Laboratoire de Mathématiques Appliquées du Havre

Elliptic partial differential equations is one of the main and most active areas in mathematics. This book is devoted to the study of linear and nonlinear elliptic problems in divergence form, with the aim of providing classical results, as well as more recent developments about distributional solutions. For this reason this monograph is addressed to master's students, PhD students and anyone who wants to begin research in this mathematical field.

In this article, we study the existence of travelling waves for a class of epidemic models structured in space and with respect to the age of infection. We obtain a necessary and sufficient condition for the existence of travelling waves for such a class of problems. As a consequence of our main result, we also derive the existence of travelling waves of a class of functional partial derivative equations.

Since the 1980s, there has been a worldwide re-emergence of vector-borne diseases including Malaria, Dengue, Yellow fever or, more recently, chikungunya. These viruses are arthropod-borne viruses (arboviruses) transmitted by arthropods like mosquitoes of Aedes genus. The nature of these arboviruses is complex since it conjugates human, environmental, biological and geographical factors. Recent researchs have suggested, in particular during the Reunion Island epidemic in 2006, that the transmission by Aedes albopictus (an Aedes genus specie) has been facilitated by genetic mutations of the virus and the vector capacity to adapt to non tropical regions. In this paper we formulate an optimal control problem, based on biological observations. Three main efforts are considered in order to limit the virus transmission. Indeed, there is no vaccine nor specific treatment against chikungunya, that is why the main measures to limit the impact of such epidemic have to be considered. Therefore, we look at time dependent breeding sites destruction, prevention and treatment efforts, for which optimal control theory is applied. Using analytical and numerical techniques, it is shown that there exist cost effective control efforts.

Abstract For operational and unpredictable reasons, many small incidents occur day after day in rail transportation systems. Most of them have a local impact, but, in some cases, mainly in dense networks, minimal disruptions can spread out through the whole network and affect significantly the train schedules. In this article, we present the railway rescheduling problem as the problem of finding a new schedule of trains after one or several incidents by minimizing some measure of the effect. We investigate the solution of this problem through a mixed‐integer programming (MIP) formulation. Because of the impossibility for solving it exactly just using a standard MIP solver, we propose to limit the search space around the original nondisrupted schedule by hard and soft fixing of integer variables with local‐branching‐type cuts. Different variations of the method are compared to a right‐shift rescheduling policy in two different networks located in France and Chile. The experimental results are also used to study the impact of different objectives on the total delay. © 2010 Wiley Periodicals, Inc. NETWORKS, 2011

Nowadays, industrial symbiosis is a key concept of industrial ecology, which studies material and energy exchange flows in the local industrial systems to reduce the costs, e.g., the wastes treatment cost, and to reduce the pollution, e.g., greenhouse gas emissions. An industrial park is a set of manufacturing businesses producing different products and by-products located at the same place (city, region, etc.). As the concept of this model encourages the development of synergy and leverage of resource networks, to the advantage of all of the enterprises present in an industrial park, a general mathematical model has been proposed. The aims of this general model are: to maximize total quantity of exchanges flows, to maximize total economic benefice of an industrial park, and to reduce relative environmental pollution, industrial waste treatment cost and delivery cost. This model can assure a win-win situation for industries and environment. There are rigorous mathematical models for specific ecological industrial parks [1]. To the best of our knowledge, there is no currently other general mathematical model for designing and optimizing an ecological industrial park. In addition, there is no currently complete ecological industrial park in France.

Let <p align="center"> $\alpha( p,q,r) =$inf{$\frac{|| u'||_p}{||u||_q}:u\in W_{p e r}^{1,p}( -1,1) $\{$ 0$}, $\int_{-1}^1|u|^{r-2} u=0$} . <p align="left" class="times"> We show that <p align="center"> $\alpha( p,q,r )=\alpha ( p,q,q)$ if $q\leq rp+r-1$ <p align="left" class="times"> <p align="center"> $\alpha( p,q,r) <\alpha( p,q,q) $ if $q> ( 2r-1) p$ <p align="left" class="times"> generalizing results of Dacorogna-Gangbo-Subía and others.

