Observation of Volterra systems with scalar kernels

Haak, Bernhard Hermann; Jacob, Birgit

doi:10.1216/jie-2011-23-3-421

Cited by 2 publications

(3 citation statements)

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“…As for the control operators, and due to the expression of ( T V (t)) t≥0 (see the proof of Proposition 4 for its definition), it is clear that one can obtain finite-time or infinitetime L p -admissibility of observation operator C for (S(t)) t≥0 from that of C (see (31) for its definition) for ( T V (t)) t≥0 . Recall, that for some classes of kernel functions a(•) and k(•), the authors in [45,34,16] related the L 2 -admissibility of the observation operators for sVIEs (5) and sVIDPs ( 4)) from the L 2 -admissibility of the observation operators for CP (20). In the same direction, the following proposition establish a links between the L padmissibility of observation operators for the nsVIDPs (16) for p ∈ (1, ∞), and for sVIDPs (4) with p = 1, and the CP (20).…”

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

Bounit,

Tismane

2023

Math. Control Signals Syst.

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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 L p -admissible observation operators kernel. Sufficient and/or necessary conditions for L p -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) L p -admissibility and the uniform L p -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 L p -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. KeywordsC 0 -semigroups • Volterra integro-differential problems • Resolvent operators • Admissibility • Weiss conjecture • Regular linear systems

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“…The idea of treating finite-time L 2 -admissibility of control or observation operators for sVIPs (5) has been exploited past years by several authors (see e.g. [45], [46], [33], [34], [25], [10]). It is legitimate to ask if there is a relation between the L p -admissibility with respect to CPs and sVIPs, sVIDPs respectively.…”

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

Bounit,

Tismane

2023

Math. Control Signals Syst.

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

confidence: 99%

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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Observation of Volterra systems with scalar kernels

Abstract: Volterra observations systems with scalar kernels are studied. New sufficient conditions for admissibility of observation operators are developed. The obtained results are applied to time-fractional diffusion equations of distributed order.

Cited by 2 publications

References 27 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

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

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