Journées internationales des sciences de l'ingénieur et de l'intelligence artificielle 2026

This work focuses on the problem of structural break detection in time series, which consists in identifying points where the statistical properties of a process change. We first review classical statistical methods such as CUSUM tests, likelihood ratio tests, and change-point regression models. These approaches are well established and interpretable, but they may become limited when dealing with complex, noisy, or nonlinear data. To overcome these limitations, modern machine learning techniques such as LSTM networks and autoencoders have been intro- duced, allowing the modeling of complex temporal dependencies without strong distributional assumptions. The main idea of this work is to adopt a hybrid approach that combines classical statistical theory with modern artificial intel- ligence methods. This combination aims to benefit from the interpretability of statistical models and the flexibility of machine learning approaches in order to improve detection performance and robustness. Finally, we highlight the importance of this problem in several real-world domains, especially in healthcare, where early detection of structural changes can play a crucial role in diagnosis and monitoring systems.

The aim of this work is to study the commutativity of a prime ring R with involution of the second kind provided with two en- domorphisms satisfying certain algebraic identities. Finally, we provide examples to show thatvarious restrictions imposed in the hypotheses of our theorems are not superfluous..

in our article, we study the existence of nonnegative solutions for a class of nonpositone problems in a ball when the nonlinearity is convex from its last zero.

In this work, we propose a nonlinear parabolic equation with variable exponents, the model of which is described as follows $$ begin{cases}frac{partial u}{partial t}-operatorname{div}left(|nabla u|^{p(x)-2} nabla u+omega(x)|nabla u|^{q(x)-2} nabla uright)=lambda(f-u) & text { in } Q_T u(0, x)=f(x) & text { in } Omega u=0 & text { on } Sigma_T,end{cases} $$ where $Omega$ is an open bounded subset of $mathbb{R}^N$, with smooth boundary $partial Omega, T$ is a positive constant, $Q_T:=(0, T) times Omega$ and $Sigma_T:=(0, T) times partial Omega$. The initial data $f$ is assumed to be a measurable function belonging to $L^2(Omega)$ and the variable exponents $p(cdot)$ and $q(cdot)$ are continuous functions on $bar{Omega}$ satisfying $inf _{x in bar{Omega}} p(x)>max left{1, frac{2 N}{N+2}right}, p(x)<N$ and $p(x)<q(x)$. The function $0leomega(cdot)in L^{infty}left( Omegaright) $ and $lambda$ is a nonnegative parameter.

Let $S$ be a semigroup, $\mu$ a discrete measure on $S$ and $\sigma:S \longrightarrow S$ is an involutive automorphism. We determine the complex-valued solutions of the integral Kannappan-Sine addition law $$\int_{S}f(x\sigma(y)t)d\mu(t)=f(x)g(y)+(y)g(x),\; x,y \in S,$$

Abstract—This work investigates graph constructions associated with finite commutative rings and studies how algebraic structure is reflected through combinatorial and spectral invariants. Zero-divisor graphs provide a fundamental bridge between commutative algebra and graph theory (Anderson and Livingston, 1999). More generally, ideal-based con- structions extend adjacency conditions by replacing the relation xy = 0 with xy ? I for a suitable ideal I (Redmond, 2003; Maimani et al., 2006). In this setting, we aim to extend the approaches introduced by Redmond and Maimani et al. by studying ideal-based graph models attached to finite commutative rings and by analyzing how algebraic data influences graph invariants and spectral behavior. The adjacency spectrum yields information on the global structure of these graphs and provides explicit control of eigenvalue multiplicities. Representative examples, including rings of the form Z/nZ, prime-power quotients, and truncated polynomial rings, illustrate how algebraic structure determines graph geometry. The objective is to develop a unified approach connecting finite ring structure, graph encoding, and spectral analysis.

