Discussion: “The Stability of a Second Order Linear Periodic System” (Davison, E. J., 1969, ASME J. Basic Eng., 91, p. 207–210)

Necessary and sufficient conditions are obtained for the stability of the following second order linear system: x˙=θ(t)x,θ(t)=θt+∑i=1lTi and θ(t) =A1,0<t<T1=A2,T1<t<T1+T2⋮=Al,∑i=1l−1Ti<t<∑i=1lTi in terms of the eigenvalues and elements of the matrices Ai, i = 1, 2…l. The conditions become very simple for the case that l = 2. An example of a pendulum with a vibrating support is included.

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Closure to “Discussion of ‘The Stability of a Second Order Linear Periodic System’” (1969, ASME J. Basic Eng., 91, p. 210)

Journal of Basic Engineering ◽

10.1115/1.3571066 ◽

1969 ◽

Vol 91 (2) ◽

pp. 210-210

Author(s):

E. J. Davison

Keyword(s):

Periodic System ◽

Second Order ◽

Order Linear ◽

The Stability ◽

Linear Periodic System

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Aboodh transform and the stability of second order linear differential equations

Advances in Difference Equations ◽

10.1186/s13662-021-03451-4 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Ramdoss Murali ◽

Arumugam Ponmana Selvan ◽

Choonkil Park ◽

Jung Rye Lee

Keyword(s):

Differential Equations ◽

Integral Transform ◽

Second Order ◽

Linear Differential Equations ◽

Ulam Stability ◽

Order Linear ◽

Linear Differential ◽

The Stability ◽

Rassias Stability

AbstractIn this paper, we introduce a new integral transform, namely Aboodh transform, and we apply the transform to investigate the Hyers–Ulam stability, Hyers–Ulam–Rassias stability, Mittag-Leffler–Hyers–Ulam stability, and Mittag-Leffler–Hyers–Ulam–Rassias stability of second order linear differential equations.

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Uniformly asymptotic stability of second-order linear time-varying systems

Advances in Mechanical Engineering ◽

10.1177/1687814020955099 ◽

2020 ◽

Vol 12 (9) ◽

pp. 168781402095509

Author(s):

Da-Ke Gu ◽

Chao Lu

Keyword(s):

Asymptotic Stability ◽

Stability Criterion ◽

Linear Time ◽

Second Order ◽

State Feedback Control ◽

Time Varying ◽

Order Linear ◽

Time Varying Systems ◽

Uniformly Asymptotic Stability ◽

The Stability

This paper is concerned with the stability of second-order linear time-varying systems. By utilizing the Lyapunov approach, a generally uniformly asymptotic stability criterion is obtained by adding the system matrices into the quadratic Lyapunov candidate function. In the case of the derivative of the Lyapunov candidate function is semi-positive definite, the stability criterion is also efficient. Based on the proposed results, the systems with uncertain disturbances such as structured independent and structured dependent perturbations are considered. Using the matrix measure and the singular value theory, the bounds of the uncertainties are obtained that guarantee the system uniformly asymptotically stable, while the bounds of state feedback control input are also derived to stabilize the second-order linear time-varying systems. Finally, several numerical examples are given to prove the validity and correctness of the proposed criteria with existing ones.

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The stability of heavy nuclei and the limit of the periodic system of elements

Uspekhi Fizicheskih Nauk ◽

10.3367/ufnr.0100.197001b.0045 ◽

1970 ◽

Vol 100 (1) ◽

pp. 45-92 ◽

Cited By ~ 15

Author(s):

G.N. Flerov ◽

V.A. Druin ◽

A.A. Pleve

Keyword(s):

Periodic System ◽

Heavy Nuclei ◽

The Stability

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Stability of radical anions derived from substituted 5-nitrofurans investigated by ESR spectroscopy

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19851594 ◽

1985 ◽

Vol 50 (7) ◽

pp. 1594-1601 ◽

Cited By ~ 5

Author(s):

Jiří Klíma ◽

Larisa Baumane ◽

Janis Stradinš ◽

Jiří Volke ◽

Romualds Gavars

Keyword(s):

Radical Anion ◽

High Stability ◽

Esr Spectroscopy ◽

Reaction Path ◽

Order Reaction ◽

Second Order ◽

Radical Anions ◽

Second Order Reaction ◽

Unpaired Electron Density ◽

The Stability

It has been found that the decay in dimethylformamide and dimethylformamide-water mixtures of radical anions in five of the investigated 5-nitrofurans is governed by a second-order reaction. Only the decay of the radical anion generated from 5-nitro-2-furfural III may be described by an equation including parallel first- and second-order reactions; this behaviour is evidently caused by the relatively high stability of the corresponding dianion, this being an intermediate in the reaction path. The presence of a larger conjugated system in the substituent in position 2 results in a decrease of the unpaired electron density in the nitro group and, consequently, an increase in the stability of the corresponding radical anions.

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