Conditions for local almost sure asymptotic stability

Nishimura, Yûki

doi:10.1016/j.sysconle.2016.05.001

Cited by 7 publications

(8 citation statements)

References 22 publications

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“…Here "almost surely" is exchangeable with "with probability one", and we sometimes use the shorthand notation "a.s.". We now introduce some stability concepts for stochastic discretetime systems analogous to those in [5] and [34] for continuoustime systems 1 .…”

Section: Finite-step Stochastic Lyapunov Functionsmentioning

confidence: 99%

Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation

Qin

Cao

Anderson

2020

IEEE Trans. Automat. Contr.

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This paper presents new sufficient conditions for convergence and asymptotic or exponential stability of a stochastic discrete-time system, under which the constructed Lyapunov function always decreases in expectation along the system's solutions after a finite number of steps, but without necessarily strict decrease at every step, in contrast to the classical stochastic Lyapunov theory. As the first application of this new Lyapunov criterion, we look at the product of any random sequence of stochastic matrices, including those with zero diagonal entries, and obtain sufficient conditions to ensure the product almost surely converges to a matrix with identical rows; we also show that the rate of convergence can be exponential under additional conditions. As the second application, we study a distributed network algorithm for solving linear algebraic equations. We relax existing conditions on the network structures, while still guaranteeing the equations are solved asymptotically.

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Section: Finite-step Stochastic Lyapunov Functionsmentioning

confidence: 99%

Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation

Qin

Cao

Anderson

2020

IEEE Trans. Automat. Contr.

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“…Proof. Condition (12) implies that there exists a sufficiently small positive scalar γ such thatΞ ijℓ (γ) < 0, whereΞ ijℓ (γ) is derived fromΞ ijℓ , in which 0.5c 2 + ln(µ) σ ℓ is replaced by γ + 0.5c 2 + ln(µ) σ ℓ . It follows thatΞ…”

Section: Sampling-time-dependent Discretized Lyapunov Function Approachmentioning

confidence: 99%

“…. , N , ℓ = 0, 1, a matrixK ∈ R n b ×nc , and a scalar c such that (12) and the following LMI holds:…”

Section: Controller Designmentioning

confidence: 99%

“…Compared with [7,10], Huang [11] proposed improved results on the stochastic stabilization and destabilization of nonlinear differential equations. Nishimura [12] utilized Gaussian white noise to control dynamical systems to meet the locally almost sure asymptotic stability. Based on the stochastic control Lyapunov function, a stabilizing controller together with the Wiener process for deterministic nonlinear systems was designed in [4].…”

Section: Introductionmentioning

confidence: 99%

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Stochastic stabilization using aperiodically sampled measurements

Luo

Deng

Zhao

et al. 2019

Sci. China Inf. Sci.

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This paper addresses the stabilization problem of sector-bounded nonlinear systems with sampled measurements via discrete-time stochastic feedback control. Unlike the previous studies, the closed-loop system is modeled as an impulsive stochastic differential equation. By developing a quasi-periodic polynomial Lyapunov function and sampling-time-dependent Lyapunov function based methods, two sufficient conditions for almost sure exponential stability are derived in terms of differential matrix inequalities (DMIs) and linear matrix inequalities (LMIs). It is shown that the DMI-based conditions can be formulated as a sum of squares (SOSs). Moreover, the obtained results are adapted to sampled-data stochastic/deterministic systems. The numerical examples illustrate the theoretical results.

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“…In the existing literature, there exist many stability concepts of stochastic dynamical systems, such as mean square stability, almost sure stability, stability in probability, and asymptotic stability in probability . Interesting results on this topic can also be found in . It is obvious that the stability concepts of stochastic dynamical systems are more complicated than those of deterministic dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic stability in probability of singular stochastic systems with Markovian switchings

Ding

Zhong

Long

2017

Intl J Robust & Nonlinear

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Summary This paper investigates the problem of asymptotic stability in probability for singular stochastic systems with Markovian switchings. A stochastic Lyapunov theorem on asymptotic stability in probability for the considered systems is provided. Also, we show that the original system has the same stability property as its difference‐algebraic form based on singular value decomposition. By utilizing the earlier results, a sufficient condition is obtained in terms of linear matrix inequalities, which is easy to check by using standard software. Copyright © 2017 John Wiley & Sons, Ltd.

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Conditions for local almost sure asymptotic stability

Cited by 7 publications

References 22 publications

Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation

Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation

Stochastic stabilization using aperiodically sampled measurements

Asymptotic stability in probability of singular stochastic systems with Markovian switchings

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