Abstract In this paper we consider two linear Hamiltonian differential systems on an open unbounded interval. We assume that the systems are completely controllable and satisfy the Sturmian majorant condition and the Legendre condition. We derive a singular Sturmian comparison theorem for the case when the minorant system has a solution, which is principal at both endpoints of the considered interval. The main result is new even for the second order differential equations and it generalizes the singular comparison theorem obtained by Aharonov and Elias (2010). We illustrate our new theory by several examples.

中文翻译：

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奇异情况下完全可控线性哈密顿系统的Sturmian比较定理

摘要 在本文中，我们考虑了开放无界区间上的两个线性哈密顿微分系统。我们假设系统是完全可控的并且满足 Sturmian Majorant 条件和 Legendre 条件。对于次要系统有一个解的情况，我们推导出奇异的 Sturmian 比较定理，该解在所考虑的区间的两个端点上都是主要的。即使对于二阶微分方程，主要结果也是新的，它概括了 Aharonov 和 Elias (2010) 获得的奇异比较定理。我们通过几个例子来说明我们的新理论。