arXiv subjects

S. Hadd

Publications and source records attributed to S. Hadd.

5 recordsLinked to original sources

arXiv2021

Controllability of vertex delay type problems by the regular linear systems approach

In this paper, we study the well-posedness and approximate controllability of a class of network systems having delays and controls at the boundary conditions. The particularity of this work is that the network system is defined on infinite metric graphs. This fact offers many difficulties in applying the usual methods. In fact, the well-posedness of the delay network system is obtained by using a semigroup approach on product spaces which is based on the concept of feedback theory of infinite-dimensional linear systems. This technique allows us to reformulate the delay system into a free-delay distributed control system. From this transformation, we deduce necessary and sufficient conditions for the boundary approximate controllability of such systems. Furthermore, a Rank condition for the approximate controllability is also obtained. This condition coincides with the usual Kalman controllability criterion in the case of a simple transport process on a finite graph. Finally, by applying our approach to a linear Eulerian model (with airborne delays) for (ATFM), we provide a new algebraic condition for the controllability of such a model in terms of the generic rank of the so-called extended controllability matrix.

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arXiv2020

Staffans-Weiss perturbations for Maximal $L^p$-regularity in Banach spaces

In this paper we show that the concept of maximal $L^p$-regularity is stable under a large class of unbounded perturbations, namely Staffans-Weiss perturbations. To that purpose, we first prove that the analyticity of semigroups is preserved under this class of perturbations, which is a necessary condition for the maximal regularity. In UMD spaces, $\mathcal{R}$-boundedness conditions are exploited to give conditions guaranteing the maximal regularity. For non-reflexive Banach space, a condition is imposed to the Dirichlet operator associated to the boundary value problem to prove the maximal regularity. A Pde example illustrating the theory and an application to a class of non-autonomous perturbed boundary value problems are presented.

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arXiv2020

On the maximal regularity for a class of Volterra integro-differential equations

We propose an approach based on perturbation theory to establish maximal $L^p$-regularity for a class of integro-differential equations. As the left shift semigroup is involved for such equations, we study maximal regularity on Bergman spaces for autonomous and non-autonomous integro-differential equations. Our method is based on the formulation of the integro-differential equations to a Cauchy problems, infinite dimensional systems theory and some recent results on the perturbation of maximal regularity (see \cite{AmBoDrHa}). Applications to heat equations driven by the Dirichlet (or Neumann)-Laplacian are considered.

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arXiv2018

On norm continuity, differentiability and compactness of perturbed semigroups

The main purpose of this paper is to treat semigroups properties, like norm continuity, compactness and differentiability for perturbed semigroups in Banach spaces. In particular, we investigate three large classes of perturbations, Miyadera-Voigt, Desch-Schappacher and Staffans-Weiss perturbations. Our approach is mainly based on feedback theory of Salamon-Weiss systems. Our results are applied to abstract boundary integro-differential equations in Banach spaces.

math.FA↗

arXiv2018

Maximal $L^p$-regularity for perturbed evolution equations in Banach spaces

The main purpose of this paper is to investigate the concept of maximal $L^p$-regularity for perturbed evolution equations in Banach spaces. We mainly consider three classes of perturbations: Miyadera-Voigt perturbations, Desch-Schappacher perturbations, and more general Staffans-Weiss perturbations. We introduce conditions for which the maximal $L^p$-regularity can be preserved under these kind of perturbations. We give examples for a boundary perturbed heat equation in $L^r$-spaces and a perturbed boundary integro-differential equation. We mention that our results mainly extend those in the works: [P. C. Kunstmann and L. Weis, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), 415-435] and [B.H. Haak, M. Haase, P.C. Kunstmann, Adv. Differential Equations 11 (2006), no. 2, 201-240].

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