# Single and Multi-Objective Performance Optimization of an Algal-Bacterial Synthetic Process

Rand Asswad, Jean-Luc Gouzé, Eugenio Cinquemani

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Rand Asswad, Jean-Luc Gouzé, Eugenio Cinquemani. Single and Multi-Objective Performance Optimization of an Algal-Bacterial Synthetic Process. CDC 2025 - 64th IEEE Conference on Decision and Control, IEEE, Dec 2025, Rio de Janeiro, Brazil. pp.1-7. hal-05229706v2

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Distributed under a Creative Commons Attribution 4.0 International License
Single- and multi-objective performance optimization of an
algal-bacterial synthetic process

Rand Asswad<fnref n="1" />,<fnref n="2" />, Jean-Luc Gouzé<fnref n="3" />,†, Eugenio Cinquemani<fnref n="1" />,<fnref n="2" />,†

*Abstract*— Microalgae are an important source of precursors (e.g. lipids) for a variety of biosynthetic processes (e.g. biofuel production). Their co-culturing with other organisms providing essential substrates for growth may reduce cost and provide new handles to control and robustify the production process. In previous work, we have introduced a nonlinear ordinary differential equation model for an optogenetically controllable algal-bacterial consortium, and studied maximization of algal biomass productivity in a continuous-flow bioreactor relative to optogenetic action and dilution rate.

In this work, we expand the investigation of steady-state production performance for different objective criteria and control knobs. We additionally consider a yield criterion and a cost criterion, as well as a multiobjective optimization problem whose solution is shown to directly relate with a notion of net process profit. We investigate dependence of the optimal solutions on all the available bioprocess control knobs (optogenetics, dilution rate, richness of input medium), providing analytical results to characterize the solutions from different criteria and the relations among them, as well as simulations illustrating our results for a realistic set of biological system parameters.

# I. INTRODUCTION

Microalgae play a crucial role in biotechnology, offering diverse applications across industrial and environmental sectors, including animal feed, fertilizers, and pharmaceutical production [1]. Notably, their high lipid content makes them a promising source for biofuel production. Amid the ongoing global energy crisis, the demand for renewable energy sources has intensified, driving increased interest in biofuels as a sustainable alternative to conventional fossil fuels [2], [3]. The highlighted potential of algal biofuel raises the need to optimize biofuel production processes. Biotechnological and economical criteria are to be considered to render algae fuel a viable option [4], [5]. While mono-cultures remain dominant in industrial biotechnology, synthetic microbial consortia have several advantages over single species such as increased performance and resilience, compartmentalization, and modular functionality. In particular, synthetic or catalytic properties of present strains can be employed to replace external nutrient or catalyst input feeds [6], [7]. Despite the fact that coexistence of different microbial strains is ubiquitous in nature, maintaining coexistence in synthetic co-cultures presents a major challenge due to the added complexity to the system dynamics [7], [8].

We have presented in a previous work [9] a synthetic algal-bacterial consortium model describing the growth of *Auxenochlorella protothecoides* microalgae along with a strain of *Escherichia coli* bacteria in a continuous-flow stirred-tank bioreactor. Bacteria are modified in order to synthesize *thiamine* vitamin along its growth, which is a limiting substrate to the growth of the algae. Vitamin secretion is controlled via optogenetics [10], allowing modulating the proportion of bacterial metabolites dedicated to vitamin synthesis and that for bacterial growth [11]. We have extensively studied in [9] the system's steady states and provided mathematical characterization for existence and stability conditions. We considered the maximization of a single criterion: microalgal productivity (i.e. harvested algal biomass from the bioreactor) via the dilution rate and the optogenetic control. Two optimal control problems were investigated: a static optimal control problem where the productivity is maximized at the coexistence steady state using constant control inputs, and a dynamic optimal control problem where productivity is maximized over a finite time horizon using time-varying control functions.

Other authors have been dedicating attention to model-based analysis, optimization and even feedback control of microbial consortia. To give some examples, in [12], an in-depth mathematical analysis of division of labor in a microbial consortium is provided, with the goal of explaining observed enhanced productivity of consortia relative to single species. In [13], optimization of biogas production at steady-state in a two-stage anaerobic digestion model is explored. In [14], [15], feedback control strategies are considered with the ultimate goal to optimize cooperative bioproduction of prototypical microbial consortia. A discussion of recent advances, challenges and perspectives of control of microbial populations from a modelling and optimization viewpoint is provided in [16]. Despite the higher productivity that can be achieved with dynamic control [9], optimization of productivity or other objective criteria at steady state is interesting from a practical viewpoint as constant controls are easy to implement in biotechnological applications. The relative simplicity of the resulting problems compared to dynamic counterparts simplifies exploration of problems in several control variables, and it allows for the study of multi-objective optimization problems. It is also worth remarking that solution of a steady-state problem is often the starting point for the implementation of a robust control strategy around the optimal state sought [14], [15].

