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Different populations of vehicles travel along a network. Each population has its origin, destination and travel costs - which may well be unbounded. Under the only requirement of the continuity of the travel costs, we prove the existence of a Nash equilibrium for all populations. Conditions for its uniqueness are also provided. A few cases are treated in detail to show specific situations of interest., Comment: 23 pages

The formation, movement and gluing of clusters can be described through a system of non local balance laws. Here, the well posedness of this system is obtained, as well as various stability estimates. Remarkably, qualitative properties of the solutions are proved, providing information on stationary solutions and on the propagation speed. In some cases, fragmentation leads to clusters developing independently. Moreover, these equations may serve as an encryption/decryption tool. This poses new analytical problems and asks for improved numerical methods., Comment: 28 pages, 7 figures

Consider the inverse design problem for a scalar conservation law, i.e., the problem of finding initial data evolving into a given profile at a given time. The solution we present below takes into account localizations both in the final interval where the profile is assigned and in the initial interval where the datum is sought, as well as additional a priori constraints on the datum's range provided by the model. These results are motivated and can be applied to data assimilation procedures in traffic modeling and accidents localization.

We prove the well posedness of a class of non linear and non local mixed hyperbolic-parabolic systems in bounded domains, with Dirichlet boundary conditions. In view of control problems, stability estimates on the dependence of solutions on data and parameters are also provided. These equations appear in models devoted to population dynamics or to epidemiology, for instance.

Recently, results regarding the Inverse Design problem for Conservation Laws and Hamilton-Jacobi equations with space-dependent convex fluxes were obtaine. More precisely, characterizations of attainable sets and the set of initialdata evolving at a prescribed time into a prescribed profile were obtained. Here, wepresent an explicit example that underlines deep diff erences between the space-dependentand space-independent cases. Moreover, we add a detailed analysis of the time asymptoticsolution of this example, again underlining diff erences with the space-independent case., Comment: arXiv admin note: text overlap with arXiv:2304.05092

Consider a Conservation Law and a Hamilton-Jacobi equation with a ux/Hamiltonian depending also on the space variable. We characterize rst the attainable set of the two equations and, second, the set of initial data evolving at a prescribed time into a prescribed prole. An explicit example then shows the deep dierences between the cases of x-independent and x-dependent uxes/Hamiltonians.

Consider a balance law where the flux depends explicitly on the space variable. At jump discontinuities, modeling considerations may impose the defect in the conservation of some quantities, thus leading to non conservative products. Below, we deduce the evolution in the smooth case from the jump conditions at discontinuities. Moreover, the resulting framework enjoys well posedness and solutions are uniquely characterized. These results apply, for instance, to the flow of water in a canal with varying width and depth, as well as to the inviscid Euler equations in pipes with varying geometry.

We present a unified framework ensuring well posedness and providing stability estimates to a class of Initial Boundary Value Problems for renewal equations comprising a variety of biological or epidemiological models. This versatility is achieved considering fairly general -- possibly non linear and/or non local -- interaction terms, allowing both low regularity assumptions and independent variables with or without a boundary. In particular, these results also apply, for instance, to a model for the spreading of a Covid like pandemic or other epidemics. Further applications are shown to be covered by the present setting., Comment: 32 pages

Consider the coupling of $2$ evolution equations, each generating a global process. We prove that the resulting system generates a new global process. This statement can be applied to differential equations of various kinds. In particular, it also yields the well posedness of a predator-prey model, where the coupling is in the differential terms, and of an epidemiological model, which does not fit previous well posedness results., Comment: 46 pages

We propose a framework for the description of the effects of vaccinations on the spreading of an epidemic disease. Different vaccines can be dosed, each providing different immunization times and immunization levels. Differences due to individuals' ages are accounted for through the introduction of either a continuous age structure or a discrete set of age classes. Extensions to gender differences or to distinguish fragile individuals can also be considered. Within this setting, vaccination strategies can be simulated, tested and compared, as is explicitly described through numerical integrations., Comment: 23 pages

We develop a time and space dependent predator-prey model. The predators' equation is a non local hyperbolic balance law, while the diffusion of prey obeys a parabolic equation, so that predators "hunt" for prey, while prey diffuse. A control term allows to describe the use of predators as parasitoids to limit the growth of prey-parasites. The general well posedness and stability results here obtained ensure the existence of optimal pest control strategies, as discussed through some numerical integrations. The specific example we have in mind is that of Trichopria drosophilae used to fight against the spreading of Drosophila suzukii.

