - \delta_M^L)M \geq 0, \\ (\delta_s^U - \omega)M_s^L - (\delta_s - \omega)M_s = (\delta_s^U - \omega)(M_s^L - M_s) + (\delta_s^U - \delta_s)M_s \geq 0. $$

Under these conditions, (27) implies (24), which is just inequality (19). The proof is achieved by use of Theorem 4, using the fact that $\alpha > \frac{1}{\gamma L}(\mathcal{N}^U - 1) \geq \frac{1}{\gamma}(\mathcal{N} - 1)$. $\square$

## 5.2 Dynamic uncertainties: Allee effect

A variant of system (2) is considered in [11], with (2c) replaced by

$$ \dot{F} = \nu\nu_E E \left(1 - e^{-\beta(M+\gamma M_s)}\right) \frac{M}{M + \gamma M_s} - \delta_F F, \quad (2e) $$

for some $\beta > 0$ (so that formally, (2c) corresponds to (2e) with $\beta = +\infty$). This factor introduces an Allee effect [18], that is a positive correlation between population density and growth rate, for small values of the density. The following result handles such a case.

**Theorem 6.** Let $\alpha > \alpha_{crit} = \frac{1}{\gamma}(\mathcal{N} - 1)$, $\omega > 0$. Then the exponential convergence property (21) holds for every trajectory of system (2a)-(2b)-(2e)-(2d), provided that $u(t)$ fulfils (19) for any $t \geq 0$. $\square$

*Proof.* First, (19) implies that (13) holds. Putting $\theta = \frac{\mathcal{N}}{1+\gamma\alpha} < 1$, one thus gets asymptotically

$$ \dot{F} \leq \frac{\nu\nu_E}{1+\gamma\alpha}E - \delta_F F = \frac{(\nu_E + \delta_E)\delta_F}{\beta_E}\theta E - \delta_F F, \quad t \geq T. $$

Using comparison system as in the proof of Proposition 3, one then deduces the exponential convergence towards $\mathbf{E}_0$. $\square$

# 6 Output feedback synthesis

In this section, we first review in Section 6.1 the quantities that may be available for measurement. Then we provide in Section 6.2 a stabilization result as an interval-observer based output-feedback control.

## 6.1 Quantities measured in practice

Measurements are produced by using a variety of traps, either aimed at the aquatic phase ('ovitraps') [22] or the adult (male and female) one [5, 21], possibly boosted by mark-release-recapture method. In absence of prior marking of the sterile males, it is quite expensive to distinguish between the sterile and the fertile males. It is even more complex to determine whether a given female has been inseminated by a sterile or a fertile male. Therefore, one simply assumes here, as in [6, 7], that two global quantities are available, namely the total number of male insects $M_{tot}(t)$, and the total number of female insects $F_{tot}(t)$. Notice that there also exist counting methods for mosquito eggs deposited on ovitrap sticks. The compartment E models here all the aquatic phases, so the use of such data would require us to revisit the model. This topic is not discussed here.

* The male insects either come from the wild population, or are sterile males still alive after their previous introduction in the field. Thus, $M_{tot}(t) = M(t) + M_s(t)$. When both equation parameters and released amounts are perfectly known, using an observer allows to reconstruct exactly the value of $M_s$ by using equation (2d). In other words, measuring $M_{tot}$ should allow us to get access to $M$.

* On the other hand, it is reasonable to assume that every female insect is fertilized once immediately after hatching, and that the adult phase takes place identically depending on whether the female has been fertilized by a fertile male or a sterile male. Therefore, by analogy with (2c), the number $F_s := F_{tot} - F$ of females fertilized by a sterile evolves according to

$$ \dot{F}_s = \nu\nu_E E \frac{\gamma M_s}{M + \gamma M_s} - \delta_F F_s. $$
Consequently, the total number $F_{tot} = F_s + F$ of female insects present in the field fulfils equation (1c). This definition is quite different from the static model $F_s = \frac{M_s}{M}F$ considered in [7], which amounts to taking $F_{tot} := \frac{M+M_s}{M}F$ (when $\gamma = 1$). In any case, adding the measurement $F_{tot}$ implies addition of a new state component, decoupled from the rest. For this reason, this quantity is not used in the sequel.

