Said Hadd

• Professor
• Professor at University Ibn Zohr - Agadir

The main purpose of this paper is to use ideas from systems theory to investigate the concept of maximal \(L^p\)-regularity for some perturbed autonomous and non-autonomous evolution equations in Banach spaces. We mainly consider two classes of perturbations: Miyadera–Voigt perturbations and Desch–Schappacher perturbations. We introduce conditions...

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...

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,...

In this paper, we are going to present a functional analytic approach to the well-posedness of input-output infinite dimensional stochastic linear systems. This is in fact a stochastic version of the known Salamon-Weiss linear systems. We also prove controllability and observability properties of such stochastic systems. On the other hand, we use t...

This paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the...

In this paper, we study evolution equations that are perturbed at the boundary by both noise and an unbounded perturbation. First, using the theory of regular linear systems, we prove the existence of solutions to this equation. Second, we investigate the long-time behavior of the solutions, such as the absolute continuity and the existence of an i...

The work is concerned with the concept of stochastic maximal regularity. In fact, we proved that such a regularity is stable under admissible observation operators.

The aim of this study is twofold. Initially, by employing a perturbation semigroup approach and admissible observation operators, a novel variation of constants formula is presented for the mild solutions of a specific set of integrodifferential equations in Banach spaces. Subsequently, utilizing this formula, an examination of the maximal \(L^p\)-...

We prove a new variation of constants formula for the mild solutions of a class of functional integrodifferential equations of neutral type with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{...

We study admissible observation operators for perturbed evolution equations using the concept of maximal regularity. We first show the invariance of the maximal Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlengt...

We present a novel approach to address integro-differential systems incorporating state, input, and output delays. Our approach leverages product spaces and employs a boundary perturbation technique. Initially, we focus on state-delay equations, wherein we introduce a variation of constants formula for the mild solution. Additionally, we establish...

This paper focuses on the study of integro-differential equations with delays, presenting a novel perturbation approach. The primary objective is to introduce the concepts of classical and mild solutions for these equations and establish their existence and uniqueness, under suitable assumptions. Furthermore, we provide a variation of constants for...

In this paper, we delve into the study of evolution equations that exhibit white-noise boundary conditions. Our primary focus is to establish a necessary and sufficient condition for the existence of solutions, by utilizing the concept of admissible observation operators and the Yosida extension for such operators. By employing this criterion, we c...

Let A,C,P:D(A)⊂X→X be linear operators on a Banach space X such that -A generates a strongly continuous semigroup on X, and F:X→X be a globally Lipschitz function. We study the well-posedness of semilinear equations of the form u˙(t)=G(u(t)), where G:D(A)→X is a nonlinear map defined by G=-A+C+F∘P. In fact, using the concept of maximal Lp-regularit...

Let $A,C,P:D(A)\subset X\to X$ be linear operators on a Banach space $X$ such that $-A$ generates a strongly continuous semigroup on $X$, and $F:X\to X$ be a globally Lipschitz function. We study the well-posedness of semilinear equations of the form $\dot{u}(t)=G(u(t))$, where $G:D(A)\to X$ is a nonlinear map defined by $G=-A+C+F\circ P$. In fact,...

We study admissible observation operators for perturbed evolution equations using the concept of maximal regularity. We first show the invariance of the maximal $L^p$-regularity under non-autonomous Miyadera-Voigt perturbations. Second, we establish the invariance of admissibility of observation operators under such a class of perturbations. Finall...

In this paper, we study the positivity and (uniform) exponential stability of a large class of perturbed semigroups. Our approach is essentially based on feedback theory of infinite-dimensional linear systems. The obtained results are applied to the stability of hyperbolic systems including those with a delay at the boundary conditions. The efficie...

In this paper, we show that the concept of maximal Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}-regularity is stable under a large class of unb...

This work focuses on the well-posedness of abstract stochastic linear systems with boundary input delay and unbounded observation operators. We use product spaces and a semigroup approach to reformulate such delay systems into free-delay distributed stochastic systems with unbounded control and observation operators. This gives us the opportunity t...

In this work, we present a way of treating an infinite-dimensional stochastic bacterial population system with delay in state space. This model is a variant of the transport equation, in which each bacteria is described by its degree of maturity and its maturation velocity. Here we use the operator theory to reformulate the boundary deterministic b...

In this paper, we consider a class of semilinear non-autonomous equations, where the nonlinear term does not perfectly satisfy the Lipschitz condition. It means that the nonlinear function is defined as the composition of a global Lipschitz function and non-autonomous unbounded linear operators. This case is beyond the scope of standard Cauchy-Lips...

