In this paper we study the properties of conjoined bases of a general linear Hamiltonian system without any controllability condition. When the Legendre condition holds and the system is nonoscillatory, it is known from our previous work that conjoined bases with eventually the same image form a special structure called a genus. In this work we extend the theory of genera of conjoined bases to arbitrary systems, for which the Legendre condition is not assumed and/or the system may be oscillatory. We derive a classification of all genera of conjoined bases and show that they form a complete lattice. These results are based on the relationship between subspaces of solutions of a linear control system and orthogonal projectors satisfying a certain Riccati type differential equation. The presented theory is applied in our paper (Šepitka in Discrete Contin Dyn Syst 39(4):1685–1730, 2019) to general Riccati matrix differential equations for possibly uncontrollable linear Hamiltonian systems.

中文翻译：

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（非）振荡线性哈密顿系统的连基的属：扩展理论

在本文中，我们研究了没有任何可控性条件的一般线性哈密顿系统的联合基的性质。当Legendre条件成立且系统不振荡时，从我们以前的工作中可以知道，将碱基与最终具有相同图像的联合起来，会形成一个称为属的特殊结构。在这项工作中，我们将联合碱基的属理论扩展到任意系统，在这些系统中不假设勒让德条件和/或系统可能具有振荡性。我们推导了所有联合碱基的分类，并表明它们形成了一个完整的晶格。这些结果基于线性控制系统的解的子空间与满足某个Riccati型微分方程的正交投影仪之间的关系。