论文标题
时间尺度的最佳二分法以及时变多维非线性系统的稳定性和稳定性
Optimal Dichotomy of Temporal Scales and Boundedness and Stability of Time-Varying Multidimensional Nonlinear Systems
论文作者
论文摘要
本文开发了一种新的方法来估计具有时间依赖性非周期系数的多维非线性系统的界限程度或稳定性 - 在各种工程和自然科学应用中的重要任务。评估此类系统稳定性的已知方法取决于Lyapunov函数的实用性和Lyapunov首先近似方法，通常为该类别的多维系统提供保守和计算上的详尽标准。在当代文献中很少有解决此类非均匀系统的界限的适当标准。最近，我们通过匹配[1]，[2]和[3]中介绍的标量辅助方程的解决方案来开发一种新的方法，该方法基于将解决方案规范界定到初始系统到初始系统的发展。尽管如此，[3]中先进的技术仍取决于以下假设：基础系统的线性组件的平均值是由稳定的一般位置矩阵定义的。当前的论文大大扩大了这种方法的应用域。仅假定，基础系统的时间依赖性线性块可以通过应用任何平滑技术将其分为慢速和快速变化的组件。时间尺度的这种二分法取决于最佳标准降低了我们估计的保守性。反过来，我们通过lyapunov变换的连续应用以对角度主导形式的慢变化矩阵转换线性子系统。这促使了新型标量辅助方程的发展，该方程涵盖了对我们初始系统的解决方案规范的估计。接下来，我们使用分析和删节的数值推理来制定有界性或稳定性标准，并估算基础系统的相关区域。
This paper develops a new approach to the estimation of the degree of boundedness or stability of multidimensional nonlinear systems with time-dependent nonperiodic coefficients-an essential task in various engineering and natural science applications. Known approaches to assessing the stability of such systems rest on the utility of Lyapunov functions and Lyapunov first approximation methodologies, typically providing conservative and computationally elaborate criteria for multidimensional systems of this category. Adequate criteria of boundedness of solutions to nonhomogeneous systems of this kind are rare in the contemporary literature. Lately, we develop a new approach to these problems which rests on bounding the evolution of the norms of solutions to initial systems by matching solutions of a scalar auxiliary equation we introduced in [1], [2] and [3]. Still, the technique advanced in [3] rests on the assumption that the average of the linear components of the underlying system is defined by a stable matrix of general position. The current paper substantially amplifies the application domain of this approach. It is merely assumed that the time-dependent linear block of the underlying system can be split into slow and fast varying components by application of any smoothing technique. This dichotomy of temporal scales is determined by the optimal criterion reducing the conservatism of our estimates. In turn, we transform the linear subsystem with slow-varying matrix in a diagonally dominant form by successive applications of the Lyapunov transforms. This prompts the development of novel scalar auxiliary equations embracing the estimation of the norms of solutions to our initial systems. Next, we formulate boundedness or stability criteria and estimate the relevant regions of the underlying systems using analytical and abridged numerical reasoning.