In this paper, we are concerned with unbounded observation operators for perturbed non-autonomous evolution equations. First, we briefly survey the necessary background about the concept of maximal regularity. Second, we establish the admissibility invariance of observation operators under non-autonomous Miyadera-Voigt perturbations. Finally, we il...

This work focuses on the well-posedness of abstract stochastic linear systems with boundary input delay and unbounded observation operators. We use product spaces and a semigroup approach to reformulate such delay systems into free-delay distributed stochastic systems with unbounded control and observation operators. This gives us the opportunity t...

This present paper is mainly devoted to investigate the property of spectral decomposition of neutral differential equations in infinite dimensional setting, that is the exponential dichotomy. In fact, we prove that the exponential dichotomy of the associated semigroup to such equations does not depend on that of their associated difference equatio...

In this paper, we are interested in the concept of controllability for a class of transport processes on infinite metric graphs with a delay term at the boundary conditions. Using the feedback theory of well-posed and regular linear systems of infinite dimension, we provide the necessary and sufficient conditions for the approximate controllability...

In this paper, we study the positivity and (uniform) exponential stability of a large class of perturbed semigroups. Our approach is essentially based on the feedback theory of infinite-dimensional linear systems. The obtained results are applied to the stability of hyperbolic systems including those with a delay at the boundary conditions.

The purpose of this paper is to introduce a semigroup approach to linear integro-differential systems with delays in state, control and observation parts. On the one hand, we use product spaces to reformulate state-delay integro-differential equations to a standard Cauchy problem and then use a perturbation technique (feedback) to prove the well-po...

In this paper, we study the concept of approximate controllability of retarded network systems of neutral type. On one hand, we reformulate such systems as free-delay boundary control systems on product spaces. On the other hand, we use the rich theory of infinite-dimensional linear systems to derive necessary and sufficient conditions for approxim...

In this paper, we study the concept of approximate controllability of retarded network systems of neutral type. On one hand, we reformulate such systems as free-delay boundary control systems on product spaces. On the other hand, we use the rich theory of infinite-dimensional linear systems to derive necessary and sufficient conditions for the appr...

The main purpose of this paper is to treat semigroup 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...

The well-posedness of abstract boundary control systems with dynamic boundary conditions (i.e., boundary conditions evolving according to an operator semigroup acting on the boundary space) is established. Here, the boundary feedback operator is unbounded which makes the investigation more interesting in many applications. The positivity of such pr...

In this paper, we cross the boundary between semigroup theory and general infinite-dimensional systems to bridge the isolated research activities in the two areas. Indeed, we first give a chronological history of the development of the semigroup approach for control theory. Second, we use the feedback theory to prove the well-posedness of a class o...

The purpose of this paper is to investigate the robustness of exact controllability of perturbed linear systems in Banach spaces. Under some conditions, we prove that the exact controllability is preserved if we perturb the generator of an infinite-dimensional control system by appropriate Miyadera–Voigt perturbations. Furthermore, we study the rob...

In this paper, the rich feedback theory of regular linear systems in the Salamon-Weiss sense as well as some advanced tools in semigroup theory are used to formulate and solve control problems for network systems. In fact, we derive necessary and sufficient conditions for approximate controllability of such systems. These criteria, in some particul...

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 o...

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...

This article is devoted to the error feedback regulation problem (EFRP) for linear distributed parameter systems. The plant, assumed to be a regular linear system, is driven by an exosystem via a disturbance signal. The exosystem has its spectrum in the imaginary axis and also generates the reference signal to be tracked. The EFRP is to design an e...

Let X;U and Z be Banach spaces such that Z ⊂ X (with continuous and dense embedding), L : Z → X be a closed linear operator and consider closed linear operators G;M : Z → U . Putting conditions on G and M we show that the operator A = L with domain D (A) = { z ∈ Z : Gz = Mz} generates a C0 {semigroup on X . Moreover, we give a variation of constant...

In this paper we prove, under suitable hypotheses, eventually norm continuity and compactness of the solution semigroups of certain neutral differential equations in Banach spaces. Our approach is based on a general perturbation theorem obtained from closed-loop systems of infinite dimensional control systems with unbounded control and observation...

A new functional analytic approach to the concept of feedback stabilizability of infinite dimensional linear neutral systems is presented. We first reformulate this systems as an infinite dimensional open-loop systems with appropriate semigroups and unbounded control operators. We introduce conditions for which such semigroups are eventually compac...

In this paper, admissibility of control operators for perturbed semigroups is considered. We prove that the set of admissible control operators for a semigroup generator is unchanged if we perturb this generator by an appropriate class of unbounded operators. The result is applied to prove the well-posedness of a class of time-delay systems.

