On the Compactness of Solutions to Certain Operator Inequalities Arising from the Likhtarnikov–Yakubovich Frequency Theorem

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On the Compactness of Solutions to Certain Operator Inequalities Arising from the Likhtarnikov–Yakubovich Frequency Theorem. / Anikushin, M. M.

In: Vestnik St. Petersburg University: Mathematics, Vol. 54, No. 4, 10.2021, p. 301-310.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{81204d690bc64b5b9eaac303c4902718,

title = "On the Compactness of Solutions to Certain Operator Inequalities Arising from the Likhtarnikov–Yakubovich Frequency Theorem",

abstract = "Abstract: The compactness property of operator solutions to certain operator inequalities arising from the Likhtarnikov–Yakubovich frequency theorem for C0-semigroups is studied. It is shown that the operator solution can be described through solutions to an adjoint problem, as was previously known under a certain regularity condition. Thus, some regularity properties of the semigroup are connected with the compactness of the operator in the general case. Several results are proved that are useful for checking the noncompactness of operator solutions to Lyapunov inequalities and equations, into which the operator Riccati equation degenerates in certain cases that arise in applications. As an example, these theorems are applied for a scalar delay equation posed in a proper Hilbert space and it is shown that the operator solution cannot be compact. These results are related to the author{\textquoteright}s recent work on the nonlocal reduction principle of cocycles (nonautonomous dynamical systems) in Hilbert spaces.",

keywords = "compact operator, delay equations, frequency theorem, Lyapunov inequality",

author = "Anikushin, {M. M.}",

note = "Publisher Copyright: {\textcopyright} 2021.",

year = "2021",

month = oct,

doi = "10.1134/S1063454121040026",

language = "English",

volume = "54",

pages = "301--310",

journal = "Vestnik St. Petersburg University: Mathematics",

issn = "1063-4541",

publisher = "Pleiades Publishing",

number = "4",

}

RIS

TY - JOUR

T1 - On the Compactness of Solutions to Certain Operator Inequalities Arising from the Likhtarnikov–Yakubovich Frequency Theorem

AU - Anikushin, M. M.

PY - 2021/10

Y1 - 2021/10

N2 - Abstract: The compactness property of operator solutions to certain operator inequalities arising from the Likhtarnikov–Yakubovich frequency theorem for C0-semigroups is studied. It is shown that the operator solution can be described through solutions to an adjoint problem, as was previously known under a certain regularity condition. Thus, some regularity properties of the semigroup are connected with the compactness of the operator in the general case. Several results are proved that are useful for checking the noncompactness of operator solutions to Lyapunov inequalities and equations, into which the operator Riccati equation degenerates in certain cases that arise in applications. As an example, these theorems are applied for a scalar delay equation posed in a proper Hilbert space and it is shown that the operator solution cannot be compact. These results are related to the author’s recent work on the nonlocal reduction principle of cocycles (nonautonomous dynamical systems) in Hilbert spaces.

AB - Abstract: The compactness property of operator solutions to certain operator inequalities arising from the Likhtarnikov–Yakubovich frequency theorem for C0-semigroups is studied. It is shown that the operator solution can be described through solutions to an adjoint problem, as was previously known under a certain regularity condition. Thus, some regularity properties of the semigroup are connected with the compactness of the operator in the general case. Several results are proved that are useful for checking the noncompactness of operator solutions to Lyapunov inequalities and equations, into which the operator Riccati equation degenerates in certain cases that arise in applications. As an example, these theorems are applied for a scalar delay equation posed in a proper Hilbert space and it is shown that the operator solution cannot be compact. These results are related to the author’s recent work on the nonlocal reduction principle of cocycles (nonautonomous dynamical systems) in Hilbert spaces.

KW - compact operator

KW - delay equations

KW - frequency theorem

KW - Lyapunov inequality

UR - http://www.scopus.com/inward/record.url?scp=85121435995&partnerID=8YFLogxK

U2 - 10.1134/S1063454121040026

DO - 10.1134/S1063454121040026

M3 - Article

AN - SCOPUS:85121435995

VL - 54

SP - 301

EP - 310

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 4

ER -

ID: 92060434