On the maximal regularity for perturbed autonomous and non-autonomous evolution equations

Amansag, A.; Bounit, Hamid; Driouich, A.; Hadd, Saïd

doi:10.1007/s00028-019-00514-8

Cited by 14 publications

(24 citation statements)

References 29 publications

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“…The feedback theory has been successfully used recently in Amansag et al [45] to prove the concept of the maximal L p -regularity for perturbed evolution equations in Banach spaces. More recently, Lahbri & Hadd [46] have extended the Salamon–Weiss theory to infinite dimensional stochastic systems.…”

Section: A Semigroup Approach To Infinite-dimensional Control Linear Systemsmentioning

confidence: 99%

Feedback theory to the well-posedness of evolution equations

Boulite

Hadd

Maniar

2020

Phil. Trans. R. Soc. A.

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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 of dynamic boundary problems. Third, the obtained results are applied to the well-posedness of neutral equations with non-autonomous past. We will also see that the strong connection between semigroup and control theories lies in feedback theory, where different kinds of perturbations appear. This article is part of the theme issue ‘Semigroup applications everywhere’.

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Section: A Semigroup Approach To Infinite-dimensional Control Linear Systemsmentioning

confidence: 99%

Feedback theory to the well-posedness of evolution equations

Boulite

Hadd

Maniar

2020

Phil. Trans. R. Soc. A.

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show abstract

“…where the integral is taken in X −1 . Formally, the well-posedness of the system (5) means that the state satisfies x(t) ∈ X for any t ≥ 0, the observation function y is extended to a locally p-integrable function y ∈ L p loc ([0, ∞), U ) satisfying the following property: for any τ > 0, there exists a constant c τ > 0 such that…”

Section: Feedback Theory Of Infinite Dimensional Linear Systemsmentioning

confidence: 99%

“…Remark 5. In the case of Desch-Schappacher perturbation (bounded perturbation at the boundary) we have obtained in [5,Theorem ] the maximal regularity without appealing to the condition (40). However for Staffans-Weiss perturbation (unbounded perturbation at the boundary), our proof of maximal regularity is essentially based on this condition.…”

Section: Define the Operatormentioning

confidence: 99%

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

Amansag¹,

Bounit²,

Driouich³

et al. 2020

Preprint

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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, 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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“…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 [2]). Applications to heat equations driven by the Dirichlet (or Neumann)-Laplacian are considered.…”

mentioning

confidence: 99%

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

Amansag¹,

Bounit²,

Driouich³

et al. 2020

Preprint

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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 [2]). Applications to heat equations driven by the Dirichlet (or Neumann)-Laplacian are considered.

show abstract

On the maximal regularity for perturbed autonomous and non-autonomous evolution equations

Cited by 14 publications

References 29 publications

Feedback theory to the well-posedness of evolution equations

Feedback theory to the well-posedness of evolution equations

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

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

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