非线性系统时滞输入状态稳定性：无穷导数Lyapunov-Krasovskii方法

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"该资源是一篇研究论文，标题为'Indefinite derivative Lyapunov-Krasovskii functional method for input to state stability of nonlinear systems with time-delay'，主要探讨了非线性系统中具有时滞的输入到状态稳定性问题。作者通过引入不定导数的Lyapunov-Krasovskii泛函方法来研究这一主题。另一篇提及的论文是'Liu, Tan, and Yang关于随机函数种群动力学的研究，涉及带有跳跃的随机函数Kolmogorov型模型，讨论了全局解的存在性和渐近路径估计。" 这篇论文聚焦于非线性系统的输入到状态稳定性（Input-to-State Stability, ISS）问题，特别是在存在时间延迟的情况下。输入到状态稳定性是控制理论中的一个重要概念，它衡量了系统在受到外部输入和内部扰动时的稳定性。时间延迟通常会增加系统的复杂性，并可能导致系统不稳定。Lyapunov-Krasovskii泛函方法是一种常用的技术，用于证明系统稳定性或设计控制器，它基于Lyapunov函数的构造，以分析系统的动态行为。论文中提到的“不定导数”可能指的是在处理具有不确定性的系统时，导数的定义可能不是常规的，这可能涉及到随机过程或者模糊系统的分析。不定导数Lyapunov-Krasovskii泛函方法可能是一种扩展的传统Lyapunov方法，以适应这种不确定性，从而提供对系统稳定性的更全面理解。此外，摘要中还提及了另一篇论文，由Liu, Tan和Yang合作完成，研究的是带有跳跃的随机函数种群动力学。他们利用一种特殊的随机Kolmogorov型模型来描述种群动态的演化，这种模型考虑了随机事件（跳跃）的影响。通过构建特定的Lyapunov函数，他们证明了与该模型相关的随机函数微分方程在正半轴上有唯一全球解。他们还利用带有跳跃的指数鞅不等式来讨论模型的渐近路径估计，这是分析系统长期行为的重要工具。这两篇论文共同展示了Lyapunov方法在处理复杂动态系统，尤其是那些包含随机性和时滞因素的系统时的灵活性和有效性。这些研究对于理解和控制非线性系统，尤其是在生物、工程和经济等领域中具有广泛的应用价值。

725 Tan, Hou, Yang

Y of [0, ∞) such that



(1 ∧ u

)λ(du) < ∞,

N(dt, du) := N(dt, du) − λ(du)dt,

F : D([−τ, 0]; R

) → R

and H : D([−τ, 0]; R

) × Y → R

For population dynamics with jumps, we refer to Bao et al. [3] and Bao and

Yuan [4], where both F and H are independent of the past history, and, in

particular, are linear in [3].

In reference to the existing results in the literature, our contributions are as

follows:

• We use stochastic functional Kolmogorov-type model with jumps to

describe the evolutions of population dynamics, which suﬀer sudden

environmental shocks;

• We show by the classical Lyapunov function argument that stochas-

tic functional diﬀerential equation associated with our model admits a

unique global solution in the positive orthant. In particular, our estab-

lished theory also demonstrates that jump processes can also be applied

to suppress solution explosion for functional diﬀerential equations.

• We verify by the exponential martingale inequality with jumps that

population size is at most of polynomial growth almost surely.

2. Global positive solution

Since the vector X = (X

, X

, · · · , X

)

denotes the population sizes of the

n interacting species, it is natural to require the solution of (1.2) not only to

be positive but also not to explode in ﬁnite time. Therefore, in this section we

intend to claim that (1.2) has a unique global solution in the positive orthant.

Throughout the paper, we impose the following assumptions.

(H1) There exists L

> 0 such that

|F (φ) − F(ϕ)|



|H(φ, u) − H(ϕ, u)|

λ(du) ≤ L

∥φ − ϕ∥

∞

where φ, ϕ ∈ D([−τ, 0]; R

) with ∥φ∥

∞

∨ ∥ϕ∥

∞

≤ k.

(H2) H

(φ, u) > −1 for φ ∈ D([−τ, 0]; R

), u ∈ Y and i = 1, · · · , n.

Since the coeﬃcients don’t satisfy linear growth condition or weak coercivity

condition, e.g., [14, Theorem 2.3], though they satisfy local Lipschitz condition,

the solutions of (1.2) may explode in ﬁnite time. Khasminskii [5, Theorem

4.1, p. 85] gave a Lyapunov function argument, which is a powerful test for

nonexplosion of solutions and generalized to delay SDEs in, e.g., [7, 8] . In the

sequel, we shall follow similar arguments to those of [5, 7, 8] to show that (1.2)

has a unique global positive solution {X(t)}

t≥0

under appropriate conditions.

To this end, we further assume that, for some p ∈ (0, 1) and any φ ∈

D([−τ, 0]; R

), there exist α

, α

, β

, γ

, δ

> 0, j = 1, · · · , 5, where

≥ α

, β

> β

, γ

≥ γ

and β

> max{p, α

, γ

}, and probability mea-

sures ρ

, ρ

on [−τ, 0] such that

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非线性系统时滞输入状态稳定性：无穷导数Lyapunov-Krasovskii方法

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