## 6.2 Output-based stabilizing control laws

We show here how to use interval observers [16] as a complement to Theorem 4 to obtain upper estimates of the state variables when the latter is not completely measured. More precisely, we consider the case where only the total number of males $M_{tot} = M + M_s$ is measured.

The space $\mathbb{R}^4$ is endowed with the so-called competitive ordering [24], denoted $\succeq$, generated by the cone $\mathbb{R}_+^3 \times \mathbb{R}_-$. In other terms, for any $x \in \mathbb{R}^4$, $x \succeq 0_4$ iff $x_i \ge 0$, $i = 1, 2, 3$ and $x_4 \le 0$. One shows without difficulty that, for any input $u$, system (2) is monotone [23] with respect to the competitive ordering. In complement to (9), denote

$$X := (x^\top \quad M_s)^\top = (E \quad M \quad F \quad M_s)^\top, \qquad (28)$$

and $f$ the function such that (2) is represented as $\dot{X} = f(X, u)$. Let $X_0 \ge 0_4$ be the initial value of (2), and denote $e_i$, $i = 1, 2, 3, 4$, the $i$-th vector of the canonical basis of $\mathbb{R}^4$.

Consider, for some scalar gain $\omega_M$, the system

$$\dot{X} = f(X, u), \quad X(0) = X_0, \qquad (29a)$$

$$\dot{\hat{X}} = f(\hat{X}, u) + \omega_M e_2(e_2 + e_4)^\top(X - \hat{X}), \quad \hat{X}(0) = \hat{X}_0, \qquad (29b)$$

where, similarly to $X$ in (9)-(28), the observer state $\hat{X}$ is decomposed according to

$$\hat{x} := (\hat{E} \quad \hat{M} \quad \hat{F})^\top, \quad \hat{X} := (\hat{x}^\top \quad \hat{M}_s)^\top.$$

One checks that the error term $(e_2 + e_4)^\top(X - \hat{X}) = M + M_s - \hat{M} - \hat{M}_s = M_{tot} - \hat{M} - \hat{M}_s$ depends only upon the measurement $M_{tot}$ and the state estimate $\hat{X}$. The value of the output $u := u(\hat{X})$ will be defined later, transforming (29) into a closed-loop system.

We first provide a useful result on the solutions of (29).

**Theorem 7.** Let $X_0, \hat{X}_0 \in \mathbb{R}_+^4$ and $u \in L_{\text{loc}}^1(0, +\infty)$ with $u \ge 0$ a.e. Assume $\omega_M \ge 0$ and $X_0 \preceq \hat{X}_0$. Then

* system (29) admits a unique solution $(X, \hat{X})$ on $[0, +\infty)$, which is nonnegative$^2$;

* for any $t \ge 0$, $X(t) \preceq \hat{X}(t)$;

Moreover, $x$ and $\hat{x}$ are ultimately bounded and $M_s - \hat{M}_s$ converges exponentially towards 0, uniformly with respect to $u$, for all solutions obtained with an initial condition $(X_0, \hat{X}_0)$ in a bounded set where $X_0 \preceq \hat{X}_0$. ■

We are now in position to state the result of global exponential stabilization of the equilibrium $E_0$ by dynamic output feedback.

**Theorem 8.** Let $\alpha > \alpha_{crit} = \frac{1}{\gamma}(N - 1)$, $\omega > 0$, $\omega_M \ge 0$. Assume $X_0 \preceq \hat{X}_0$. Then the exponential convergence property (21) holds for $x(t)$ and for $\hat{x}(t)$ along any solution of system (29), provided that

$$u(t) \ge (\alpha \dot{M}^+ + \omega \alpha \hat{M}(t) + (\delta_s - \omega) \hat{M}_s(t))^+ , \quad t \ge 0. \qquad (30)$$

Moreover, if $\omega \ge \delta_s$ and there exists $C \ge 1$ such that

$$u(t) \le C (\alpha \dot{M}^+ + \omega \alpha \hat{M}(t) + (\delta_s - \omega) \hat{M}_s(t))^+ , \quad t \ge 0, \qquad (31)$$

then $M_s, \hat{M}_s, u$ vanish exponentially and (12) holds. The same holds if $\omega < \delta_s$ and (23) is fulfilled. ■