Maritime terminals need more efficiency in their handling operations due to the phenomenal evolution of world container traffic, and to the increase of the container ship capacity. In this work, we propose a new integrated modeling considering the optimization of maritime container terminals using straddle carriers. The problem is considered at import. We study a combination between two known problems, the first is the storage location assignment problem, and the second is the straddle carrier scheduling problem. This approach, which combines between two chronologically successive problems, leads to the use of multi‐objective optimization. In fact, we study the multi‐objective integrated problem of location assignment and Straddle carrier Scheduling (IPLASS) in maritime container terminal at import. We prove that the problem is NP‐Complete. The objective is to minimize the operating cost which we evaluate according to eight components: the date of last task called makespan, the total vehicle operating time, the total storage bay occupation time, the number of vehicles used, the number of storage bays used, the number of storage locations used, and two different costs of storage location assignment. The location assignment costs are evaluated in order to facilitate the containers transfer for deliveries. We assume that the operating cost is a function of these components and that the influence of each component is variable and dependent on different parameters. These parameters are essentially: the number of quays in the terminal, the straddle carrier traffic layout, the number of container ships to serve in the terminal, the influence of concurrent operations in the terminal, the storage space configuration, the number of free storage bays, the number of free straddle carriers, the number of free quay cranes, the mobility of quay cranes, etc. To solve IPLASS efficiently, we propose an adapted multi‐objective Tabu Search algorithm. Lower‐bound evaluations are introduced to perform approximation of Pareto Front. To explore efficiently the non‐convex Pareto Front Region, we evaluate also a maximized distance adapted to the set of objectives. Indicators of efficiency are developed to propose distinguished solutions to operator. 2D‐projections of approximated Pareto Frontier are given to more understand the efficiency of proposed solutions.

This paper focuses on the development of a deterministic approach for optimum sizing of the hybrid power systems (PV/wind/battery/diesel and PV/wind/diesel) based on the DIviding RECTangles (DIRECT) algorithm, which can attain the optimum values of commercially available system devices ensuring that the system total investment cost is minimized while guaranteeing the electricity requirements of the customers and the safety of the system. The hybrid power systems are assumed to be installed at an Experimental Remote Ecological Area (EREA), France, with 5-year period of average hourly data (solar radiation, wind speed, ambient temperature and electrical power demand of the load). Finally, the optimum values obtained of the system components during a period of 20-year are obtained including the number of PV modules, the PV module surface area, the number of wind turbines, the wind turbine installation height, the battery bank number and the diesel generator operating hours with their lowest system total investment costs. Additionally, a detailed analysis of the System Total Investment Cost (STIC), structure of the hybrid PV/wind/battery/diesel power system and an impact of optimum system configuration on system performance are compared and discussed in the case studied.

Under the hypothesis of convergence in probability of a sequence of càdlàg processes $(X^n)$ to a càdlàg process $X$, we are interested in the convergence of corresponding values in optimal stopping and also in the convergence of optimal stopping times. We give results under hypothesis of inclusion of filtrations or convergence of filtrations.

Distribution of non-medical products is a very interesting problem in healthcare supply chain logistics. It has a considerable impact on profit, especially when products have to be transported firstly from the manufacturing centres to intermediate healthcare facilities then to the final healthcare establishments. This transportation is ensured at two levels by selecting the best intermediate facilities and the best trips. In this paper, we address the two-echelon location-distribution problem and we develop a multi-objective mathematical formulation for minimising two objective functions. The first is the total distribution cost which is the sum of transportation costs, the opening cost of intermediate healthcare facilities and the usage cost of vehicles. The second is the total horizon time of the distribution of products. For solving this new problem, we propose two solution approaches, the multi-objective particle swarm optimisation algorithm improved by variable neighbourhood search heuristics and the non-dominated sorting genetic algorithm combined also with the same heuristics. They are tested on thirty problems and compared with different related works. Moreover, a real case study of non-medical products distribution is studied. The validation and efficiency of the algorithms are based on several performance metrics which are presented later.

This work addresses the spread of a disease within an urban system, definedas a network of interconnected cities. The first step consists of comparing two differentapproaches: a macroscopic one, based on a system of coupled Ordinary DifferentialEquations (ODE) Susceptible-Infected-Recovered (SIR) systems exploiting populations onnodes and flows on edges (so-called metapopulational model), and a hybrid one, couplingODE SIR systems on nodes and agents traveling on edges. Under homogeneous conditions(mean field approximation), this comparison leads to similar results on the outputs on whichwe focus (the maximum intensity of the epidemic, its duration and the time of the epidemicpeak). However, when it comes to setting up epidemic control strategies, results rapidlydiverge between the two approaches, and it appears that the full macroscopic model is notcompletely adapted to these questions. In this paper, we focus on some control strategies,which are quarantine, avoidance and risk culture, to explore the differences, advantages anddisadvantages of the two models and discuss the importance of being hybrid when modelingand simulating epidemic spread at the level of a whole urban system.

This paper is concerned with some mathematical analysis and numerical aspects of a reaction–diffusion system with cross-diffusion. This system models a modified version of Leslie–Gower functional response as well as that of the Holling-type II. Our aim is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The conditions of boundedness, existence of a positively invariant set are proved. Criteria for local stability/instability and global stability are obtained. By using the bifurcation theory, the conditions of Hopf and Turing bifurcation critical lines in a spatial domain are proved. Finally, we carry out some numerical simulations in order to support our theoretical results and to interpret how biological processes affect spatiotemporal pattern formation which show that it is useful to use the predator–prey model to detect the spatial dynamics in the real life.