This study develops and analyzes an eco-epidemiological predator–prey model describing the interaction between copepods and Atlantic horse mackerel (Trachurus trachurus) along the Moroccan coast. The predator population is stratified into susceptible and infected subclasses, with nonlinear prey refuge mechanisms and selective harvesting incorporated for each group. Predation follows a Holling type I functional response for each class, while disease transmission occurs through contact between susceptible and infected predators. Rigorous mathematical analysis establishes the existence, uniqueness, positivity, and boundedness of solutions, guaranteeing the biological well-posedness of the system. Stability analysis demonstrates that the disease-free equilibrium is globally asymptotically stable whenever R0 < 1, while a transcritical bifurcation at R0 = 1 gives rise to a stable endemic equilibrium. Disease persistence or elimination is shown to be critically governed by the transmission coefficient, prey refuge intensity, and harvesting pressures applied to each predator class. Hopf bifurcation analysis further reveals the emergence of sustained oscillatory dynamics under certain parameter regimes. To account for environmental stochasticity, the deterministic framework is extended to a stochastic differential equation system, for which the existence of a unique global positive solution and sufficient conditions for disease extinction are rigorously derived. Numerical simulations corroborate the theoretical results, demonstrating that harvesting strategies, prey refuge mechanisms, and stochastic perturbations collectively govern the long-term dynamics and resilience of marine predator–prey systems. The proposed harvesting and refuge strategies contribute to reducing infection prevalence while maintaining ecological balance, providing insights for sustainable and climate-resilient fisheries management under environmental variability.

Plastic waste pyrolysis has emerged as a promising approach for obtaining energy and achieving sustainability. Nevertheless, the output of pyrolysis products is heavily influenced by the values of certain process parameters, including temperature, heating rate, residence time, and feedstock characteristics. This study proposes a data-driven approach based on the application of machine learning methods to create and improve predictions of bio-oil yield during pyrolysis of plastic waste. The input dataset for training different supervised learning algorithms, such as Random Forest, XGBoost, and Artificial Neural Network is formed by the experimental data collected under specific conditions. An exploratory data analysis (EDA) is conducted to identify the most influential factors affecting bio-oil production and to better understand the relationships between process parameters and output performance. The results demonstrate that machine learning models provide accurate predictions of bio-oil yield and offer valuable insights into optimizing pyrolysis conditions.

The analysis of influence in social media has become progressively important to understand information dynamics in complex online environments. However, current approaches last limited because they do not effectively combine semantic content, user interaction behaviors and uncertainty in heterogeneous social signals. This paper proposes a hybrid decision-support framework that integrates probabilistic topic modeling, behavioral interaction analysis and evidence-based uncertainty model for influence estimation. Latent Dirichlet Allocation (LDA) is first used to extract latent thematic structures from user-generated content. This method supports the identification of topics in massive textual corpora. In parallel, interaction signals such as retweets, likes and mentions serve to characterize user engagement and information diffusion structures. To address ambiguity and incompleteness in social media data, the theory of belief functions is adopted as a mechanism to combine heterogeneous sources of evidence and represent uncertainty and conflict. The proposed framework also introduces a risk-oriented decision layer. This layer supports influential users document ranking based on semantic relevance, behavioral impact and results of uncertainty-aware fusion. This integration enables the system to go beyond descriptive analysis and to provide actionable decision support. Experimental validation demonstrates that the proposed approach is robust and adaptable. It is appropriate for influence estimation in dynamic social media environments. The system supports applications such as user ranking, monitoring processes and alert generation. It can be utilized in intelligent social media analytics systems.