<footnote>$^2$for the usual order.</footnote>
Theorem 8 proposes a way to stabilize the extinction equilibrium $\mathbf{E}_0$ of system (2) by a dynamic output feedback computed from the interval observer providing $\hat{X}$ in (29). Notice that the correction term $(e_2+e_4)^T(X-\hat{X})$ is the sum of two terms, namely $M_s - \hat{M}_s$ and $M - \hat{M}$. By construction the first term is nonnegative and converges naturally to zero with rate $\delta_s$, as $\dot{M}_s - \dot{\hat{M}}_s = -\delta_s(M_s - \hat{M}_s)$. On the other hand, the second term is nonpositive, and when $\omega_M > 0$ its contribution is expected to speed up the convergence of the estimate $\hat{X}$ towards $X$ and of both variables to zero, compared to the case where $\omega_M = 0$ and no measurement is used. In this latter case, stabilization also occurs, but taking $\omega_M > 0$ is expected to yield faster estimation.

Initialization of the observer is achieved according to the uncertainty range. The definition of the invariant set $[\mathbf{E}_0, \mathbf{E}^U]$ in (3) or the value of $\mathbf{E}^*$ in (7) (as the order interval $[\mathbf{E}_0; \mathbf{E}^*]$ is also forward-invariant) may serve as a basis for this.

Here, condition (30) reads (compare with (24))

$$u(t) \geq \max \left\{ \alpha(1-\nu)\nu_E\hat{E} + \alpha(\omega - \delta_M - \omega_M)\hat{M} + \alpha\omega_M M_{\text{tot}} + (\delta_s - \omega - \alpha\omega_M)\hat{M}_s; \\ \qquad \alpha\omega\hat{M} + (\delta_s - \omega)\hat{M}_s; 0 \right\}, \quad t \geq 0.$$

As a last remark, notice that when only $M$ or only $M_s$ is measured, the same argument shows that the conclusions of Theorem 8 remain valid when replacing $(M_{\text{tot}} - M_s^L - M^U)$ by $(M - M^U)$ or $(M_s - M_s^L)$ in the feedback term. The availability of such measures being dubious, this aspect is left aside.

*Proof of Theorem 7.* The existence, uniqueness and nonnegativity of the solution $X$ of (29a) have been shown in Theorem 1.

Due to the term $e_4^T(X - \hat{X}) = M_s - \hat{M}_s$, where $M_s$ bears the wrong sign, system (29a)-(29b) is not monotone with respect to the product ordering on $(\mathbb{R}^4, \preceq) \times (\mathbb{R}^4, \preceq)$, as may be checked directly. One shows without difficulty the local existence and the uniqueness of solutions of (29a)-(32).

A key point is that, as long as $X(\cdot) \preceq \hat{X}(\cdot; \hat{X}_0)$, one has $e_4^T(X - \hat{X}) = M_s - \hat{M}_s \geq 0$. Thus $(e_2 + e_4)^T(X - \hat{X}) = M_{\text{tot}} - \hat{M} - \hat{M}_s \geq M - \hat{M} = e_2^T(X - \hat{X})$, so that

$$e_2(e_2 + e_4)^T(X - \hat{X}) \geq e_2 e_2^T(X - \hat{X}).$$

Therefore, due to the monotonicity of system (29a)-(32), one may use it for comparison, and one deduces that

$$\hat{X}(t; \hat{X}_0) \geq X'(t; \hat{X}_0),$$

as long as $X(\cdot) \preceq \hat{X}(\cdot; \hat{X}_0)$. On the other hand, due to the monotonicity of (29a)-(32) and the hypothesis on the initial conditions, one has

$$X'(t; \hat{X}_0) \geq X'(t; X_0),$$

where one may recognize

$$X'(t; X_0) = X(t).$$

One thus deduces that $\hat{X}(t; \hat{X}_0) \geq X'(t; \hat{X}_0) \geq X'(t; X_0) = X(t)$ for any $t \geq 0$, whenever $\hat{X}_0 \geq X_0$. In particular $\hat{x}(t) \geq x(t) \geq 0_3$ and $\hat{M}_s(t) \leq M_s(t)$, while the fact that $\hat{M}_s(t) \geq 0$ is directly deduced from equation (2d), with the help of the fact that $\hat{M}_s(0) \geq 0$ and $u \geq 0$. Therefore, $\hat{X}(t) \geq 0$ for any $t \geq 0$.