In this article, we extend the steady-state optimization analysis of our previous work [9] in several directions. We

<footnote>†These authors contributed equally to the work</footnote>

<footnote n="1">¹Université Grenoble Alpes, Inria, 38000 Grenoble, France
rand.asswad@inria.fr; eugenio.cinquemani@inria.fr</footnote>

<footnote n="2">²Université Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France</footnote>

<footnote n="3">³Université Côte d'Azur, Inria, INRAE, CNRS, MACBES Team, 06902
Sophia Antipolis, France jean-luc.gouze@inria.fr</footnote>
introduce several single and multiple optimization criteria, and study the optimal solutions relative to all optimization parameters entering the model for the microbial consortium biosynthesis process (namely dilution rate, optogenetic control and richness of the bacterial growth substrate provided as input). In more detail, we first review the consortium model and the conditions for algal-bacterial coexistence in Section II. In Section III, we introduce the steady-state optimization problems and formalize the relevant decision variables. We review the criterion of productivity explored in [9], and further consider bioreactor yield as well as a multi-objective optimization framework that is shown to be directly related with a concept of bioreactor's running net profit. In Section IV, we analyze these optimization problems with respect to two decision variables (dilution rate and optogenetic control), establishing properties of existence and uniqueness of solutions (scalar criteria) and of the Pareto solutions (multiple criteria). We support our findings with numerical simulations based on reference parameters. Section V extends this analysis by considering the remaining decision variable (substrate richness). We investigate its impact on the mathematical properties established in Section IV and its influence on the solutions. Simulations and mathematical analysis are provided to guide the appropriate selection of this added variable. Finally, we summarize our findings and discuss future research perspectives in Section VI.

## II. ALGAL-BACTERIAL CONSORTIUM MODEL

The model we consider is the consortium presented in [9], describing a co-culture of *Escherichia coli* bacteria and of a microalgal strain of the family of chlorella (*Auxenochlorella protothecoides*) in a continuously stirred-tank bioreactor. This algal strain has the peculiarity to need vitamin B1 (thiamine) for growth. In this consortium, the vitamin is synthesized by the (suitably engineered) bacteria in dependence of the action of optogenetic control.

We denote by $s$ [$g \cdot L^{-1}$], $e$ [$g \cdot L^{-1}$], $v$ [$mg \cdot L^{-1}$], and $c$ [$g \cdot L^{-1}$] the glucose substrate, *E. coli* biomass, secreted vitamin, and chlorella biomass concentration in the (fixed volume) bioreactor, in the same order. We further denote by $q$ [$mg \cdot g^{-1}$] the internal algal quota of the vitamin. The state of the system at time $t$ is defined as $\mathbf{x}(t) = (s(t), e(t), v(t), q(t), c(t))^{\top}$. The system dynamics are given by a cascade of a Monod model for *E. coli* growth on glucose, modified to account for the synthesis of vitamins under optogenetic control, and of a so-called variable yield Droop model (see [9] and references therein) for algal growth as a function of vitamin availability. They take the form

$$ \dot{s} = -\frac{1}{\gamma}\varphi(s)e + d(s_{\text{in}} - s) \quad (1) $$

$$ \dot{e} = (1 - \alpha)\varphi(s)e - de \quad (2) $$

$$ \dot{v} = \alpha\beta\varphi(s)e - \rho(v)c - dv \quad (3) $$

$$ \dot{q} = \rho(v) - \mu(q)q \quad (4) $$

$$ \dot{c} = \mu(q)c - dc \quad (5) $$

<table><caption>TABLE I
BIOLOGICAL MODEL PARAMETERS FROM [9]</caption><tr><td>$k_v$</td><td>0.57</td><td>mg·L<sup>-1</sup></td><td>$k_s$</td><td>0.09</td><td>g·L<sup>-1</sup></td></tr><tr><td>$\rho_{\text{max}}$</td><td>27.3</td><td>mg·g<sup>-1</sup>·day<sup>-1</sup></td><td>$\varphi_{\text{max}}$</td><td>6.48</td><td>day<sup>-1</sup></td></tr><tr><td>$q_{\text{min}}$</td><td>2.76</td><td>mg·g<sup>-1</sup></td><td>$\gamma$</td><td>0.44</td><td>g·g<sup>-1</sup></td></tr><tr><td>$\mu_{\text{max}}$</td><td>1.02</td><td>day<sup>-1</sup></td><td>$\beta$</td><td>23</td><td>mg·g<sup>-1</sup></td></tr></table>

where the functions $\varphi$ and $\rho$, defined over $[0, \infty)$, and $\mu$, defined over $[q_{\text{min}}, \infty)$, have the expressions

$$ \varphi(s) = \frac{\varphi_{\text{max}}s}{k_s + s}, \quad \rho(v) = \frac{\rho_{\text{max}}v}{k_v + v}, \quad \mu(q) = \mu_{\text{max}} \left(1 - \frac{q_{\text{min}}}{q}\right). $$

Let $\Omega = \mathbb{R}_+^5 \setminus \{q < q_{\text{min}}\}$ be the state space. In fact, $\mathbf{x}(0) \in \Omega$ implies that $\mathbf{x}(t) \in \Omega$ for all $t \ge 0$.

