Investigation op stability of solutions of some nonlinear systems of differential equations

Persidskii, S. K.

doi:10.1016/0021-8928(68)90045-2

Cited by 2 publications

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“…Then it is easy to see [2] that the corresponding cone K ( α 1 , … , α n ) for system (2) is a closed sector. Using this fact, one can easily obtain the following theorem on the exponential instability in a cone: Theorem 3.…”

Section: Theorem 2 Suppose That the Continuous Functions ϕ S ( X S )mentioning

confidence: 97%

“…To prove the sufficiency, we note that the condition of the quasipositivity of the matrix P implies that the corresponding cone K ( α 1 , … , α n ) is a closed sector [2]. We determine the numbers b 1 , … , b n from the system of equations By virtue of system (2) and conditions (9), in the cone under consideration we have…”

Section: Theorem 2 Suppose That the Continuous Functions ϕ S ( X S )mentioning

confidence: 98%

“…Using this fact, one can easily obtain the following theorem on the exponential instability in a cone: Theorem 3. Suppose that system (2) with quasipositive matrix P is such that the functions ϕ s ( x s ) satisfy conditions (4) and, in the corresponding cone K ( α 1 , … , α n ), the inequalities…”

Section: Theorem 2 Suppose That the Continuous Functions ϕ S ( X S )mentioning

confidence: 99%

“…Assume that K ( α 1 , … , α n ) : x s α s ≥ 0, s = 1, … , n, is a convex cone in the space R n and α 1 , … , α n are the parameters of the cone [2]. For example, the positive octant K N + of the space R n has unit parameters.…”

Section: On Exponential Stability Of Nonlinear Systems Of Differentiamentioning

confidence: 99%

“…are positive [2]. Multiplying the right-hand sides of system (6) by the corresponding parameters α 1 , … , α n of the cone K ( α 1 , … , α n ), we obtain the system …”

Section: On Exponential Stability Of Nonlinear Systems Of Differentiamentioning

confidence: 99%

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On the Exponential Stability of Some Nonlinear Systems

Persidskii

2005

Ukr Math J

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By using Lyapunov functions, we obtain, for the first time, necessary and sufficient conditions for the exponential stability of some nonlinear systems of differential and difference equations.In the present paper, we obtain necessary and sufficient conditions for the global exponential stability and exponential instability for one class of nonlinear systems of differential and difference equations. We investigate the exponential stability of perturbed systems with nonlinear first approximation.Note that the problem of the exponential stability of nonlinear systems is connected with the work by Grujic [1], in which sufficient conditions were obtained for the global exponential stability of one nonlinear complicated system with nonlinear isolated subsystems, which are assumed to be either exponentially stable or exponentially unstable.

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Section: Theorem 2 Suppose That the Continuous Functions ϕ S ( X S )mentioning

confidence: 97%

Section: Theorem 2 Suppose That the Continuous Functions ϕ S ( X S )mentioning

confidence: 98%

Section: Theorem 2 Suppose That the Continuous Functions ϕ S ( X S )mentioning

confidence: 99%

Section: On Exponential Stability Of Nonlinear Systems Of Differentiamentioning

confidence: 99%

“…are positive [2]. Multiplying the right-hand sides of system (6) by the corresponding parameters α 1 , … , α n of the cone K ( α 1 , … , α n ), we obtain the system …”

Section: On Exponential Stability Of Nonlinear Systems Of Differentiamentioning

confidence: 99%

See 3 more Smart Citations