Let us now demonstrate the claimed asymptotic properties. Theorem 1 ensures that $x$ is ultimately uniformly bounded. One shows straightforwardly that $\limsup_{t \to +\infty} \hat{E}(t) \leq K$, so that $\hat{E}$, and thus $\hat{F}$, have the

same property. On the other hand, from the equations ruling the evolution of $M_s$ and $\hat{M}_s$, one deduces $\hat{M}_s - \hat{M}_s = -\delta_s(M_s - \hat{M}_s)$, so that

$$\lim_{t \to +\infty} (M_s(t) - \hat{M}_s(t)) = 0. \qquad (33)$$
Up to terms vanishing exponentially, one thus has asymptotically $\dot{M} = (1-\nu)\nu_E\hat{E} - \delta_M \hat{M} + \omega_M(M-\hat{M})$. One then deduces from the boundedness of $M$ and $\hat{E}$ the same property for $\hat{M}$. This achieves the proof of Theorem 7. $\square$

*Proof of Theorem 8.* The result is true if $x(0) = \mathbf{E}_0$. Otherwise, the quantity $M(t)$ is positive for any $t > 0$, and the ratio $\frac{M_s(t)}{M(t)}$ is thus well-defined, as well as the apparent offspring number (10).

Under the hypotheses of Theorem 8, Theorem 7 applies, so that for any $t \ge 0$, $X(t) \le \hat{X}(t)$. One deduces in particular that

$$ \frac{\hat{M}_s(t)}{\hat{M}(t)} \le \frac{M_s(t)}{M(t)}, \quad t > 0. $$

On the other hand, (30) is just (19) for the variables $\hat{M}, \hat{M}_s$. Proposition 3 thus applies, in such a way that

$$ \liminf_{t \to +\infty} \frac{\hat{M}_s(t)}{\hat{M}(t)} \ge \alpha. \qquad (34) $$

From this one gets

$$ \liminf_{t \to +\infty} \frac{M_s(t)}{M(t)} \ge \liminf_{t \to +\infty} \frac{\hat{M}_s(t)}{\hat{M}(t)} \ge \alpha, $$

and one then deduces by Theorem 4 that

$$ \lim_{t \to +\infty} x(t) = \mathbf{E}_0 $$

for every trajectory. The same arguments as in Theorem 4 is used to show that this convergence obeys formula (21), ensuring exponential stability. In particular, $M(t)$ vanishes exponentially.

From (33), one then gets

$$ \lim_{t \to +\infty} (M_{\text{tot}}(t) - \hat{M}_s(t)) = \lim_{t \to +\infty} M(t) + \lim_{t \to +\infty} (M_s(t) - \hat{M}_s(t)) = 0, $$

with exponential convergence. As $\dot{M}(t) = (1-\nu)\nu_E\hat{E} - \delta_M \hat{M} + \omega_M(M_{\text{tot}} - \hat{M} - \hat{M}_s)$, one thus has asymptotically

$$ \dot{M}(t) \le (1-\nu)\nu_E\hat{E} - (\delta_M + \omega_M)\hat{M} \le (1-\nu)\nu_E\hat{E} - \delta_M \hat{M}, $$

up to exponentially decaying terms. Using (34) and a comparison argument, one then concludes that

$$ \lim_{t \to +\infty} \hat{x}(t) = \mathbf{E}_0. $$

Last, (12) is ensured under the same conditions when (31) holds, as well as the exponential convergence of $\hat{M}$ and $\hat{M}_s$ to zero. This completes the proof of Theorem 8. $\square$

# 7 Simulations

Section 7 is devoted to the presentation of numerical simulations related to the state feedback law presented in Theorem 4. Section 7.1 presents an elementary example of application. Section 7.2 analyzes the influence of the two design parameters $\alpha$ and $\omega$ on the closed-loop characteristics.

