Examinando por Autor "Warma, Mahamadi"

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  • Miniatura
    Ítem
    Addendum: Local elliptic regularity for the Dirichlet fractional Laplacian
    (Walter de Gruyter GmbH, 2017) Biccari, Umberto; Warma, Mahamadi; Zuazua, Enrique
    In [1], for 1 < p < 1
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    Ítem
    Control and numerical approximation of fractional diffusion equations
    (Elsevier B.V., 2022) Biccari, Umberto; Warma, Mahamadi; Zuazua, Enrique
    The aim of this chapter is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls, though not forgetting to recall other relevant contributions which can be currently found in the literature of this prolific field. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain Ω. The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the L2 and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset ω⊂Ω. Secondly, we move our attention to the exterior control problem, in which the control region O⊂Ωc is located outside Ω. This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on ∂Ω. We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigation.
  • Miniatura
    Ítem
    Local elliptic regularity for the Dirichlet fractional Laplacian
    (Walter de Gruyter GmbH, 2017) Biccari, Umberto; Warma, Mahamadi; Zuazua, Enrique
    We prove the Wloc2s,p local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of RN. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.
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    Ítem
    Local regularity for fractional heat equations
    (Springer International Publishing, 2018) Biccari, Umberto; Warma, Mahamadi; Zuazua, Enrique
    We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set Ω ⊂ ℝN. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.
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    Ítem
    Null-controllability properties of a fractional wave equation with a memory term
    (American Institute of Mathematical Sciences, 2020-06) Biccari, Umberto; Warma, Mahamadi
    We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. The problem being equivalent to a coupled nonlocal PDE-ODE system, in which the ODE component has zero velocity of propagation, we are required to use a moving control strategy. Assuming that the control is acting on an open subset ω(t) which is moving with a constant velocity c ∈ R, the main result of the paper states that the equation is null controllable in a sufficiently large time T and for initial data belonging to suitable fractional order Sobolev spaces. The proof will use a careful analysis of the spectrum of the operator associated with the system and an application of a classical moment method.