Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations

Самойленко, А. М.; Stanzhytskyi, Oleksandr

doi:10.1142/9789814329071

Cited by 15 publications

(17 citation statements)

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“…It should be mentioned that the conditions given in [28] imply that the existence of a solution requires that system (2) should satisfy the global Lipschitz condition (Theorem 1.1 in [28]). However, the global Lipschitz condition generally fails to hold in many applications.…”

Section: Remarkmentioning

confidence: 99%

“…We will not list here all the possible stability definitions; instead, we will confine ourselves to those that are in our view of important practical interest. One can refer to recent works in [24,27,28,36], where a series of stability concepts were introduced.…”

Section: Remarkmentioning

confidence: 99%

“…Nevertheless, the aforementioned remarkable work has not yet taken into account other important factors, for example, impulsive effects. To the best of the authors' knowledge, the stability problem for random impulsive systems has been addressed only in [28] so far. But the work in [28] imposed limitations on the impulse instants and the truncated deterministic impulsive subsystem.…”

Section: Introductionmentioning

confidence: 99%

“…To the best of the authors' knowledge, the stability problem for random impulsive systems has been addressed only in [28] so far. But the work in [28] imposed limitations on the impulse instants and the truncated deterministic impulsive subsystem. Hence, it is highly desirable to remove or relax those limitations so as to develop some less conservative results for random impulsive systems.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Stability analysis for a class of random nonlinear impulsive systems

Jiao

Zheng

2016

Int. J. Robust. Nonlinear Control

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In this paper, the problem of noise-to-state stability (NSS) and globally asymptotic stability (GAS) is investigated for a class of nonlinear systems with random disturbances and impulses, where the random noises have finite second-order moments and the so-called random impulses mean that impulse ranges are driven by a sequence of random variables. First, some general conditions are given to guarantee the existence and uniqueness of solutions to random nonlinear impulsive systems. Next, when the continuous dynamics are stable but the impulses are destabilizing, the NSS and GAS of random nonlinear impulsive systems are examined by the average impulsive interval approach. Then, when the continuous dynamics are unstable but the impulses are stabilizing, it is shown that the NSS and GAS can be retained by using the reverse average impulsive interval approach. Finally, the theoretical findings are substantiated with illustrative examples. STABILITY ANALYSIS OF RANDOM IMPULSIVE SYSTEMS P .x.t/ D Q x.t/; 8t 2 OEt 0 ; T / D 1;then system (2) is said to have a unique solution. Furthermore, if for any T > t 0 , system (2) has a unique solution on the finite interval OEt 0 ; T , then it is said to have a globally unique solution.Throughout this paper, the random processes and the nonlinear functions in system (2) are supposed to satisfy the following assumptions.V .x.t kC1 /; t kC1 / 6 V x t kC1 ; t kC1 C d 2 jÁ kC1 j 2 ;then system (2) has a globally unique solution, and it is exponentially NSS-m-M for any impulsive sequence ¹t kC1 º. Now, we turn to considering the NSS-P for random impulsive system (2).

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Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability analysis for a class of random nonlinear impulsive systems

Jiao

Zheng

2016

Int. J. Robust. Nonlinear Control

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“…Recently, Wu 16 has proposed a new stability analysis framework for a class of random nonlinear systems modeled by random differential equations and reduced the conservatism compared with the early stability results of random nonlinear systems. 9,[17][18][19] Following the line of Wu, 16 a series of results about the stabilization and tracking of random nonlinear systems is obtained. [20][21][22][23] The above discussions motivate us to focus on random Markovian switching systems.…”

Section: Introductionmentioning

confidence: 99%

Adaptive tracking control for a class of random pure‐feedback nonlinear systems with Markovian switching

Yao

Zhang

2018

Intl J Robust & Nonlinear

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This note studies the tracking control problem for a class of random pure-feedback nonlinear systems with Markovian switching and unknown parameters. An adaptive tracking controller is constructed by introducing an auxiliary integrator subsystem and using the improved backstepping method such that the closed-loop system has a unique solution that is globally bounded in probability. Meanwhile, the tracking error can converge to an arbitrarily small neighborhood of zero via the parameter regulation technique. The efficiency of the tracking controller designed in this paper is demonstrated by simulation examples. KEYWORDSbackstepping, Markovian switching, random pure-feedback systems, tracking control 3112

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New noise‐to‐state stability and instability criteria for random nonlinear systems

Yao

Zhang

2019

Intl J Robust & Nonlinear

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Summary This paper investigates the noise‐to‐state stability and instability criteria for random nonlinear affine systems. Firstly, some new noise‐to‐state stability theorems, which weaken the sufficient conditions in the existing stability criteria on random nonlinear systems, are given by means of the uniformly asymptotically stable function. Secondly, the noise‐to‐state instability definitions are introduced and the sufficient conditions of noise‐to‐state instability are provided based on a new established lemma and the uniformly asymptotically stable function. Finally, some examples show the feasibility of theoretical findings.

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Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations

Cited by 15 publications

References 0 publications

Stability analysis for a class of random nonlinear impulsive systems

Stability analysis for a class of random nonlinear impulsive systems

Adaptive tracking control for a class of random pure‐feedback nonlinear systems with Markovian switching

New noise‐to‐state stability and instability criteria for random nonlinear systems

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