The numerical values of the biological constants are extracted from [11], and similar to the choices of [6]: $\beta_E = 10$ day$^{-1}$, $\gamma = 1$, $\nu_E = 0.05$ day$^{-1}$, $\delta_E = 0.03$ day$^{-1}$, $\delta_M = 0.1$ day$^{-1}$, $\delta_F = 0.04$ day$^{-1}$, $\delta_s = 0.12$ day$^{-1}$, $\nu = 0.49$. Notice that the set of solutions of (1) depends linearly upon $K$, and that the same is true for the solutions of the controlled systems$^3$. Therefore we normalize the results and take $K = 1$.

One deduces, see (6), that $\mathcal{N} = \frac{\beta_E \nu \nu_E}{(\nu_E + \delta_E) \delta_F} = 76.6 > 1$. In these conditions, the positive equilibrium is given as $\mathbf{E}^* = (0.987, 0.252, 0.605)$ and the upper bound defined in (3) is $\mathbf{E}^U = \frac{\mathcal{N}}{\mathcal{N}-1} \mathbf{E}^* = (1, 0.255, 0.613)$. The critical value of $\alpha$ defined in (13) is $\alpha_{\text{crit}} = \frac{1}{\gamma} (\mathcal{N}-1) = 75.6$.

<footnote>$^3$More precisely: if $(E, M, F, M_s)$ is a solution of (2) with $K = 1$, then $K(E, M, F, M_s)$ is a solution of (2) for any $K > 0$, provided that the definition of $u$ is also linear in $K$. In particular, this is the case when taking equality in (19).</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 1: Evolution as function of time $t$ (in days) of the preliminary phase $E(t)$, the wild males $M(t)$, the fertilized females $F(t)$, the sterile males $M_s(t)$, the apparent offspring number $N_{app}(t)$, during application of state-feedback control $u(t) = \max\{\alpha(1 - \nu)\nu_E E + \alpha(\omega - \delta_M)M + (\delta_s - \omega)M_s; \omega\alpha M + (\delta_s - \omega)M_s; 0\}$ to system (2). This value corresponds to the lower bound of condition (24) in Theorem 4. In Figure 1e, the dashed line represents the control goal $\theta$. Detailed explanation of the figure is in Section 7.1.1.</figcaption></figure>

# 7.1 State-feedback control : measurement of all population compartments

## 7.1.1 Basic example of the release campaign

System (2) is simulated with $u$ equal to the right-hand side of (19) (i.e. (24)). The design parameters are $\alpha = 1.5 \alpha_{crit} = 113$, corresponding to $\theta = \frac{N}{1+\alpha\gamma} = 0.670$ in (11); and $\omega = 1.1 \delta_s = 0.132$ day$^{-1}$. The results are presented in Figure 1.

At time $t = 0$, the system departs from the equilibrium $E^*$, and the control is switched on at time $t = 20$ days. The control input $u$ then jumps to a large value, triggering a sharp increase of the number of sterile males $M_s$ and a rapid drop of the apparent offspring number $N_{app}$. This reduces the emergence of females $F$ inseminated by fertile males. This trend then induces an even slower reduction of the number of mosquitoes $E$ in aquatic phase and of wild males $M$. The input variable $u$ decreases accordingly.

The evolution of $E$ and $M$ are slow, compared to $F$, so we choose a stopping criterion in terms of reduction of $M$ in order to express and compare the control performances. To fix the ideas, we consider by convention the smallest value $T_{stop}$ fulfilling the condition

$$ \forall t > T_{stop}, \quad \frac{M(t)}{M^*} < 10^{-2}. \qquad (35) $$

Analyzing the simulation results yields $T_{stop} = 473$ days (not shown in the figure). The corresponding total number of released mosquitoes $\int_{0}^{T_{stop}} u(t) \cdot dt$ is 532, that is approximately $2110 M^*$. The following sections 7.1.2 and 7.2 show several variations on this basic example.
<figure><img src="image_4.png" /><figcaption>Figure 2: Same experiment as in Section 7.1.1, but with control interruption (= no release) for $100 \le t \le 140$. In Figure 2e, the dashed line represents the control goal $\theta$. Detailed explanation of the figure is in Section 7.1.2.</figcaption></figure>

## 7.1.2 Effects of release interruption

We reproduce the previous experiment with the same parameter values, adding an interruption between $t = 100$ and $t = 140$ days, during which the control is set to zero. This emulates a failure of application of the control for any reason. The latter is switched on again afterwards. The results are presented in Figure 2.

