Solution to the Common Lyapunov-Function Problem via Fusion of Numerical and thalytical Aspec

1.我们首先测试了几种可用于求解LMI的方法，其中常见的Lyapunov函数问题可以被重塑为LMI，并找到了最适合当前目的的所谓内点法。基于这种方法，我们自行开发了计算程序，并在整个研究过程中使用了它。根据上述计算过程中所获得的知识或经验，我们成功地找到了一些具有二次Lyapunov函数的连续时间或离散时间系统的子类。还发现了一些低阶系统的子类。总的来说，我们注意到，在连续时间和离散时间的情况下，存在着一个平行的情况。这将使我们对这个问题有进一步的了解。由于该问题与系统的二次稳定性问题有着密切的联系，我们研究了区间多项式的二次稳定性问题，得到了一些结果。

1. We first tested several methods available to solve LMI,into which the common Lyapunov function problem can be recast, and found the so-called interior-point method most suitable for the present purpose. Based on this method, we developed a computation code by ourselves and used it throughout the study.2. With knowledge or experience attained in the above computation process, we are successful in finding some subclasses of systems, continuous-time or discrete-time, that have a quadratic Lyapunov function in common. Some subclasses for low-order systems have also been found. In general, we noticed, there exists a parallel between continuous-time case and discrete-time one. This would give a further insight into the problem.3. Since there exists a close connection between the problem in question and the quadratic stability problem of systems, we looked into the quadratic stability property of interval polynomials and obtained some results.