The variables $d$ and $s_{\text{in}}$ are operational parameters of the bioreactor representing the dilution rate and the input substrate feed respectively. As for $\alpha \in [0, 1]$ appearing in (2)-(3) represent the proportion of glucose resources going towards vitamin synthesis while the remaining $1 - \alpha$ goes toward bacterial growth. The factor $\gamma$ represents the bacterial growth yield and $\beta$ represents the vitamin synthesis yield. The functions $\varphi(s)$ and $\mu(q)$ are the growth rates per capita for bacteria and algae respectively, and $\rho(v)$ is the vitamin uptake rate per capita of algae. In fact, the yields constants $\gamma$ and $\beta$ as well as the parameters defining $\varphi$, $\rho$, and $\mu$ are intrinsic biological parameters that are reported in Table I taken from [9].

### A. Coexistence at steady states

In our previous work [9], we have proved the existence of a washout equilibrium at $\mathbf{x}_0 = (s_{\text{in}}, 0, 0, q_{\text{min}}, 0)^{\top}$ regardless of the system parameters. Furthermore, an algal washout equilibrium exists at $\mathbf{x}_{1,0} = (s^*, e^*, v_{\text{in}}^*, q_0, 0)^{\top}$ if $d < (1 - \alpha)\varphi(s_{\text{in}})$ with

$$ \begin{aligned} s^* &= \varphi^{-1}\left(\frac{d}{1-\alpha}\right), & e^* &= (1-\alpha)\gamma(s_{\text{in}} - s^*), \\ v_{\text{in}}^* &= \alpha\beta\gamma(s_{\text{in}} - s^*), & q_0 &= q_{\text{min}} + \rho(v_{\text{in}}^*)/\mu_{\text{max}}. \end{aligned} \quad (6) $$

Finally, a coexistence equilibrium state may exist at $\mathbf{x}_{1,1} = (s^*, e^*, v^*, q^*, c^*)^{\top}$ with

$$ v^* = \rho^{-1}(d \cdot \mu^{-1}(d)), \quad q^* = \mu^{-1}(d), \quad c^* = \frac{v_{\text{in}}^* - v^*}{q^*}. \quad (7) $$

The coexistence equilibrium exists if $d < \psi_{\alpha}(s_{\text{in}})$ where $\psi_{\alpha}$ is an increasing function defined such that

$$ \psi_{\alpha}^{-1}(y) = \varphi^{-1}\left(\frac{y}{1-\alpha}\right) + \frac{\rho^{-1}(y \cdot \mu^{-1}(y))}{\alpha\beta\gamma}. \quad (8) $$

<caption>TABLE II</caption>

EXISTENCE OF EQUILIBRIA OVER $\Omega$ AND THEIR STABILITY OVER

$$ \Omega \setminus \{e = 0 \cup c = 0\} \text{ WITH } d_1(\alpha, s_{\text{in}}) = \psi_{\alpha}(s_{\text{in}}) \text{ AND} \\ d_2(\alpha, s_{\text{in}}) = (1 - \alpha)\varphi(s_{\text{in}}) $$

<table><tr><td></td><td>$0 < d < d_1$</td><td>$d_1 < d < d_2$</td><td>$d_2 < d$</td></tr><tr><td>$\mathbf{x}_0$</td><td>unstable</td><td>unstable</td><td>GAS</td></tr><tr><td>$\mathbf{x}_{1,0}$</td><td>unstable</td><td>GAS</td><td>-</td></tr><tr><td>$\mathbf{x}_{1,1}$</td><td>GAS</td><td>-</td><td>-</td></tr></table>
Stability and existence of the equilibria, as reported in Table II, are studied in [9].

### III. SINGLE- AND MULTI-OBJECTIVE OPTIMIZATION PROBLEMS

In this section we introduce the optimization problems of interest. Having established the necessary and sufficient conditions for the existence of the coexistence equilibrium $x_{1,1}$, referred to from this point onward as $x^*$, we have yet to introduce the decision variables that constitute the control $u$ from the appropriate set of admissible controls $\mathcal{U}$ that ensure the coexistence at the functional steady state $x^*(u)$. Then we consider the optimization criteria to take into account that are relevant in biotechnology.

#### A. Decision variables

Existence of equilibria as well as their values are characterized by the system's parameters as reported in the previous section (see Table II). While the parameters reported in Table I are intrinsic to the system, relating to the bacterial and algal biological properties, the dilution rate $d$ and the substrate feed $s_{in}$ are operational parameters of the bioreactor that can be adjusted as best for the biotechnological process. Finally, the $\alpha \in (0,1)$ models the amount of resources that are allocated for bacterial growth or for vitamin synthesis. In this sequel, we consider optogenetic control through $\alpha$ following [9], [10].