Reinforcement learning (RL) has demonstrated measurable success in agricultural applications, with recent agents cutting water use by 29% and raising profit by 9% in simulation [1]. However, current systems typically optimize single scalar rewards such as water conservation or financial profit. In contrast, the Agriculture 5.0 framework necessitates a broader multi-dimensional approach encompassing carbon footprint, soil health, equity, and human-centered decision-making [3]. We critically examine how agricultural RL systems define their objectives and identify existing gaps. By synthesizing recent work in crop management RL [5, 11], MORL theory [4, 9], and Agriculture 5.0 frameworks [14], we propose a taxonomy of objectives used across published studies. Under 12% optimize more than one objective simultaneously [8]; socioenvironmental dimensions are nearly absent; and no study applies MORL formalism — Pareto front optimization or constrained Markov decision processes — in integrated crop management. We identify four research gaps and propose a conceptual MORL framework aligned with Agriculture 5.0 as a roadmap for future work [10, 20].

This work investigates an optimal control problem for an SEIR epidemic model incorporating multiple time delays in both state and control variables. The model considers three preventive intervention strategies: mask-wearing, active screening and testing, and vaccination. These controls are introduced with delays in order to represent realistic situations where public health measures are implemented late or become effective only after a certain time. The objective is to minimize the number of exposed and infected individuals while maximizing the number of recovered individuals, taking into account the cost of applying the control measures. The delayed optimal control problem is formulated using a cost functional and analyzed through Pontryagin’s Maximum Principle adapted to systems with delays. The corresponding adjoint system and characterization of the optimal controls are derived. Numerical simulations based on the forward-backward sweep method illustrate the influence of delayed interventions on epidemic propagation. The results show that when preventive measures are delayed, immediate strengthening of mask-wearing and vaccination after the delay phase is crucial, followed by active screening and testing to further reduce transmission. This study highlights the importance of optimal timing in epidemic intervention strategies.

Let R be a commutative ring with identity, and let S be a multiplicative subset of R. In this paper, we introduce the notion of S-injective modules as a weak version of injective modules. Among other results, we provide an S-version of Baer's characterization of injective modules. We also present an S-version of Lambek's characterization of flat modules: an R-module M is S-flat if and only if its character, $Hom_\mathbb{Z}(M, \mathbb{Q}/\mathbb{Z})$, is an S-injective R-module. As applications, we establish, under certain conditions, S-counterparts of the Cartan-Eilenberg-Bass and Cheatham-Stone characterizations of Noetherian rings.

In this work, we study a class of inequality problems that arise in the quasistatic analysis of frictional contact between a thermo-electro-viscoelastic body and a rigid foundation that is both thermally and electrically conductive. The model is formulated as a system of three coupled variational inequalities describing the evolution of the displacement, the electric potential, and the temperature. The mechanical interaction at the contact surface is governed by the Signorini condition together with Coulomb-type friction. The thermal exchange at the interface depends on the normal stress, while the electrical behavior is modeled through a constant prescribed surface charge. To analyze the problem, we first establish its variational formulation. Then, by applying a fixed-point approach, we prove the existence and uniqueness of the solution.

In the present paper, we study some commutativity criteria for a prime ring with involution (R, $ast$) which admits generalized derivations $F$ and $G$ satisfying various identities. Further, examples are given to demonstrate that the restrictions imposed on the hypothesis of our results are not superfluous.

This study presents a mathematical analysis of a dynamic model describing the transmission of monkeypox in a human population. The model is formulated as a system of nonlinear ordinary differential equations based on a compartmental framework [1].The positivity and boundedness of solutions are established to ensure the biological relevance of the model. The existence of the disease-free and endemic equilibria is investigated, and their stability properties are analyzed. Local stability is examined using Jacobian matrix techniques, while global stability is established [2][3]. Through appropriate Lyapunov functions and LaSalle’s invariance principle. These results provide a clear qualitative understanding of the long-term behavior of the system and contribute to the development of effective strategies for controlling the spread of monkeypox [4][5].

In this paper, we propose a novel epidemic model describing the transmission dynamics of three interacting influenza strains using nonlinear saturated incidence rates. First, the deterministic model is analyzed to establish the existence, uniqueness, and stability of the disease-free equilibrium. To incorporate environmental variability, a stochastic version of the model is formulated by introducing white noise perturbations. We derive sufficient conditions for the extinction and persistence in mean of each strain and obtain stochastic reproduction numbers that determine the threshold dynamics of the system. Finally, numerical simulations are presented to support and illustrate the theoretical results.

