We formulate an immuno-epidemiological model of coupled “within-host” model of ODEs and“between-host” model of ODE and PDE, using the Human Immunodeficiency Virus (HIV) forillustration. Existence and uniqueness of solution to the “between-host” model isestablished, and an explicit expression for the basic reproduction number of the“between-host” model derived. Stability of disease-free and endemic equilibria isinvestigated. An optimal control problem with drug-treatment control on the within-hostsystem is formulated and analyzed; these results are novel for optimal control of ODEslinked with such first order PDEs. Numerical simulations based on the forward-backwardsweep method are obtained.

This paper investigates the output controllability problem of temporal Boolean networkswith inputs (control nodes) and outputs (controlled nodes). A temporal Boolean network isa logical dynamic system describing cellular networks with time delays. Using semi-tensorproduct of matrices, the temporal Boolean networks can be converted into discrete timelinear dynamic systems. Some necessary and sufficient conditions on the outputcontrollability via two kinds of inputs are obtained by providingcorresponding reachable sets. Two examples are given to illustrate the obtainedresults.

We consider a damped abstract second order evolution equation with an additionalvanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, wedo not assume the operator defining the main damping to be bounded. First, using aconstructive frequency domain method coupled with a decomposition of frequencies and theintroduction of a new variable, we show that if the limit system is exponentially stable,then this evolutionary system is uniformly − with respect to the calibration parameter −exponentially stable. Afterwards, we prove uniform polynomial and logarithmic decayestimates of the underlying semigroup provided such decay estimates hold for the limitsystem. Finally, we discuss some applications of our results; in particular, the case ofboundary damping mechanisms is accounted for, which was not possible in the earlier workmentioned above.

Galerkin reduced-order models for the semi-discrete wave equation, that preserve thesecond-order structure, are studied. Error bounds for the full state variables are derivedin the continuous setting (when the whole trajectory is known) and in the discrete settingwhen the Newmark average-acceleration scheme is used on the second-order semi-discreteequation. When the approximating subspace is constructed using the proper orthogonaldecomposition, the error estimates are proportional to the sums of the neglected singularvalues. Numerical experiments illustrate the theoretical results.

We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phaseflow model, that is able to cope with arbitrarily small values of the statistical phasefractions. The solver relies on a relaxation approximation of the model for which theRiemann problem is exactly solved for subsonic relative speeds. In an original manner, theRiemann solutions to the linearly degenerate relaxation system are allowed to dissipatethe total energy in the vanishing phase regimes, thereby enforcing the robustness andstability of the method in the limits of small phase fractions. The scheme is proved tosatisfy a discrete entropy inequality and to preserve positive values of the statisticalfractions and densities. The numerical simulations show a much higher precision and a morereduced computational cost (for comparable accuracy) than standard numerical schemes usedin the nuclear industry. Finally, two test-cases assess the good behavior of the schemewhen approximating vanishing phase solutions.

Inspired by the growing use of non linear discretization techniques for the lineardiffusion equation in industrial codes, we construct and analyze various explicit nonlinear finite volume schemes for the heat equation in dimension one. These schemes areinspired by the Le Potier’s trick [C. R. Acad. Sci. Paris, Ser. I348 (2010) 691–695]. They preserve the maximum principle and admita finite volume formulation. We provide a original functional setting for the analysis ofconvergence of such methods. In particular we show that the fourth discrete derivative isbounded in quadratic norm. Finally we construct, analyze and test a new explicit nonlinear maximum preserving scheme with third order convergence: it is optimal on numericaltests.

We prove uniform continuity ofradially symmetric vector minimizersuA(x) = UA(|x|)to multiple integrals ∫BRL**(u(x), |Du(x)|) dx on aball BR ⊂ ℝd,among the Sobolev functions u(·) in A+W01,1 (BR, ℝm), using ajointly convex lsc L∗∗ : ℝm×ℝ → [0,∞]withL∗∗(S,·) evenand superlinear. Besides such basic hypotheses,L∗∗(·,·) is assumed to satisfy alsoa geometrical constraint, which we callquasi − scalar; the simplest example being thebiradial caseL∗∗(|u(x)|,|Du(x)|).Complete liberty is given forL∗∗(S,λ)to take the ∞ value, so that our minimization problem implicitly also representse.g. distributed-parameteroptimal control problems, onconstrained domains, under PDEs or inclusions inexplicit or implicit form. While generic radial functionsu(x) = U(|x|) inthis Sobolev space oscillate wildly as |x| → 0, our minimizingprofile-curve UA(·) is, incontrast, absolutely continuous andtame, in the sense that its“static level” L∗∗(UA(r),0)always increases with r, a original feature of our result.

The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? The characteristic of our viscous term is that it contains the fractional power α of the Dirichlet Laplace operator. Through the parameter α we may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term, being multiplied by a small parameter ε devoted to tend to zero, vanishes in the limit. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to ε, under the assumption that α∈[0,1)\{½}. It follows that, under this assumption, our starting question has a positive answer.

This note considers an established reaction–diffusion model for a combustion system, in which there are competing endothermic and exothermic reaction pathways. A combustion front is assumed to move at constant speed through the medium. An asymptotic theory is presented for solid fuels in which material diffusion is ignored, and it allows a simple and complete analysis of the approximate system in the phase plane. Both the adiabatic and nonadiabatic cases are discussed.

Microbial competition for nutrients is a common phenomenon that occurs between species inhabiting the same environment. Bioreactors are often used for the study of microbial competition since the number and type of microbial species can be controlled, and the system can be isolated from other interactions that may occur between the competing species. A common type of competition is the so-called “pure and simple” competition, where the microbial populations interact in no other way except the competition for a single rate-limiting nutrient that affects their growth rates. The issue of whether pure and simple competition under time-invariant conditions can give rise to chaotic behaviour has been unresolved for decades. The third author recently showed, for the first time, that chaos can theoretically occur in these systems by analysing the dynamics of a model where both competing species grow following the biomass-dependent Contois model while the yield coefficients associated with the two species are substrate-dependent. In this paper we show that chaotic behaviour can occur in a much simpler model of pure and simple competition. We examine the case where only one species grows following the Contois model with variable yield coefficient while the other species is allowed to grow following the simple Monod model with constant yield. We show that while the static behaviour of the proposed model is quite simple, the dynamic behaviour is complex and involves period doubling culminating in chaos. The proposed model could serve as a basis to re-examine the importance of Contois kinetics in predicting complex behaviour in microbial competition.