The PI controller for plants with unbounded control and observation operators is discussed. This is a generalization of pervious work considering bounded control operators. Our approach is mainly based on regular linear systems in the Salamon-Weiss sense.

The feedback stabilizability of a general class of well-posed linear systems with state and input delays in Banach spaces is studied in this paper. Using the properties of infinite dimensional linear systems, a necessary condition for the feedback stabilizability of delay systems is presented, which extends the well-known results for finite dimensi...

A semigroup approach for the well-posedness of perturbed nonhomogeneous abstract boundary value problems is developed in this paper. This allows us to introduce a useful variation of constant formula for the solutions. Drawing from this formula, necessary and sufficient conditions for the approximate controllability of such systems are obtained, us...

In this paper, necessary and sufficient conditions for the approximate controllability of a large class of neutral systems in reflexive Banach spaces in investigated. The approach is based on the transformation of such systems into appropriate boundary control systems.

In this note we present sufficient conditions to guarantee the passivity of linear systems with state, input and output delays in Hilbert spaces. Our approach is mainly based on the transformation of such systems into distributed parameter systems.

The well-posedness of a large class of singular partial differential equations of neutral type is discussed. Here the term singularity means that the difference operator of such equations is nonatomic at zero. This fact offers many difficulties in applying the usual methods of perturbation theory and Laplace transform technique and thus makes the s...

We show that the class of regular time-varying systems is invariant under perturbations by time-varying state and input delays. In particular, we give explicit formulas of the resulting input, output and input-output maps. This result is used to solve the feedback problem for the delayed system. The relationship between the open- and the closed-loo...

Semigroup approach to feedback stabilization of systems with state delay has already been introduced in the literature. The aim of this work is to apply the semigroup approach to the feedback stabilization of partial differential systems with state and input delays. Here we shall work with general state Banach spaces and a general class of state an...

In this paper, the well-posedness of neutral equations with atomic difference operators in infinite-dimensional Hilbert spaces is studied. Some classes of difference and delay operators which are not necessarily represented as Riemann--Stieltjes integrals are considered. The general approach is based essentially on representation of closed loop sys...

This paper studies the well-posedness of a class of non-autonomous neutral control systems in Banach spaces. We prove that such systems are represented by absolutely regular non-autonomous linear systems in the sense of Schnaubelt [R. Schnaubelt, Feedback for non-autonomous regular linear systems, SIAM J. Control Optim. 41 (2002) 1141–1165]. This p...

The aim of this paper is to prove that a class of distributed parameter systems governed by neutral FDEs provides regular linear systems. Employing the well established theory of representation, transfer function and feedback of these later we then give new representations of the state and the output function of the neutral systems.

This paper concerns linear infinite-dimensional systems with unbounded observation operators. We study the invariance of admissibility for observation operators under some unbounded perturbations of the generator. We give also some relations between the Λ-extensions of such observation operators with respect to the original generator and the pertur...

We investigate the infinite dimensional control linear systems with delays in the state and input. We give a new variation of constants formula when the state and control delay operators are unbounded. We prove the existence of mild and classical solutions of such systems. Our approach is based on the theory of abstract and regular linear systems i...

We propose a new approach which brings nonautonomous linear systems with state, input, and output delays in the line with the standard theory of nonautonomous linear systems. To this purpose, we establish, using the concept of Lebesgue extensions, a new variation of constants formula for nonhomogenous delay equations. From this we deduce another ne...

This paper studies the concept of controllability for infinite-dimensional linear control systems in Banach spaces. First, we prove that the set of admissible control operators for the semigroup generator is unchanged if we perturb the generator by the Desch–Schappacher perturbations. Second we show that exact controllability is not changed by such...

In this paper, we give a new reformulation of linear systems with delays in input, state and output. We show that these systems can be written as a regular linear system without delays. The technique used here is essentially based on the theory recently developed by Salamon and Weiss and the shift in semigroup properties. Our framework can be appli...

In this work we study the asymptotic behavior of the solutions of some population equations with diffusion in unbounded domains by using the notion of critical spectrum introduced recently by R. Nagel and J. Poland. To do this, we extend the abstract results of Brendle-Nagel-Poland, concerning the persistence under perturbations of the critical spe...

In this paper we prove a perturbation result for strongly continuous semigroups extending one of G. Weiss from Hilbert to Banach spaces. This allows us to establish a new variation of constants formula for non-homogeneous perturbed Cauchy problems. From this formula we deduce another new one for non-homogeneous functional differential equations wit...

We show that the class of regular non–autonomous systems is invariant under perturbation by time–varying state and input delays. In particular, we give explicit formulas of the resulting input, output, and input–output maps. This result is used to solve the feedback problem for the delayed system. The relationship between the open and the closed lo...