In this work, we investigate the Wiener index, a well-known topological index in graph theory, for a specific class of corona graphs formed by paths and cycles. The corona product of a graph G with a graph H, denoted G ? H, is constructed by taking one copy of G and attaching a copy of H to each vertex of G. We derive closed-form expressions for the Wiener index of Pn and Cn ? H, where Pn and Cn represent path and cycle graphs with n vertices, respectively, and H is a graph of order m. Special attention is given to the case where H is the singleton graph K1. Our results generalize existing findings and provide a deeper understanding of the distance-based properties of corona graphs. These expressions can be applied in chemical graph theory, network analysis, and other fields where distance-based descriptors are of interest.

The rapid growth of e-learning platforms has produced vast amounts of learner behavioral data, enabling increasingly sophisticated educational recommendation systems. These have demonstrated their capacity to personalize learning pathways and recommend resources, but a critical question remains largely overlooked: does recommending a resource actually guarantee meaningful learning? A systematic review of the current literature reveals that the majority of approaches, whether they are based on collaborative filtering, deep learning, or even reinforcement learning, maximize engagement metrics like completion or assessment scores, without necessarily reflecting lasting knowledge acquisition. Reinforcement learning, while promising to adapt the learner’s path in real-time, usually optimizes reward signals that are not related to the learner’s cognitive processes. This work, in the systematic literature review phase, aims to map the current landscape of AI-based pedagogical recommendation systems, identify their limitations, and highlight the gap between resource recommendation and real learning.

This work studies an epidemic control problem in which interventions are implemented after disease transmission has already begun and under realistic resource limitations. We formulate a stochastic SEIR model describing the dynamics of susceptible, exposed, infected, and recovered individuals, while accounting for random fluctuations in transmission. Two bounded time-dependent controls are introduced to model preventive behavior and detection-based public-health effort. The objective is to reduce the exposed and infected populations while limiting the cost of intervention. Using a Pontryagin-type maximum principle adapted to the stochastic framework, we derive the necessary optimality conditions for the controlled system. The resulting optimality system is solved numerically by combining the Forward--Backward Sweep Method with an appropriate Runge--Kutta scheme for stochastic differential equations. Numerical simulations show that the joint use of practical intervention strategies can significantly reduce epidemic burden compared with uncontrolled dynamics or single-intervention approaches. These results suggest that simple and low-cost public-health measures can remain effective even after the beginning of an outbreak, especially in settings where expensive strategies are difficult to implement.