When $t \in [100, 140]$, the number of sterile males decreases rapidly and the apparent offspring number increases. This is sufficient to trigger significant growth of $F$, $E$ and $M$. After $t = 140$ days the control starts again (i.e., releases are resumed), from a value larger than at $t = 100$ days, and the population reduction occurs in a way similar to the situation after $t = 20$ days.

## 7.2 Influence of the design parameter values on the control performance

In this section, we study rapidly the influence of the two design parameters $\alpha$ and $\omega$ on the performance of the state-feedback law introduced in Theorem 4. As in Section 7.1.1, the solution of system (2) with state-feedback control $u(t) = \max\{\alpha(1-\nu)\nu_E E + \alpha(\omega-\delta_M)M + (\delta_s-\omega)M_s; \omega\alpha M + (\delta_s-\omega)M_s; 0\}$ departing from the equilibrium $E^*$ is computed, for several values of the parameters. We consider and analyze the characteristics of this trajectory, relative to the value of three relevant criteria: the *time to practical elimination* (here the control is switched on at time $t=0$), the *total number of sterile males released*, and the *control peak value*. The results are plotted in Figure 3 as functions of $\omega$, for different values of $\alpha > \alpha_{crit}$.

**Time to practical elimination (Figure 3a).** As in Section 7.1, we define for each configuration the stopping time $T_{stop}$ as the smallest value fulfilling (35). For small values of $\omega$, that is for a slow control law, the time to reach elimination goes to infinity. When $\omega$ increases, the value of $T_{stop}$ decreases and increases again after a certain point, located approximately around 0.1 to $0.15\delta_s$. For large values of this parameter, the control law ensures that $M_s$ converges rapidly to $\alpha M$, and the system then behaves
essentially like the autonomous system

$$ \dot{E} = \beta_E F \left(1 - \frac{E}{K}\right)^+ - (\nu_E + \delta_E)E, \quad (36a) $$

$$ \dot{M} = (1 - \nu)\nu_E E - \delta_M M, \quad (36b) $$

$$ \dot{F} = \nu\nu_E \frac{1}{1 + \gamma\alpha} E - \delta_F F, \quad (36c) $$

where on $(0, T_{\text{stop}}]$ the control $u = \dot{M}_s + \delta_s M_s$ is now given as the output

$$ u = \alpha(\dot{M} + \delta_s M). \quad (36d) $$

The limiting time to elimination may then be assessed numerically from the corresponding value for the solution of (36). As a last remark, notice that the time to elimination is a decreasing function of the parameter $\alpha$, in accordance with the fact that the value of $N_{\text{app}}$ converges towards a decreasing value $\theta = \frac{N}{1+\gamma\alpha}$.

**Total number of sterile males released until practical elimination (Figure 3b).** This quantity is defined as

$$ \int_{0}^{T_{\text{stop}}} u(t) \cdot dt. $$

It becomes very large for small values of $\omega$, and converges to some constant depending upon $\alpha$ for large values. This asymptotic value corresponds to the one obtained with system (36) (adding to $u$ a Dirac function of amplitude $\alpha M(0)$ at the origin). Additionally, the total release appears to be increasing as a function of $\alpha$, a property which is not self-evident.

There seems to exist a fixed value of $\omega$ that minimizes the total release. This is of course of great interest from the perspective of reducing the production of sterile males. For the displayed parameter values, the minimal release until practical elimination is attained approximately at $\omega \approx 0.025\delta_s$, whatever the value of $\alpha$. The minimal value of the integral is approximately 365, for $\alpha = 1.2\alpha_{\text{crit}}$ — about 1450 times the equilibrium value $M^*$.

**Control peak value (Figure 3c).** The last performance value considered is equal to $\max_{t \in [0, T_{\text{stop}}]} u(t)$.

The picture suggests that it is increasing and goes to infinity for large values of the control gain $\omega$, and also that it is increasing as a function of $\alpha$. The noticeable changes in the slopes of the curve occur around the line $\omega = \delta_s$. The linear behaviour that emerges beyond this point reflects the convergence of the control to a Dirac function at the time instant where the control is switched on (when $M_s = 0$ and $\dot{M} = 0$ at such a point, then one must have $\dot{M}_s = u \ge \omega\alpha M(0)$).