In short, we consider the parameters $\alpha, d$, and $s_{in}$ as decision variables for the control problem. We show later that performance improves as $s_{in}$ increases [8], [17], we therefore consider in section IV the problem for a fixed $s_{in}$ with free control variables $u = (\alpha, d)$ to be chosen in $\mathcal{U}$, such as $\mathcal{U}$ is the set of all values of $(\alpha, d)$ for which the coexistence steady state $x^*(u)$ is well-defined. We expand the scope in section V to study the effect of $s_{in}$ on the problem properties and its solution, where $w = (\alpha, d, s_{in}) \in W$ for the appropriate $W$.

#### B. Optimization criteria

Naturally, we aim to maximize the amount of microalgae that is harvested from the bioreactor which is the quantity

$$P_{\text{out}} = d \cdot c^* = \frac{\alpha \beta \gamma d (s_{\text{in}} - \psi_{\alpha}^{-1}(d))}{\mu^{-1}(d)}, \quad (9)$$

and is commonly referred to as the **microalgal productivity**. Another quantity to consider is the glucose feed that enters the bioreactor, characterized as $P_{in} = d \cdot s_{in}$. The **nutrient feed** being a cost to the bioprocess, we minimize this amount. A common metric for the efficiency of a bioreactor synthesis process is the **bioreactor yield**, which is the ratio of the harvested target biomass with respect to the nutrient feed [8], here simply $P_{yield} = P_{out}/P_{in} = c^*/s_{in}$.

While the bioreactor yield combines the input and the target output of the bioreactor, it only takes into account the volumetric efficiency of the process. However, this criterion does not account for the running economical cost and the profitability of the bioprocess. From a biotechnological standpoint, the bioreactor efficiency is optimized by

maximizing the microalgal productivity $P_{out}$ and minimizing the nutrient feed $P_{in}$ simultaneously. Which corresponds essentially to a multiobjective optimization problem (MOP) defined as

$$\max_{u \in \mathcal{U}} \{P_{\text{out}}(u), -P_{\text{in}}(u)\}, \quad (10)$$

where an admissible solution $u \in \mathcal{U}$ that maximizes both $P_{out}$ and $-P_{in}$ is sought.

#### C. Bioreactor net profit

We prove in sections IV and V that there is no feasible point that optimizes both criteria $P_{out}$ and $P_{in}$ simultaneously. This calls for a weaker notion of optimality that is based on compromise and an external decision maker [18].

**Definition 1 (Pareto optimality).** A feasible solution is Pareto optimal if none of the objective functions could be improved without deteriorating at least one other objective. This means that for the maximization of $\{f_1(z), \dots, f_k(z)\}$ functions for $z \in \mathcal{Z}$, a solution $z^* \in \mathcal{Z}$ is Pareto optimal if there is no $z \in \mathcal{Z}$ such that $f_i(z^*) \le f_i(z)$ for all $i \in \{1, \dots, k\}$ with $f_j(z^*) < f_j(z)$ for some $j \in \{1, \dots, k\}$ [18].

**Definition 2 (Pareto optimal front).** The Pareto optimal front (POF) is the set of Pareto optimal solutions.

Solving the MOP consists of constructing the POF. We achieve this by using the weighting method, which constructs a scalar objective function that is a linear combination of the objective functions, where the coefficients (or weights) are positive and add up to 1. In the case of two objectives, the weights can be simply $\theta$ and $1-\theta$ with $\theta \in [0, 1]$, [18]. The weighting problem for (10) is given as

$$\max_{u \in \mathcal{U}} P_{\theta}(u) = \theta P_{\text{out}}(u) + (1-\theta)[-P_{\text{in}}(u)]. \quad (11)$$

The weighting method allows obtaining locally Pareto optimal solutions. Nevertheless, stronger properties of the solutions are discussed in sections IV and V.

We note that the corresponding weighting problem (11) corresponds to maximizing the **net profit** of the bioreactor. Let $w_{in}$ and $w_{out}$ be the costs of a gram of glucose and a gram of microalgae respectively. The net profit of the bioreactor is the running cost difference between the harvested microalgae and the bioreactor feed which is $w_{out}P_{out} - w_{in}P_{in}$. The expression of $P_{\theta}$ can be found by dividing this expression by $w_{in} + w_{out}$ and setting $\theta = w_{out}/(w_{in} + w_{out})$ which gives $\theta \in [0, 1]$ for all positive $w_{in}$ and $w_{out}$.

Apart from constructing the POF, the net profit provides insight for the industrial biotechnology operators allowing to choose the optimal values for the decision values for any given set of costs $w_{out}$ and $w_{in}$.