Résumé pour participation au colloque : Sciences de l’ingénieur et intelligence artificielle Thème : Autres applications de l’intelligence artificielle Axe d’intervention : Intelligence artificielle en finance et Fintech Auteur : Sara EZZAOUTANE de l’université HASSAN PREMIER Doctorante à la faculté d’économie et de gestion Laboratoire : Laboratoire de recherche en modélisation mathématique et de calcul économique Encadrant de thèse : Professeur Mostapha KHABOUZ Résumé : La succession des crises financières démontre que les institutions bancaires traditionnelles sont incapables de garantir la stabilité financière. La recherche d’alternatives a servi à l’essor des Fintech. Alimentées par l’intelligence artificielle, qui de son coté, s’impose comme l’une des technologies les plus transformatrices présente dans plusieurs domaines d’activité notamment en finance. Les solutions Fintech révolutionnent le système bancaire classique en offrant des services financiers innovants et plus accessibles. Ces start-ups qui viennent révolutionner le modèle entrepreneurial en finance se fixent les objectifs de simplifier les activités financières en les rendant plus efficaces, plus sécurisées, et moins chers. Apparues depuis les années 50, le grand tournant de la technologie Fintech a eu lieu après la crise de 2008. Elles sont présentes dans tous les segments du marché financier : transfert d’argent, crédit, financement participatif, assurances, gestion financière... et bien plus. Leur cœur de métier s’appuie sur l’intelligence artificielle, le big data et la blockchain. Cette industrie ne se limite pas aux marchés développés et trouve de plus en plus place dans les économies émergentes. Le Maroc, pays en développement, qui connait de très grandes mutations au niveau de son secteur financier, n’est pas épargné de cette dynamique. Les Fintech deviennent partie prenante du système. Elles se positionnent, en collaboration avec les banques, pour améliorer l’efficacité et la sécurité des services offerts. L’écosystème est composé par des start-ups spécialisées dans le domaine de la finance numérique, des organismes incubateurs publics et privés, des associations représentatives, et de tout professionnel en rapport avec le secteur financier. Sous le contrôle de Bank al Maghrib et de l’autorité marocaine des marchés de capitaux, ils ont pour rôle l’accompagnement de cette transformation. Cette évolution accompagne les initiatives gouvernementales encourageant l’innovation financière et les paiements numériques. En effet, le Maroc se projette comme acteur clé sur le plan régional en matière de technologie financière. Au troisième trimestre 2025, le pays enregistre une dynamique exceptionnelle avec 14.5 millions de dollars. Il se positionne ainsi cinquième de la région MENA. Le marché marocain attire davantage de nouveaux investisseurs, malgré la grande concurrence des pays du golfe comme l’Arabie Saoudite et les Emirats Arabes Unis. Il bénéficie d’une conjoncture économique favorable et de la volonté du Morocco Fintech Center (MFC) soutenue par Bank Al Maghrib. L’union Fintech et intelligence artificielle présente certes différentes opportunités financières. Toutefois, elle soulève des challenges et fait face à différents défis notamment sur le plan législatif, le traitement des données personnelles, et le cyber sécurité.

In this work, we develop a stochastic multi-variant compartmental model in which transmission rates are perturbed by independent Brownian motions. Building on this framework, we formulate a stochastic optimal control problem that combines preventive vaccination and variant-specific treatment in order to reduce infection levels while accounting for implementation costs. The optimality system is derived via Pontryagin’s stochastic maximum principle, and numerical simulations are provided for a two-variant setting motivated by SARS-CoV-2 dynamics.

In this talk, we explore the properties of rings and their prime ideals within the framework of homoderivations. We present four important theorems that define conditions under which certain algebraic structures display specific behaviors. These findings deepen our understanding of the relationship between homoderivations and the structural characteristics of rings and their quotients, offering valuable insights for future research in ring theory and algebra.

La prévision des prix des métaux représente un défi majeur en raison de leur forte volatilité et de la multiplicité des facteurs qui les influencent, notamment les dynamiques offre–demande, les conditions macroéconomiques et le comportement des investisseurs. Ces prévisions jouent un rôle central dans l’industrie minière, en particulier pour l’optimisation des stratégies d’extraction et la gestion des risques à long terme. Dans ce travail, nous proposons un modèle stochastique avancé pour la modélisation des prix des métaux, fondé sur une extension du processus d’Ornstein–Uhlenbeck. Le modèle considéré s’écrit sous la forme suivante : dXt = ? ?1(t) + ?2(t)Xt ? dt + ?1(t)dBt, où ?1, ?2 et ?t sont des fonctions dépendantes du temps, et {Bt}t?0 désigne un processus à accroissements indépendants et stationnaires suivant une loi ?-stable. Contrairement aux approches classiques reposant sur des hypothèses gaussiennes et des paramètres constants, cette formulation permet de mieux capturer les caractéristiques empiriques des données financières, telles que l’asymétrie, les queues épaisses et l’inhomogénéité temporelle. Une procédure d’estimation des paramètres, construite étape par étape, est développée afin de tenir compte de ces spécificités. La performance de cette approche est évaluée à l’aide de simulations de Monte Carlo, puis illustrée sur des données réelles de prix de l’or, considéré comme un facteur de risque clé dans le secteur minier. Les résultats obtenus montrent que le modèle proposé améliore significativement la qualité de la modélisation et la précision des prévisions `a long terme par rapport aux méthodes traditionnelles, confirmant ainsi sa pertinence pour l’analyse des marchés des matières premières.

