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5 Research products, page 1 of 1

  • Other research products
  • Lecture
  • French National Research Agency (ANR)
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  • English
    Authors: 
    Boulier, François;
    Publisher: HAL CCSD
    Country: France
    Project: ANR | SYMBIONT (ANR-17-CE40-0036)

    Doctoral

  • English
    Authors: 
    Ruiz-Balet, Domènec; Zuazua, Enrique;
    Publisher: HAL CCSD
    Country: France
    Project: EC | DYCON (694126), EC | ConFlex (765579), ANR | ICON (ANR-16-ACHN-0014)

    Doctoral; These lecture notes address the controllability under relevant state constraints of reaction-diffusion equations. Typically the quantities modeled by reaction-diffusion equations in socio-biological contexts (e.g. population, concentrations of chemicals, temperature, proportions etc) are positive by nature. The uncontrolled models intrinsically preserve this nature thanks to the maximum principle. For this reason, any control strategy for such systems has to preserve these state constraints. We restrict our study in the case of scalar equations with monostable and bistable nonlinearities. The presence of constraints produces new phenomena such as a possible lack of controllability, or existence of a minimal controllability time. Furthermore , we explain general ways for proving controllability under state constraints. Among different strategies, we discuss how to use traveling waves and connected paths of steady states to ensure controllability. We devote particular attention to the construction of such connected paths of steady-states. Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to provide intuition to the reader.

  • Other research product . Lecture . 2016
    Open Access English
    Authors: 
    Golenia, Sylvain;
    Publisher: HAL CCSD
    Country: France
    Project: ANR | GeRaSic (ANR-13-BS01-0007)

    Master; This aim of this course is to give an overview to the study of the continuous spectrum of bounded self-adjoint operators and especially those coming from the setting of graphs. For the sake of completeness, a short course in spectral theory is given with proofs. The continuous and Borelian functional calculi are also developed.

  • Other research product . Lecture . 2016
    French
    Authors: 
    Idmhand, Fatiha;
    Publisher: HAL CCSD
    Country: France
    Project: ANR | CHispa (ANR-13-JSH3-0006)

    Master; Intervention réalisée à Lorient, le 22-09-2016, pour le Master mention Métiers du Livre et Humanités Numériques

  • English
    Authors: 
    Coulombel, Jean-François;
    Publisher: HAL CCSD
    Country: France
    Project: ANR | INTOCS (ANR-08-JCJC-0132)

    DEA; The aim of these notes is to present some results on the stability of finite difference approximations of hyperbolic initial boundary value problems. We first recall some basic notions of stability for the discretized Cauchy problem in one space dimension. Special attention is paid to situations where stability of the finite difference scheme is characterized by the so-called von Neumann condition. This leads us to the important class of geometrically regular operators. After discussing the discretized Cauchy problem, we turn to the case of initial boundary value problems. We introduce the notion of strongly stable schemes for zero initial data. The first main result characterizes strong stability in terms of a solvability property and an energy estimate for the resolvent equation. This first result shows that the so-called Uniform Kreiss-Lopatinskii Condition is a necessary condition for strong stability. The main result of these notes shows that the Uniform Kreiss-Lopatinskii Condition is also a sufficient condition for strong stability in the framework of geometrically regular operators. We illustrate our results on the Lax-Friedrichs and leap-frog schemes and check strong stability for various types of boundary conditions. We also extend a stability result by Goldberg and Tadmor for Dirichlet boundary conditions. In the last section of these notes, we show how to incorporate nonzero initial data and prove semigroup estimates for the discretized initial boundary value problems. We conclude with some remarks on possible improvements and open problems.

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