We introduce a formalism to deal with the microscopic modeling of vehicular traffic on a road network. Traffic on each road is uni-directional, and the dynamics of each vehicle is described by a Follow-the-Leader model. From a mathematical point of view, this amounts to define a system of ordinary differential equations on an arbitrary network. A general existence and uniqueness result is provided, while priorities at junctions are shown to hinder the stability of solutions. We investigate the occurrence of the Braess paradox in a time-dependent setting within this model. The emergence of Nash equilibria in a non-stationary situation results in the appearance of Braess type paradoxes, and this is supported by numerical simulations., Comment: 15 pages

SIR models, also with age structure, can be used to describe the evolution of an infective disease. A vaccination campaign influences this dynamics immunizing part of the susceptible individuals, essentially turning them into recovered individuals. We assume that vaccinations are dosed at prescribed times or ages which introduce discontinuities in the evolutions of the S and R populations. It is then natural to seek the 'best' vaccination strategies in terms of costs and/or effectiveness. This paper provides the basic well posedness and stability results on the SIR model with vaccination campaigns, thus ensuring the existence of optimal dosing strategies., Comment: 31 pages

Consider two hyperbolic systems of conservation laws in one space dimension with the same eigenvalues and (right) eigenvectors. We prove that solutions to Cauchy problems with the same initial data differ at third order in the total variation of the initial datum. As a first application, relying on the classical Glimm-Lax result, we obtain estimates improving those in by Saint Raymond on the distance between solutions to the isentropic and non-isentropic inviscid compressible Euler equations, under general equations of state. Further applications are to the general scalar case, where rather precise estimates are obtained, to an approximation by Di Perna of the p-system and to a traffic model.

We introduce the use of conservation laws to develop strategies in multi-player consensus games. First, basic well posedness results provide a reliable analytic setting. Then, a general non anticipative strategy is proposed through its rigorous analytic definitions and then tested by means of numerical integrations.

The well posedness for a class of non local systems of conservation laws in a bounded domain is proved and various stability estimates are provided. This construction is motivated by the modelling of crowd dynamics, which also leads to define a non local operator adapted to the presence of a boundary. Numerical integrations show that the resulting model provides qualitatively reasonable solutions.

The initial boundary value problem for a class of scalar non autonomous conservation laws in one space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity assumptions on the flow are extended to a merely $\mathbf{L}^{\infty}$ dependence on time. These results ensure, for instance, the well posedness of a class of vehicular traffic models with time dependent speed limits. A traffic management problem is then shown to admit an optimal solution.

We show the existence of the Braess paradox for a traffic network with nonlinear dynamics described by the Lighthill-Whitham-Richards model for traffic flow. Furthermore, we show how one can employ control theory to avoid the paradox. The paper offers a general framework applicable to time-independent, uncongested flow on networks. These ideas are illustrated through examples.

A system of renewal equations on a graph provides a framework to describe the exploitation of a biological resource. In this context, we formulate an optimal control problem, prove the existence of an optimal control and ensure that the target cost function is polynomial in the control. In specific situations, further information about the form of this dependence is obtained. As a consequence, in some cases the optimal control is proved to be necessarily bang--bang, in other cases the computations necessary to find the optimal control are significantly reduced.

Consider two compressible immiscible fluids in 1D in the isentropic approximation. The first fluid is surrounded and in contact with the second one. As the Mach number of the first fluid vanishes, the coupled dynamics of the two fluids results as the compressible to incompressible limit and is known to satisfy an ODE--PDE system. Below, a characterization of this limit is provided, ensuring its uniqueness.

Inspired by the works of Hughes [17, 18], we formalize and prove the well posedness of a hyperbolic--elliptic system whose solutions describe the dynamics of a moving crowd. The resulting model is here shown to be well posed and the time of evacuation from a bounded environment is proved to be finite. This model also provides a microscopic description of the individuals' behaviors.