### IV. PROPERTIES OF THE OPTIMIZATION PROBLEMS FOR A FIXED $s_{in}$

In this section we restrict the problem to the control $u = (\alpha, d)^{\top} \in \mathcal{U}$, for a fixed $s_{in}$, where the set of admissible controls is defined as the set defining presence of the coexistence equilibrium $x^*$, formalized as

$$\mathcal{U} = \{(\alpha, d) \in (0, 1) \times \mathbb{R}_{+}^{*}, \psi_{\alpha}^{-1}(d) < s_{\text{in}}\}. \quad (12)$$
<figure><img src="image_2.png" /><figcaption>Fig. 1. Logarithm of the objective function $P_{\text{out}}(\alpha, d)$ contours (borders of the domain of interest, ensuring nonzero algal biomass, indicated in solid red line), as well as its global maximum.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. Objective function $P_{\text{yield}}(\alpha, d)$ contours, and $\max_{\alpha \in \mathcal{A}} P_{\text{yield}}(\alpha, d)$ for all $d$ (dashed red line).</figcaption></figure>

In [9], we have shown that the $P_{\text{out}}(\alpha, d)$ is logarithmically bi-concave (i.e. log-concave with respect to $\alpha$ and to $d$ separately [19]) over $\mathcal{U}$. In [20] (an extended version of [9]), we have strengthened the result by showing that $\mathcal{U}$ is a convex set, and that $P_{\text{out}}(\alpha, d)$ is logarithmically concave, therefore admits a global maximum on $\mathcal{U}$, as shown in Figure 1. In contrast, the nutrient feed $P_{\text{in}}(d) = d \cdot s_{\text{in}}$ is an affine function (for a fixed $s_{\text{in}}$), and therefore has its optima at the border of $\mathcal{U}$.

### A. Bioreactor yield

The bioreactor yield function $P_{\text{yield}}(\alpha, d)$ does not enjoy the same properties as the productivity $P_{\text{out}}$. We can, nevertheless, investigate its properties and exploit them to find its global maximum analytically.

The expression of the yield is given as

$$P_{\text{yield}}(\alpha, d) = \frac{\alpha\beta\gamma(s_{\text{in}} - \psi_{\alpha}^{-1}(d))}{s_{\text{in}}\mu^{-1}(d)} \quad (13)$$

Let $\mathcal{A} = (0, 1)$ be an open interval, for a fixed $\alpha \in \mathcal{A}$ we define the open interval $\mathcal{D}_{\alpha} = (0, \psi_{\alpha}^{-1}(s_{\text{in}}))$.

**Proposition 3.** For each fixed $\alpha \in \mathcal{A}$, the function $P_{\text{yield}}(\alpha, d)$ is strictly decreasing for all $d \in \mathcal{D}_{\alpha}$.

*Proof.* We first notice that the affine function $1/\mu^{-1}(d) = (\mu_{\max} - d)/q_{\min}\mu_{\max}$ is positive and strictly decreasing on $\mathcal{D}_{\alpha}$. Furthermore, $\psi_{\alpha}^{-1}(d)$ is strictly increasing for all $d \in \mathcal{D}_{\alpha}$ because it is the positive linear combination of strictly increasing functions composed with increasing functions in $d$, therefore $d \mapsto s_{\text{in}} - \psi_{\alpha}^{-1}(d)$ is positive and strictly decreasing on $\mathcal{D}_{\alpha}$. Finally, $d \mapsto P_{\text{yield}}(\alpha, d)$ is the product of two positive strictly decreasing functions therefore it is also strictly decreasing for all $d \in \mathcal{D}_{\alpha}$. ■

It follows that for each $\alpha \in \mathcal{A}$, the bioreactor yield increases as $d$ decreases. Let $\bar{D}_{\alpha} = [0, \psi_{\alpha}^{-1}(s_{\text{in}})]$ be the closure of $\mathcal{D}_{\alpha}$, it follows that for all $\alpha \in \bar{\mathcal{A}} = [0, 1]$,

$$\max_{d \in \bar{D}_{\alpha}} P_{\text{yield}}(\alpha, d) = P_{\text{yield}}(\alpha, 0). \quad (14)$$

Consequently, the maximization problem of $P_{\text{yield}}(\alpha, d)$ over $\bar{U}$ reduces to the maximization $P_{\text{yield}}(\alpha, 0)$ over $\bar{\mathcal{A}}$, since

$$\max_{(\alpha, d) \in \bar{D}_{\alpha} \times \bar{\mathcal{A}}} P_{\text{yield}}(\alpha, d) = \max_{\alpha \in \bar{\mathcal{A}}} \max_{d \in \bar{D}_{\alpha}} P_{\text{yield}}(\alpha, d). \quad (15)$$

From (13) and (8) we have

$$P_{\text{yield}}(\alpha, 0) = \frac{\alpha\beta\gamma}{q_{\min}}, \quad (16)$$

which is affine and increasing with respect to $\alpha$ and reaches its maximum at $\alpha = 1$. Finally, the bioreactor yield maximization problem over $\bar{U}$ has a trivial solution at $(1, 0)$ as shown in Figure 2.

**Remark 4.** For all $d \in \mathcal{D}_{\alpha}$, the function $P_{\text{yield}}(\alpha, d)$ is strictly concave with respect to $\alpha$ over $\mathcal{A}$. Consequently maximizing $P_{\text{yield}}$ for a fixed $d \in \mathcal{D}_{\alpha}$ gives a nontrivial solution (Figure 2).

