Jensen不等式的收敛性与等价性结果——在时滞和采样数据系统中的应用
Convergence and Equivalence results for the Jensen's inequality - Application to time-delay and sampled-data systems
AI总结：
本文利用Grüss不等式研究Jensen不等式的保守性，证明均匀分割足够大时Jensen间隙可任意小，并分析非均匀分割加速收敛的情况，最后刻画一族边界并与文献中其他边界比较，证明其等价性及良好的适定性和线性性质。
摘要：
Jensen不等式在时滞和采样数据系统的分析中起着关键作用。通过使用Grüss不等式研究其保守性。文献中报道，分割（或分区）方案能够经验性地改善结果。我们在此证明，只要均匀分割的阶数足够大，Jensen间隙可以任意小。非均匀分割方案也被证明在某些情况下能加速收敛。最后，刻画了一族边界，并与文献中的其他边界进行了比较。结果表明，其他边界与Jensen边界等价，并且它们具有有趣的适定性和线性性质，这些性质可用于获得更好的数值结果。
英文摘要：
The Jensen's inequality plays a crucial role in the analysis of time-delay and sampled-data systems. Its conservatism is studied through the use of the Grüss Inequality. It has been reported in the literature that fragmentation (or partitioning) schemes allow to empirically improve the results. We prove here that the Jensen's gap can be made arbitrarily small provided that the order of uniform fragmentation is chosen sufficiently large. Non-uniform fragmentation schemes are also shown to speed up the convergence in certain cases. Finally, a family of bounds is characterized and a comparison with other bounds of the literature is provided. It is shown that the other bounds are equivalent to Jensen's and that they exhibit interesting well-posedness and linearity properties which can be exploited to obtain better numerical results.
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