In this paper, we show that the invariant subspace method can be successfully utilized to get exact solutions for nonlinear fractional partial differential equations with generalized fractional derivatives. Using the invariant subspace method, some exact solutions have been obtained for the time fractional Hunter–Saxton equation, a time fractional nonlinear diffusion equation, a time fractional thin-film equation, the fractional Whitman–Broer–Kaup-type equation, and a system of time fractional diffusion equations.

Dans un contexte marqué par la montée rapide des usages de l’intelligence artificielle (IA) dans l’enseignement supérieur et l’entrepreneuriat étudiant, cette recherche analyse le rôle de l’IA dans le développement de deux compétences clés chez les étudiants?entrepreneurs : la gestion du temps et l’autonomie entrepreneuriale. L’objectif est de comprendre comment ces étudiants mobilisent les outils d’IA dans leurs pratiques d’apprentissage et de gestion de projets, et comment ces usages influencent leurs capacités organisationnelles et décisionnelles au sein de l’enseignement supérieur marocain. S’appuyant sur un cadre théorique combinant la théorie de l’apprentissage auto?régulé (Zimmerman, 2000) et la théorie de l’efficacité personnelle (Bandura, 1997), cette étude adopte une approche qualitative exploratoire. Les données ont été recueillies à travers des entretiens semi?directifs menés auprès de huit (08) étudiants?entrepreneurs inscrits à l’École Nationale de Commerce et de Gestion (ENCG), évoluant dans divers domaines entrepreneuriaux. Les résultats montrent que l’IA est principalement utilisée comme un outil de soutien à la planification, à l’optimisation du temps et à la priorisation des tâches, facilitant la conciliation entre exigences académiques et activités entrepreneuriales. L’usage raisonné de l’IA contribue également au renforcement du sentiment d’efficacité personnelle, favorisant une plus grande autonomie dans la prise de décision, la gestion des imprévus et la conduite de projets. Toutefois, les discours des participants révèlent une tension entre les bénéfices de l’IA et le risque d’une dépendance technologique susceptible de freiner le développement d’une autonomie entrepreneuriale réelle. Cette recherche apporte une contribution empirique contextualisée à la littérature sur l’IA appliquée à l’éducation entrepreneuriale, en mettant en lumière des pratiques concrètes, situées et réflexives dans le contexte universitaire marocain.

The rapid digitalisation of education and the widespread adoption of online learning platforms have intensified cybersecurity challenges within educational environments, creating an urgent need for intelligent and adaptive security mechanisms. This study presents a review of AI-driven cybersecurity approaches in online education, synthesising evidence from 21 primary studies to examine security frameworks, threat distributions, dataset characteristics, evaluation practices, and implementation barriers. The findings reveal that current educational cybersecurity research is predominantly identity-centric, with Multi-Factor Authentication (MFA) and Single Sign-On (SSO) emerging as the most frequently adopted security mechanisms. Impersonation, unauthorised access, and system rule violations were identified as the most prevalent threats, particularly within e-learning and remote assessment environments. Methodologically, the literature is dominated by supervised deep learning approaches, especially Convolutional Neural Networks (CNNs), reflecting the growing emphasis on biometric authentication and behavioural monitoring systems. However, the review also exposes significant limitations, including fragmented governance integration, overreliance on local datasets, limited methodological standardisation, and insufficient attention to usability and ethical implications. Furthermore, infrastructure constraints, scalability challenges, and privacy concerns remain major barriers to institutional AI adoption. The study argues that the field remains in an exploratory developmental phase characterised by strong technical innovation but limited deployment maturity.