*Proof.* This stems directly from the fact that $P_{\text{out}}(\alpha, d)$ is strictly concave with respect to $\alpha$ for any fixed $d$ (Lemma 2 in [9]) since $P_{\text{yield}}(\alpha, d) = P_{\text{out}}(\alpha, d)/P_{\text{in}}(d)$. ■

In conclusion, maximizing the bioreactor yield degenerates to the special case of a batch process ($d = 0$). If continuous production is the goal, provided a given extraction rate $d > 0$ is of interest, Remark 4 shows that one can optimally and uniquely choose $\alpha$ for the given value of $d$.

### B. Bioreactor net profit

As discussed in section III, maximizing productivity might not be the best course of action in practice, mainly for economical reasons, and considering the running cost of the bioreactor represents a matter of interest in biotechnology. While the bioreactor yield takes the input feed into account, it might not provide the needed insight in the context of an industrial bioprocess, on top of the fact that we have shown how its maximum does not correspond to our system of interest. Hence, we consider instead the MOP defined in (10) and the associated weighted scalar problem (11) corresponding to the net profit.

The weighting method scalarizes the MOP in order to obtain locally Pareto optimal solutions. We now explore stronger properties of the solutions.
<figure><img src="image_4.png" /><figcaption>Fig. 3. Contours of objective functions $P_{\text{out}}$ and $P_{\text{in}}$ over the set of admissible controls $\mathcal{U}$. The color-gradient line is the set Pareto-optimal solutions for $\theta \in [0, 1]$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. The image of $\mathcal{U}$ in the multi-objective space; the color-gradient line represents the image of the POF through $P_{\theta}$ for $\theta \in [0, 1]$, corresponding to the color-gradient line in Figure 3.</figcaption></figure>

**Definition 5.** A decision vector $\mathbf{z}^* \in \mathcal{Z}$ is locally Pareto optimal if there exists $\delta > 0$ such that $\mathbf{z}^*$ is Pareto optimal in $\mathcal{Z} \cap B(\mathbf{z}^*, \delta)$ where $B(\mathbf{z}^*, \delta)$ denotes an open ball with a center $\mathbf{z}^*$ and a radius $\delta > 0$ [18].

**Lemma 6.** Every locally Pareto optimal solution is also globally Pareto optimal if the feasible region is convex and the objective functions are quasiconcave with at least one strictly quasiconcave for the maximization problem (Theorem 2.2.4. in [18]).

**Definition 7 (Quasiconcave function).** A function $f: \mathbb{R}^n \to \mathbb{R}$ is quasiconcave if its domain $\text{dom } f$ and all its superlevel sets

$$C_{\eta}(f) = \{\mathbf{z} \in \text{dom } f, f(\mathbf{z}) \geq \eta\}$$

for $\eta \in \mathbb{R}$, are convex [21], [22].

**Proposition 8.** Solutions of the weighted problem defined in (11) are globally Pareto optimal solutions for the feasible region $\mathcal{U}$ defined in (12).

*Proof.* Indeed, $\mathcal{U}$ is a convex set, $P_{\text{out}}(\alpha, d)$ is log-concave therefore quasiconcave, and $P_{\text{in}}(d)$ is an affine function (non-constant) therefore strictly quasiconcave as well [22]. Hence, the proof follows from Lemma 6.

Proposition 8 proves that a unique global optimum exists for any given $\theta \in [0, 1]$. Figure 3 illustrates the POF in the $(\alpha, d)$ plane and Figure 4 shows the POF in the $(P_{\text{out}}, P_{\text{in}})$ plane.

## V. DEPENDENCE OF SOLUTIONS ON $s_{\text{in}}$

The substrate feed $s_{\text{in}}$ is an operational parameter of the bioreactor that is commonly used as a control input for bio-processes. We have mentioned in section III that performance improves as it increases [8], [17]. In this section we explore in detail the impact of adding $s_{\text{in}}$ as a third control variable on the optimization problem properties and to the bioprocess. We consider here the control $\mathbf{w} = (\alpha, d, s_{\text{in}}) \in \mathcal{W}$ where

$$\mathcal{W} = \{(\alpha, d, s_{\text{in}}) \in (0, 1) \times (\mathbb{R}_{+}^{*})^{2}, \psi_{\alpha}^{-1}(d) < s_{\text{in}}\} \quad (17)$$

is the set of admissible controls.

### A. Convex properties

Here we explore the convexity of the problems defined in section III with respect to the new control $\mathbf{w} \in \mathcal{W}$.

**Proposition 9.** The admissible control set $\mathcal{W}$ is nonconvex.

*Proof.* Consider the function $g_0(\alpha, d) = \psi_{\alpha}^{-1}(d)$. Its epigraph is clearly the closure of the set $\mathcal{W}$ (i.e. the smallest closed set containing $\mathcal{W}$, denoted $\overline{\mathcal{W}}$). We remind that a function is convex if and only if its epigraph is a convex set [21]. Nevertheless, for the given parameter set in Table I, $g_0$ is not a convex function since the second order convexity condition does not hold for all $(\alpha, d) \in \mathcal{U}$: As demonstrated in Figure 5, the Hessian of $g_0$ is non-definite over portions of the domain.