The Escalator Boxcar Train (EBT) is a tool widely used in the study of balance laws motivated by structure population dynamics. This paper proves that the approximate solutions defined through the EBT converge to exact solutions. Moreover, this method is rigorously shown to be effective also in computing optimal controls. As preliminary results, the well posedness of classes of PDEs and of ODEs comprising various biological models is also obtained. A specific application to welfare policies illustrates the whole procedure., Comment: 25 pages

We prove the stability with respect to the flux of solutions to initial-boundary value problems for scalar non-autonomous conservation laws in one space dimension. Key estimates are obtained through a careful construction of the solutions.

In this paper we study isentropic flow in a curved pipe. We focus on the consequences of the geometry of the pipe on the dynamics of the flow. More precisely, we present the solution of the general Cauchy problem for isentropic fluid flow in an arbitrarily curved, piecewise smooth pipe. We consider initial data in the subsonic regime, with small total variation about a stationary solution. The proof relies on the front-tracking method and is based on [1].

Two compressible immiscible fluids in 1D and in the isentropic approximation are considered. The first fluid is surrounded and in contact with the second one. As the Mach number of the first fluid vanishes, we prove the rigorous convergence for the fully non--linear compressible to incompressible limit of the coupled dynamics of the two fluids. A key role is played by a suitably refined wave front tracking algorithm, which yields precise $\mathbf{BV}$, $\mathbf{L}^1$ and weak* convergence estimates, either uniform or explicitly dependent on the Mach number.

The initial-boundary value problem for a general balance law in a bounded domain is proved to be well posed. Indeed, we show the existence of an entropy solution, its uniqueness and its Lipschitz continuity as a function of time, of the initial datum and of the boundary datum. The proof follows the general lines in Bardos, Leroux and N\'ed\'elec (1979), striving to provide a rigorous treatment and detailed references.

For a class of systems of nonlinear and nonlocal balance laws in several space dimensions, we prove the local in time existence of solutions and their continuous dependence on the initial datum. The choice of this class is motivated by a new model devoted to the description of a metal plate being cut by a laser beam. Using realistic parameters, solutions to this model obtained through numerical integrations meet qualitative properties of real cuts. Moreover, the class of equations considered comprises a model describing the dynamics of solid particles along a conveyor belt.

Nowadays, traffic monitoring systems have access to real time data, e.g. through GPS devices. We propose a new traffic model able to take into account these data and, hence, able to describe the effects of unpredictable accidents. The well posedness of this model is proved and numerical integrations show qualitative features of the resulting solutions. As a further motivation for the use of real time data, we show that the inverse problem for the Lighthill-Whitam and Richards (LWR) model is ill posed., Comment: 21 pages, 8 figures

Nonlocal conservation laws are used to describe various realistic instances of crowd behaviors. First, a basic analytic framework is established through an "ad hoc" well posedness theorem for systems of nonlocal conservation laws in several space dimensions interacting non locally with a system of ODEs. Numerical integrations show possible applications to the interaction of different groups of pedestrians, and also with other "agents"., Comment: 26 pages, 5 figures

We prove existence and uniqueness of solutions, continuous dependence from the initial datum and stability with respect to the boundary condition in a class of initial--boundary value problems for systems of balance laws. The particular choice of the boundary condition allows to comprehend models with very different structures. In particular, we consider a juvenile-adult model, the problem of the optimal mating ratio and a model for the optimal management of biological resources. The stability result obtained allows to tackle various optimal management/control problems, providing sufficient conditions for the existence of optimal choices/controls., Comment: 22 pages, 7 figures

We present a traffic flow model consisting of a gluing between the Lighthill-Whitham and Richards macroscopic model with a first order microscopic follow the leader model. The basic analytical properties of this model are investigated. Existence and uniqueness are proved, as well as the basic estimates on the dependence of solutions from the initial data. Moreover, numerical integrations show some qualitative features of the model, in particular the transfer of information among regions where the different models are used.

We present a nonlinear predator-prey system consisting of a nonlocal conservation law for predators coupled with a parabolic equation for preys. The drift term in the predators' equation is a nonlocal function of the prey density, so that the movement of predators can be directed towards region with high prey density. Moreover, Lotka-Volterra type right hand sides describe the feeding. A theorem ensuring existence, uniqueness, continuous dependence of weak solutions and various stability estimates is proved, in any space dimension. Numerical integrations show a few qualitative features of the solutions., Comment: 35 pages, 7 figures