In this paper, a network model has been proposed to control dengue disease transmission considering host-vector dynamics in n patches. The control of mosquitoes is performed by SIT. In SIT, the male insects are sterilized in the laboratory and released into the environment to control the number of offsprings. The basic reproduction number has been computed. The existence and stability of various states have been discussed. The bifurcation diagram has been plotted to show the existence and stability regions of disease-free and endemic states for an isolated patch. The critical level of sterile male mosquitoes has been obtained for the control of disease. The basic reproduction number for n patch network model has been computed. It is evident from numerical simulations that SIT control in one patch may control the disease in the network having two/three patches with suitable coupling among them.

Electric vehicles (EV) use an eco-friendly technology that limits the greenhouse gas emissions of the transport sector, but the limited battery capacity and the density of the battery are the major barriers to the widespread adoption of EV. To mitigate this, a good method seems to be the innovative wireless charging technology called ‘On-Line EV (OLEV)’, which is a contactless electric power transfer technology. This EV technology has the potential to charge the vehicle’s battery dynamically while the vehicle is in motion. This system helps to reduce not only the size of the battery but also its cost, and it also contributes to extending the driving range before the EV has to stop. The high cost of this technology requires an optimal location of the infrastructure along the route. For this reason, the objective of this paper is to study the problem of the location of the wireless charging infrastructure in a transport network composed of multiple routes between the origin and the destination. To find a strategic solution to this problem, we first and foremost propose a nonlinear integer programming solution to reach a compromise between the cost of the battery, which is related to its capacity, and the cost of installing the power transmitters, while maintaining the quality of the vehicle’s routing. Second, we adapt the multi-objective particle swarm optimization (MPSO) approach to our problem, as the particles were robust in solving nonlinear optimization problems. Since we have a multi-objective problem with two binary variables, we combine the binary and discrete versions of the particle swarm optimization approach with the multi-objective one. The port of Le Havre is presented as a case study to illustrate the proposed methodology. The results are analyzed and discussed in order to point out the efficiency of our resolution method.

International audience

Abstract. We study Mixed Mode Oscillations (MMOs) in systems of two weakly coupled slow/fast oscillators. We focus on the existence and properties of a folded singularity called FSN II that allows the emergence of MMOs in the presence of a suitable global return mechanism. As FSN II corresponds to a transcritical bifurcation for a desingularized reduced system, we prove that, under certain non-degeneracy conditions, such a transcritical bifurcation exists. We then apply this result to the case of two coupled systems of FitzHugh-Nagumo type. This leads to a non trivial condition on the coupling that enables the existence

One of the most important problems in the petroleum industry is the well-known petrol station replenishment problem with time windows, which calls for the determination of optimal routes by using a fleet of tank trucks to serve a set of petrol stations over a given planning horizon. In this paper, we introduce a model and solve a specific problem that originates from a real-life application arising in the fuel distribution where specific attention is paid to tank trucks with compartments and customers with different types of products and time windows. Literally, we call the resulting problem the multi-compartment vehicle routing problem with time windows (MCVRPTW). To solve the MCVRPTW, we begin by describing the problem, providing its mathematical formulation and discussing the sense of its constraints. As the problem is NP-hard, we propose an efficient tabu search algorithm for its solution. We introduce the Kolmogorov–Smirnov statistic into the framework of the tabu search to manage the neighbourhood size. We evaluate the performance of the algorithm on a set of vehicle routing problems with time windows instances as well as other realistic instances. Our results are compared to CPLEX, to the heuristics reported in the literature and also to those extracted from the company plans.

An SIR epidemic model is analysed with respect to the identification of its parameters and initial values, based upon reported case data from public health sources. The objective of the analysis is to understand the relationship of unreported cases to reported cases. In many epidemic diseases the reported cases are a small fraction of the unreported cases. This fraction can be estimated by the identification of parameters for the model from reported case data. The analysis is applied to the Hong Kong seasonal influenza epidemic in New York City in 1968-1969.

This study follows the new concept of the OnLine Electric Vehicle (OLEV) using the wireless charging technology as one of contactless electric power transfer technologies for Electric Vehicle (EV) charging. This electric transport system, which has been developed by Korea Advanced Institute of Science and Technology (KAIST), allows vehicles to charge their battery while in motion. In this paper, we used integer programming to find the minimum total investment cost that considers the OLEV system with a multiple route; afterwards, we introduced the resolution of this problem with the particle swarm optimization as the particles showed their robustness against nonlinear optimization problems.

The vehicle routing problem is usually treated as a static routing problem, since all information is known to the planner at the time the routing is being done. In this paper, we treat a dynamic version of the problem (DVRP) where requests arrive in a dynamic way and the solution methodology must therefore be able of promptly inserting new requests in the current vehicle rides. Due to the distributed aspect of VRPs, we propose a model based on self-organisation mechanism and a multi-agent system (MAS) to model the environment. We show that our model tackles this problem by providing vehicle agents, for a new request, to make the final decision via an auction negotiation process.