Therefore, $\text{epi}(g_0) = \overline{\mathcal{W}}$ is not a convex set.

<figure><img src="image_6.png" /><figcaption>Fig. 5. The positive-definiteness of the Hessian of the function $g_0(\alpha, d)$. Yellow regions indicate where the Hessian is non-definite, i.e. the Hessian is found having a positive and a negative eigenvalue.</figcaption></figure>

Since $\mathcal{W}$ is not necessarily a convex set, the optimization problems over $\mathcal{W}$ are not convex problems regardless of the considered objective function. We can however explore weaker properties.
<figure><img src="image_7.png" /><figcaption>Fig. 6. The top plot shows the optimal controls that corresponds to all $\theta \in [0, 1]$ values. The bottom plot illustrates the corresponding objective values for the optimal controls in the top figure.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. The image of $\overline{B(z)}$ with $z = 2$ in the multi-objective space; the color-gradient line is the POF for $\theta \in [0, 1]$. The shaded areas correspond to the set of reachable $P_{\text{out}}$ and $P_{\text{in}}$ for different values of $s_{\text{in}}$ (reachable sets for larger $s_{\text{in}}$ including those for smaller $s_{\text{in}}$).</figcaption></figure>

## B. Problem biconvexity

The set $W = \mathcal{U}(z) \times \mathbb{R}_+^*$ is biconvex with $\mathcal{U}(z) = \{(\alpha, d) \in (0, 1) \times \mathbb{R}_+^*, \psi_\alpha^{-1}(d) < z\}$ for any $z > 0$ [19]. This follows directly from the convexity of $\mathcal{U}(z)$ that corresponds to the admissible controls set defined in section IV and the trivial convexity of $\mathbb{R}_+^*$.

We now explore the biconvexity/biconcavity of the objective functions, which would ensure the existence of global maxima. Uniqueness however would not be guaranteed. Having comprehensively investigated in Section IV the optima of the objective functions $P_{\text{out}}$, $P_{\text{in}}$, $P_{\text{yield}}$, and $P_\theta$ with respect to $\mathbf{u} = (\alpha, d)$ over $\mathcal{U}$ for a fixed $s_{\text{in}} > 0$, we now study their convexity/concavity with respect to $s_{\text{in}}$ for a fixed $(\alpha, d)$.

The functions $P_{\text{out}}$, $P_{\text{in}}$, and $P_\theta$ are affine in $s_{\text{in}}$, therefore they do not have an optimum with respect to $s_{\text{in}}$ over an unbounded interval, therefore we introduce the open bounded set $\mathcal{B}(z) = \mathcal{U}(z) \times (0, z)$ and its closure $\overline{\mathcal{B}(z)} = \overline{\mathcal{U}(z)} \times [0, z]$ with $\overline{\mathcal{U}(z)} = \{(\alpha, d) \in [0, 1] \times \mathbb{R}_+, \psi_\alpha^{-1}(d) \le z\}$.

Both $P_{\text{out}}$ and $P_{\text{in}}$ are affine and increasing with respect to $s_{\text{in}}$, therefore their optima lie on the border of $\overline{\mathcal{B}(z)}$. For a fixed pair $(\alpha, d) \in \overline{\mathcal{U}(z)}$, $P_{\text{out}}(\alpha, d, s_{\text{in}})$ reaches its maximum for $s_{\text{in}} = z$ while $P_{\text{in}}(d, s_{\text{in}})$ reaches its minimum for $s_{\text{in}} = 0$. We skip the optimization of the process yield since we have proven in section IV that it has a trivial maximum that lies on the border of $\mathcal{U}(z)$ which corresponds to the singular case of a batch ($d = 0$) process.

## C. Bioreactor net profit with respect to $s_{\text{in}}$

Considering the biconvex optimization problem over $\overline{\mathcal{B}(z)}$, the scalar weighting function of the MOP defined in III has a unique solution for a fixed $s_{\text{in}}$ as established in IV. Here we explore the effect of $s_{\text{in}}$ on the bioreactor net profit

$$P_{\theta}(\alpha, d, s_{\text{in}}) = \theta P_{\text{out}}(\alpha, d, s_{\text{in}}) - (1 - \theta) P_{\text{in}}(d, s_{\text{in}}) \quad (18)$$

$$= \theta \frac{\alpha \beta \gamma d (s_{\text{in}} - \psi_{\alpha}^{-1}(d))}{\mu^{-1}(d)} - (1 - \theta) ds_{\text{in}} \quad (19)$$

for a fixed $(\alpha, d) \in \overline{\mathcal{U}(z)}$ and a given $\theta \in [0, 1]$. Since $P_\theta$ is affine with respect to $s_{\text{in}}$, it reaches therefore its maximum with respect to $s_{\text{in}}$ on the border of the closed interval $[0, z]$. The function is either increasing or decreasing, depending

on the sign of the partial derivative of $P_\theta$ with respect to $s_{\text{in}}$ (i.e. the coefficient of $s_{\text{in}}$). From (19),

$$\begin{aligned} \frac{\partial P_{\theta}}{\partial s_{\text{in}}} &= \left[ \theta \left( 1 + \frac{\alpha \beta \gamma}{\mu^{-1}(d)} \right) - 1 \right] d \\ &= (\theta - \theta_0(\alpha, d)) \left( 1 + \frac{\alpha \beta \gamma}{\mu^{-1}(d)} \right) d \quad (20) \\ \text{with } \theta_0(\alpha, d) &= \frac{1}{1 + \alpha \beta \gamma / \mu^{-1}(d)}. \end{aligned}$$

