Lyapunov's first method: Estimates of characteristic numbers of functional matrices

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Lyapunov's first method: Estimates of characteristic numbers of functional matrices. / Ermolin, V. S.; Vlasova, T. V.

In: Vestnik Sankt-Peterburgskogo Universiteta, Prikladnaya Matematika, Informatika, Protsessy Upravleniya, Vol. 15, No. 4, 2019, p. 442-456.

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Author

Ermolin, V. S. ; Vlasova, T. V. / Lyapunov's first method: Estimates of characteristic numbers of functional matrices. In: Vestnik Sankt-Peterburgskogo Universiteta, Prikladnaya Matematika, Informatika, Protsessy Upravleniya. 2019 ; Vol. 15, No. 4. pp. 442-456.

BibTeX

@article{cb07fe8f495842fcba2d0b828cf64dab,

title = "Lyapunov's first method: Estimates of characteristic numbers of functional matrices",

abstract = "This paper contains the development of theoretical fundamentals of the first method of Lyapunov. We analyze the relations between characteristic numbers of functional matrices, their rows, and columns. We consider Lyapunov's results obtained to evaluate and calculate characteristic numbers for products of scalar functions and prove a theorem on the generalization of these results to the products of matrices. This theorem states necessary and sufficient conditions for the existence of rigorous estimates for characteristic numbers of matrix products. Also, we prove a theorem that establishes a relationship between the characteristic number of a square non-singular matrix and the characteristic number of its inverse matrix, and the determinant. The stated relations and properties of the characteristic numbers of square matrices we reformulate in terms of the Lyapunov exponents. Examples of matrices illustrate the proved theorems.",

keywords = "Characteristic numbers, Functional matrices, Lyapunov's first method, Stability theory, The Lyapunov exponent, Первый метод Ляпунова, теория устойчивости, характеристичные числа, показатели Ляпунова, функциональные матрицы",

author = "Ermolin, {V. S.} and Vlasova, {T. V.}",

note = "Ermolin V. S., Vlasova T. V. Lyapunov{\textquoteright}s first method: estimates of characteristic numbers of functional matrices. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2019, vol. 15, iss. 4, pp. 442–456. https://doi.org/10.21638/11702/spbu10.2019.403 ",

year = "2019",

doi = "10.21638/11702/spbu10.2019.403",

language = "English",

volume = "15",

pages = "442--456",

journal = " ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. ПРИКЛАДНАЯ МАТЕМАТИКА. ИНФОРМАТИКА. ПРОЦЕССЫ УПРАВЛЕНИЯ",

issn = "1811-9905",

publisher = "Издательство Санкт-Петербургского университета",

number = "4",

}

RIS

TY - JOUR

T1 - Lyapunov's first method: Estimates of characteristic numbers of functional matrices

AU - Ermolin, V. S.

AU - Vlasova, T. V.

N1 - Ermolin V. S., Vlasova T. V. Lyapunov’s first method: estimates of characteristic numbers of functional matrices. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2019, vol. 15, iss. 4, pp. 442–456. https://doi.org/10.21638/11702/spbu10.2019.403

PY - 2019

Y1 - 2019

N2 - This paper contains the development of theoretical fundamentals of the first method of Lyapunov. We analyze the relations between characteristic numbers of functional matrices, their rows, and columns. We consider Lyapunov's results obtained to evaluate and calculate characteristic numbers for products of scalar functions and prove a theorem on the generalization of these results to the products of matrices. This theorem states necessary and sufficient conditions for the existence of rigorous estimates for characteristic numbers of matrix products. Also, we prove a theorem that establishes a relationship between the characteristic number of a square non-singular matrix and the characteristic number of its inverse matrix, and the determinant. The stated relations and properties of the characteristic numbers of square matrices we reformulate in terms of the Lyapunov exponents. Examples of matrices illustrate the proved theorems.

AB - This paper contains the development of theoretical fundamentals of the first method of Lyapunov. We analyze the relations between characteristic numbers of functional matrices, their rows, and columns. We consider Lyapunov's results obtained to evaluate and calculate characteristic numbers for products of scalar functions and prove a theorem on the generalization of these results to the products of matrices. This theorem states necessary and sufficient conditions for the existence of rigorous estimates for characteristic numbers of matrix products. Also, we prove a theorem that establishes a relationship between the characteristic number of a square non-singular matrix and the characteristic number of its inverse matrix, and the determinant. The stated relations and properties of the characteristic numbers of square matrices we reformulate in terms of the Lyapunov exponents. Examples of matrices illustrate the proved theorems.

KW - Characteristic numbers

KW - Functional matrices

KW - Lyapunov's first method

KW - Stability theory

KW - The Lyapunov exponent

KW - Первый метод Ляпунова

KW - теория устойчивости

KW - характеристичные числа

KW - показатели Ляпунова

KW - функциональные матрицы

UR - http://www.scopus.com/inward/record.url?scp=85082080314&partnerID=8YFLogxK

U2 - 10.21638/11702/spbu10.2019.403

DO - 10.21638/11702/spbu10.2019.403

M3 - Article

AN - SCOPUS:85082080314

VL - 15

SP - 442

EP - 456

JO - ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. ПРИКЛАДНАЯ МАТЕМАТИКА. ИНФОРМАТИКА. ПРОЦЕССЫ УПРАВЛЕНИЯ

JF - ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. ПРИКЛАДНАЯ МАТЕМАТИКА. ИНФОРМАТИКА. ПРОЦЕССЫ УПРАВЛЕНИЯ

SN - 1811-9905

IS - 4

ER -

ID: 52545915