On norm continuity, differentiability and compactness of perturbed semigroups

Boulouz, Abed; Bounit, Hamid; Driouich, A.; Hadd, Saïd

doi:10.1007/s00233-020-10138-x

Cited by 8 publications

(4 citation statements)

References 28 publications

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“…which is unbounded on X and it is bounded from X to X A −1 . The system (36)- (37) without memory term has been studied in detail in [64] (see e.g [4]) and it is transformed to the system (2), where it is proved that the control operator B = −A −1 D 0 ∈ L (U, (D(A))…”

Section: 1 (I)]) Now Let Us Definementioning

confidence: 99%

“…Now, thanks to the analyticity of (T (t)) t≥0 , the operator (−A) γ |D(A) is a finite-time L 2 -admissible observation for (T (t)) t≥0 , and that A has the finite-time L 2 -maximal regularity, which implies in turns that that ((T (t)) t≥0 , B, (−A) γ ) generates a L 2 -regular system (see [1]). Furthermore, the L 2 -regularity of ((T (t) t≥0 ), B,C) has been recently established in [4]). Consequently, the conditions of Theorem 2 are fulfilled and this ends the proof.…”

Section: Examplementioning

confidence: 99%

“…The L p -admissibility problem of both control and observation operators has attracted many authors during the last 30 years and there is a wealth of literature on the Cauchy problems (CPs in short) (i.e. (4) with k (t) = 0) [42,65,68,44,50,43,61,7,52]. There are too many related works to list them comprehensively here.…”

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confidence: 99%

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On the admissibility and input-output representation for a class of Volterra integro-differential systems

Bounit

Tismane

2022

Preprint

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This article studies a class of controlled-observed Volterra integro-differential problems in the case that the operator of the associated Cauchy problem generate a semigroup on a Banach space and the integral part being given by a convolution of integrable functions with Lp-admissible observation operators kernel. Sufficient and/or necessary conditions for Lp-admissibility of control and observation operators are given in term of the kernels. In particular, results on the equivalence between the finite-time (or infinite-time) Lp-admissibility and the uniform Lp-admissibility are given for both control and observation operators, extending a generalization of results known to hold for the standard Cauchy problems. Particular attention is paid to the problem of obtaining the input-output representation of such systems, providing a theory which is analogue to that of Salamon-Weiss for linear systems. We mention that our approach is mainly based on the theory of infinitedimensional Lp-regular linear systems in the Salamon-Weiss sense. These results are illustrated by an example involving heat conduction with memory given by some space fractional Laplacian kernel.

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Section: 1 (I)]) Now Let Us Definementioning

confidence: 99%

Section: Examplementioning

confidence: 99%

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On the admissibility and input-output representation for a class of Volterra integro-differential systems

Bounit

Tismane

2022

Preprint

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“…Note that the solution of the Cauchy problem (2.3) is xfalse(tfalse)=Tclfalse(tfalse)x0. Finally, we mention that the authors of [27] used a semigroup approach to prove that the operators A and A cl coincide. We also mention that norm continuity, compactness and eventual differentiability of the semigroup ( T cl ( t )) t ≥0 have been studied in the recent work [41].…”

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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Boundary perturbation of m-dissipative operators

Amansag

Boulouz

2022

Arch. Math.

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On norm continuity, differentiability and compactness of perturbed semigroups

Cited by 8 publications

References 28 publications

On the admissibility and input-output representation for a class of Volterra integro-differential systems

On the admissibility and input-output representation for a class of Volterra integro-differential systems

Feedback theory to the well-posedness of evolution equations

Boundary perturbation of m-dissipative operators

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