asymptotic stability

简明释义

渐近稳定度

英英释义

例句

1.Engineers often design feedback systems to ensure asymptotic stability in their applications.

工程师们常常设计反馈系统，以确保其应用中的渐近稳定性。

2.In control theory, a system is said to have asymptotic stability when its state approaches a steady state over time.

在控制理论中，当一个系统的状态随着时间的推移接近稳态时，它被称为具有渐近稳定性。

3.The research focused on the asymptotic stability of nonlinear systems in dynamic environments.

该研究集中于动态环境中非线性系统的渐近稳定性。

4.A Lyapunov function can be used to demonstrate the asymptotic stability of a given system.

可以使用李雅普诺夫函数来证明给定系统的渐近稳定性。

5.The mathematical model showed that the equilibrium point had asymptotic stability under certain conditions.

数学模型显示，在某些条件下，平衡点具有渐近稳定性。

作文

In the field of mathematics and control theory, the concept of asymptotic stability is crucial for understanding the behavior of dynamical systems. A system is said to be asymptotically stable if, after a disturbance, it returns to its equilibrium state as time approaches infinity. This property is essential in various applications, including engineering, economics, and ecological models. To illustrate this concept, consider a simple pendulum. When the pendulum is displaced from its vertical resting position and then released, it will oscillate back and forth before eventually coming to rest at the lowest point. This behavior demonstrates asymptotic stability because the pendulum returns to its equilibrium position over time.Mathematically, asymptotic stability can be analyzed using differential equations. For instance, if we have a linear system described by the equation dx/dt = Ax, where A is a matrix representing the system dynamics, the eigenvalues of A play a significant role in determining stability. If all eigenvalues have negative real parts, the system is considered asymptotically stable. This means that any small perturbation from the equilibrium point will decay exponentially, leading the system back to stability.Understanding asymptotic stability is not only important for theoretical purposes but also has practical implications. In engineering, for example, designing a control system requires ensuring that the system behaves predictably under various conditions. If a control system is asymptotically stable, it implies that the system can handle disturbances and still return to a desired state without oscillating indefinitely or diverging.In the context of ecological models, asymptotic stability can help predict the long-term behavior of populations. For instance, in a predator-prey model, if the population of prey is disturbed by an increase in predators, asymptotic stability indicates that the populations will eventually settle into a stable cycle. This insight is valuable for conservation efforts and managing wildlife resources effectively.Moreover, asymptotic stability can also be observed in economic models, particularly those involving market equilibrium. When an economy faces shocks, such as changes in consumer preferences or supply chain disruptions, asymptotic stability suggests that the market will eventually adjust and return to a state of balance. This understanding helps policymakers formulate strategies to mitigate the effects of such disturbances.In summary, the concept of asymptotic stability is fundamental in various scientific fields, providing insights into how systems respond to perturbations. Whether in physics, biology, or economics, recognizing and analyzing asymptotic stability allows researchers and practitioners to design more robust systems and predict their long-term behavior. As we continue to explore complex systems and their dynamics, the importance of asymptotic stability will only grow, highlighting the interconnectedness of various disciplines and the need for a comprehensive understanding of stability in our world.

在数学和控制理论领域，渐近稳定性的概念对于理解动态系统的行为至关重要。当一个系统在受到扰动后，随着时间趋于无穷大而返回到其平衡状态时，该系统被称为具有渐近稳定性。这一特性在工程、经济学和生态模型等多个应用中都是必不可少的。为了说明这个概念，我们可以考虑一个简单的摆。当摆从其垂直静止位置偏离并释放时，它会来回摆动，最终在最低点停止。这种行为展示了渐近稳定性，因为摆随时间返回到其平衡位置。在数学上，渐近稳定性可以通过微分方程进行分析。例如，如果我们有一个由方程dx/dt = Ax描述的线性系统，其中A是表示系统动态的矩阵，则A的特征值在确定稳定性方面起着重要作用。如果所有特征值的实部均为负，则该系统被认为是渐近稳定的。这意味着任何小的扰动都会以指数方式衰减，导致系统重新回到稳定状态。理解渐近稳定性不仅对理论目的重要，而且具有实际意义。在工程中，设计控制系统时需要确保系统在各种条件下表现可预测。如果一个控制系统是渐近稳定的，这意味着该系统能够处理扰动，并且仍然返回到所需状态，而不会无限振荡或发散。在生态模型的背景下，渐近稳定性可以帮助预测种群的长期行为。例如，在捕食者-猎物模型中，如果猎物的种群因捕食者的增加而受到干扰，渐近稳定性表明这些种群最终将稳定在一个循环中。这一见解对于保护工作和有效管理野生动物资源极为重要。此外，渐近稳定性还可以在经济模型中观察到，特别是在市场均衡的情况下。当经济面临冲击时，例如消费者偏好变化或供应链中断，渐近稳定性表明市场最终将调整并恢复到平衡状态。这一理解帮助政策制定者制定策略，以减轻此类扰动的影响。总之，渐近稳定性的概念在多个科学领域中是基础性的，为我们提供了关于系统如何对扰动作出反应的深刻见解。无论是在物理学、生物学还是经济学中，识别和分析渐近稳定性使研究人员和从业者能够设计出更稳健的系统，并预测其长期行为。随着我们继续探索复杂系统及其动态，渐近稳定性的重要性只会增加，突显出各学科之间的相互关联性以及对我们世界中稳定性全面理解的必要性。