随机切换常微分方程对切换率的敏感性
Sensitivity to switching rates in stochastically switched ODEs
摘要：
我们考虑由线性常微分方程驱动的随机过程，其右侧在不同矩阵的集合之间以指数时间切换。我们构建平面的例子，在两个矩阵之间切换，其中单个矩阵和两个矩阵的平均值都是Hurwitz（所有特征值都有严格的负实部），但尽管如此，该过程在大的时间去无穷大的切换率的某些值。我们进一步构建更高维度的例子，其中两个单独的矩阵和它们的平均值都是Hurwitz，但是随着切换速率的变化，该过程在大时间内有任意多个从零到无穷大的过渡。为了构造这些例子，我们首先证明，如果每个单独的矩阵是赫维茨，那么这个过程在足够慢的切换速率大的时间为零，如果平均矩阵是赫维茨，那么这个过程在足够快的切换速率大的时间为零。我们还给出了简单的条件，确保该过程在大的时间为零的所有切换率。
英文摘要：
We consider a stochastic process driven by a linear ordinary differential equation whose right-hand side switches at exponential times between a collection of different matrices. We construct planar examples that switch between two matrices where the individual matrices and the average of the two matrices are all Hurwitz (all eigenvalues have strictly negative real part), but nonetheless the process goes to infinity at large time for certain values of the switching rate. We further construct examples in higher dimensions where again the two individual matrices and their averages are all Hurwitz, but the process has arbitrarily many transitions between going to zero and going to infinity at large time as the switching rate varies. In order to construct these examples, we first prove in general that if each of the individual matrices is Hurwitz, then the process goes to zero at large time for sufficiently slow switching rate and if the average matrix is Hurwitz, then the process goes to zero at large time for sufficiently fast switching rate. We also give simple conditions that ensure the process goes to zero at large time for all switching rates.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
