可解李代数上一类差分方程的全局稳定性
Global Stability of a Class of Difference Equations on Solvable Lie Algebras
摘要：
受应用控制中普遍存在的采样数据设置的启发，我们研究了一类差分方程的稳定性，该方程是通过对矩阵李群上的右不变流或左不变流进行采样而产生的。定义这样一个差分方程的映射有三个关键性质，便于我们的分析：1）它的幂级数展开具有一种强收敛性; 2）原点是一个平衡点; 3）李代数的下中心序列中枚举的代数理想是动态不变的。我们表明，某些全局稳定性属性所隐含的稳定性的雅可比线性化的动力学在原点。特别是全局渐近稳定性。如果李代数是幂零的，则原点具有半幂指数稳定性。
英文摘要：
Motivated by the ubiquitous sampled-data setup in applied control, we examine the stability of a class of difference equations that arises by sampling a right- or left-invariant flow on a matrix Lie group. The map defining such a difference equation has three key properties that facilitate our analysis: 1) its power series expansion enjoys a type of strong convergence; 2) the origin is an equilibrium; 3) the algebraic ideals enumerated in the lower central series of the Lie algebra are dynamically invariant. We show that certain global stability properties are implied by stability of the Jacobian linearization of dynamics at the origin. In particular global asymptotic stability. If the Lie algebra is nilpotent, then the origin enjoys semiglobal exponential stability.
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