For fixed value of $\omega$, smaller values of $\alpha$ yield smaller peak value, but longer time to practical elimination. A tradeoff is therefore apparent: quick elimination requires a large value of $\alpha$, which results in a large peak value, necessitating extensive rearing and irradiation facilities to produce sterile males.

# 8 Conclusion

In this paper, we studied the implementation of the Sterile Insect Technique by feedback control against *Aedes* mosquitoes. Several state-feedback and output-feedback control laws have been proposed and their convergence established. We also studied how to deal with some parametric and dynamical uncertainties of the model. A simple illustrative example has been provided.

Biological control methods aim generally speaking at controlling specific pest or vector using another living organism. They raise original questions in control theory, and two of them are worth emphasizing:

* First, a central role is played in the article by the *basic offspring number*. This scalar quantity, defined as ‘the average number of secondary female insects produced by a single female insect’ [17] and analogous to the basic reproductive number in epidemiological context, characterizes the viability of the considered population. The survival capacity of the pest or vector population is impacted by
<figure><img src="image_5.png" /><figcaption>Figure 3: Characteristics of the time response as function of $\omega$ (day$^{-1}$) during application of state-feedback control $u(t) = \max\{\alpha(1-\nu)\nu_E E + \alpha(\omega-\delta_M)M + (\delta_s-\omega)M_s; \omega\alpha M + (\delta_s-\omega)M_s; 0\}$ to system (2). The values of $\omega$ in abscissa are expressed as multiples of $\delta_s$. The colours of the curves correspond to the following values of the parameter $\alpha$: • $\alpha = 1.2\alpha_{crit}$; • $\alpha = 1.4\alpha_{crit}$; • $\alpha = 1.6\alpha_{crit}$; • $\alpha = 1.8\alpha_{crit}$; • $\alpha = 2\alpha_{crit}$. Detailed explanation of the figure is in Section 7.2.</figcaption></figure>

the presence of the agent chosen for control (here the sterile males) and shrinks when their number increases. This impact may be quantified by the scalar state-function introduced in the text under the name *apparent offspring number* (see Definition 1), whose value is the basic offspring number *modified by the presence of the agent and the induced interactions*. Under these conditions, driving the apparent offspring number, or *a pessimistic estimate of it*, below the critical threshold of unity, under which the population is unviable, may be set as a goal to guarantee population elimination. This remark serves as a guide for control design, and control laws generated from this principle also benefit from intrinsic robustness properties.

* On the other hand, the use of monotone system theory appears quite well suited to study the competitive interaction between the harmful species and the control agent in such situations. Monotonicity appears naturally in the description of the life cycle of insects: roughly speaking, larger quantities of individuals in early life stages tend to favour the growth of the subsequent ones, yielding in turn an increased production of eggs. This property has strong consequences on the characteristics of the asymptotic behaviour, providing useful tools for analysis [23]. On the contrary, the presence of the control agent is usually of antagonistic nature, but the introduction of a *competitive ordering* [24] may lead, as in the application displayed here, to an augmented, controlled, system that is still cooperative. Control strategies that ensure convergence of the apparent offspring number to subcritical value may then be sought, based on comparison arguments. Also, introducing ‘pessimistic’ dynamical systems that upper bound the nominal one, may allow the design of interval observer-based control, and provides a framework to consider uncertainties.

We believe these two features — considering a scalar ‘apparent offspring number’ as a substitute for Lyapunov function, and using monotone system theory to design control strategies in a variety of situations — constitute interesting leads towards powerful and flexible design methodology for robust control strategies of proven effectiveness. We plan to explore in more detail the extension of these ideas in upcoming work in preparation.

## Acknowledgment

The second author expresses gratitude to Yves Dumont (CIRAD, France) and Alain Rapaport (Inrae, France) for valuable discussions and constructive comments on the results. The financial support of the STIC AmSud program is acknowledged, through the project 23-STIC-02 BIO-CIVIP ‘Biological control of insects vectors and insects pests’.
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