It follows that for a fixed $(\alpha, d) \in \mathcal{U}$, $P_\theta$ is strictly increasing with respect to $s_{\text{in}}$ if $\theta > \theta_0(\alpha, d)$ and decreasing otherwise.

**Proposition 10.** For all $(\alpha, d) \in \mathcal{U}$, $0 < \theta_0(\alpha, d) < 1$.

*Proof.* Consider the functions $g_1$ and $g_2$ defined as

$$g_1(t) = \frac{1}{1 + t} \quad \text{and} \quad g_2(\alpha, d) = \frac{\alpha \beta \gamma}{\mu^{-1}(d)}.$$

Then $\theta_0 = g_1 \circ g_2$. The function $g_2$ is strictly positive on $\mathcal{U}$. In addition, $g_1(\mathbb{R}_+^*) = (0, 1)$. Therefore, $\theta_0(\alpha, d) = g_1(g_2(\alpha, d)) \in (0, 1)$. ■

It follows from (20) and Proposition 10 that, for a fixed $(\alpha, d) \in \mathcal{U}$, $\partial P_\theta / \partial s_{\text{in}}$ changes signs once at $\theta = \theta_0(\alpha, d)$. Indeed, for values of $\theta < \theta_0$, $P_\theta(\alpha, d, s_{\text{in}})$ decreases as $s_{\text{in}}$ increases therefore the maximum of $P_\theta$ is reached at $s_{\text{in}} = 0$. Otherwise, the net profit $P_\theta$ increases with $s_{\text{in}}$ linearly until it reaches its maximum at the border for $s_{\text{in}} = z$. Figure 6 illustrates this result with respect to all values of $\theta \in [0, 1]$, for an upperbound on $s_{\text{in}}$ fixed to $z = 2$. Clearly, $P_{\text{out}}(\alpha, d, 0) = P_{\text{in}}(d, 0) = 0$ because $\mathcal{U}$ is the empty set for $s_{\text{in}} = 0$. Otherwise, $s_{\text{in}}$ takes its maximal value in the bounded interval to maximize $P_\theta$.

Figure 7 shows the image of $\overline{\mathcal{B}(z)}$ in the $(P_{\text{out}}, P_{\text{in}})$ plane. It demonstrates the reachable values of $P_{\text{out}}$ and $P_{\text{in}}$ for different fixed values of $s_{\text{in}}$, as well as the set of Pareto optimal solutions.

In conclusion, the added degree of freedom does not give optimal solutions of the problem on the open set $W$. Instead, it should be chosen a priori at its optimal boundary value within the limits of feasibility and practicality to reach
a target performance. As a result of its optimal choice saturating to boundaries, for fixed values of $\alpha$ and $d$, a discontinuity is introduced in its optimal choice as a function of $\theta$.

## VI. CONCLUSIONS

Starting from our algal-bacterial consortium model from [9] and the coexistence conditions established therein, we formulated optimization problems for the system at its coexistence steady state that are relevant to biotechnological applications. These criteria were then integrated into a multi-objective optimization problem. We demonstrated the uniqueness of solutions for these problems when controlling the dilution rate $d$ and the optogenetic control $\alpha$, holding regardless of specific biological parameter values, and validated our findings through numerical simulations for realistic biological parameter values.

Expanding the control framework, we introduced the bioreactor nutrient feed $s_{in}$ as a third control variable and analyzed its influence on the optimization problems and their solutions. In line with previous studies [8], [17], for this class of optimization criteria, our results indicated that enriching the nutrient feed enhances system performance overall. The maximum allowable $s_{in}$ should then be chosen based on different biotechnological constraints (e.g. maximal allowable culture density, etc.). Next, we defined the bioreactor net profit metric as an alternative to the bioreactor yield, offering a weighted multi-objective perspective that is relevant for industrial biotechnological applications. In the case of net profit, we showed that the role of the nutrient feed ($s_{in}$) is more complex (Proposition 10 and subsequent discussion). While this economy metric can be extended to other bioreactor models, the uniqueness of solutions would need to be reexamined in each specific case.

Numerical simulations indicate that optimal steady-state solutions generally lie near washout boundaries. For future work, we will address the challenge of designing feedback control strategies that drive the system toward its optimal coexistence equilibrium and guarantee steady-state coexistence and optimality robustly, in spite of modeling uncertainties, limited state observations and inevitable sources of noise.

## ACKNOWLEDGEMENTS

This work was supported in part by the French national research agency (ANR) via project Ctrl-AB [ANR-20-CE45-0014].

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