La prédiction de l’agitation portuaire est un enjeu important pour l’ingénierie côtière, la conception des ouvrages maritimes et l’évaluation de l’opérabilité des quais. Dans les bassins semi-fermés, le champ d’agitation résulte de phénomènes complexes tels que la réfraction, la diffraction, la réflexion sur les ouvrages, les interférences et les amplifications locales. ARTEMIS, modèle d’agitation portuaire fondé sur l’équation de pente douce de Berkhoff, permet de reproduire ces mécanismes de manière physiquement cohérente. Cependant, l’analyse d’un grand nombre de conditions incidentes peut devenir coûteuse lorsqu’il s’agit d’explorer de nombreux scénarios ou de fournir une aide à la décision rapide. Cette étude propose de coupler la modélisation numérique et l’intelligence artificielle afin de développer un modèle de substitution rapide basé sur des simulations ARTEMIS. Une base de données est générée en faisant varier les paramètres de houle incidente, tels que la hauteur, la période, la direction, le niveau d’eau et les conditions de réflexion. Chaque simulation fournit une cartographie du champ d’agitation dans le port. Ces résultats sont ensuite utilisés pour entraîner un modèle d’apprentissage automatique basé sur le deep learning, capable de prédire directement les champs bidimensionnels d’agitation. L’objectif est d’exploiter les résultats fournis par ARTEMIS afin de construire un émulateur rapide, cohérent avec le modèle physique de référence, permettant l’analyse massive de scénarios, l’évaluation des incertitudes et l’aide à la décision pour les applications portuaires et côtières.

Ensuring interoperability between heterogeneous and external information systems remains a major challenge for organizations, particularly in an open context such as the Semantic Web (SW), which requires modeling approaches capable of formalizing and harmonizing concepts from multiple information sources. UML is commonly used to represent the structure and interactions of systems; however, it is insufficient to ensure semantic interoperability. On the other hand, Ontologies enable the formal representation of knowledge and promote the sharing of a common semantics between systems [1]. By integrating ontologies into UML, the semantics of concepts can be explicitly formalized, thus enabling automatic interoperability between heterogeneous information systems. This article proposes a method for the automatic extraction of UML views from SW sources and RDF/OWL ontology graphs , based on a detection, encapsulation, and simplification process that preserves semantic constraints and data quality.

Water pollution is a major issue with serious consequences for human health and the environment, particularly in developing countries. It highlights that contaminated water is a source of waterborne diseases, such as cholera, and underscores the importance of integrating temperature variations into pollution management strategies. A discrete mathematical model is proposed, segmenting the problem into three compartments: at-risk water, polluted water, and the total sum of pollutants, accompanied by difference equations that represent their interactions. The challenge of optimal control aims to reduce pollutant concentrations through three approaches: awareness, purification, and source reduction. Numerical simulations conducted with MATLAB show that these interventions can significantly reduce water pollution. In conclusion, the article emphasizes that the application of mathematical modeling and optimal control strategies is crucial for mitigating the effects of pollution and proposing sustainable solutions for water management.

This paper investigates the null controllability of a class of coupled linear systems governed by an abstract operator with discrete spectrum. The model under consideration arises from the study of controlled evolution equations and is formulated as a system of two interacting components driven by a distributed control. The analysis is based on a spectral decomposition approach, reducing the controllability problem to an infinite family of moment problems. By exploiting suitable gap conditions on the eigenvalues of the operator and specific structural assumptions on the coupling terms, we establish sufficient conditions ensuring null controllability of the system. Two distinct frameworks are developed depending on whether the coupling coefficients vanish or not. In each case, we derive precise criteria involving spectral properties and interaction coefficients, and we characterize the minimal time required for controllability. The methodology relies on biorthogonal families and functional analytic techniques, providing a constructive way to design controls. These results contribute to the understanding of controllability issues for infinite-dimensional systems and highlight the interplay between spectral analysis and control theory.