非自治线性哈密顿系统的非Atkinson扰动：指数二分性和非振动性
Non-Atkinson perturbations of nonautonomous linear hamiltonian systems: exponential dichotomy and nonoscillation
摘要：
本文分析了定义在紧度量空间轨道上的有限维非自治线性哈密顿系统的单参数族的指数二分性和Weyl函数$M^\pm$的整体存在性，这些系统从初始值向不满足经典Atkinson条件的方向扰动：或者对任何参数值都不具有指数二分性;或者至少对所有的非实值都有，在这种情况下，外尔函数存在，并且是Herglotz。当参数在实直线上变化时，如果未扰动族满足指数二分性和$M^+$的整体存在性，则这两个性质在0的邻域中持续存在，该邻域要么与整条实直线一致，要么与开负半直线一致;在后一种情况下，ED在右端值处失败。ED的性质和$M^+$的整体存在性是保证线性二次控制过程的经典极小化问题的可解性的基础。
英文摘要：
We analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions $M^\pm$ for one-parametric families of finite-dimensional nonautonomous linear Hamiltonian systems defined along the orbits of a compact metric space, which are perturbed from an initial one in a direction which does not satisfy the classical Atkinson condition: either they do not have ED for any value of the parameter; or they have it for at least all the nonreal values, in which case the Weyl functions exist and are Herglotz. When the parameter varies in the real line, and if the unperturbed family satisfies the properties of exponential dichotomy and global existence of $M^+$, then these two properties persist in a neighborhood of 0 which agrees either with the whole real line or with an open negative half-line; and in this last case, the ED fails at the right end value. The properties of ED and of global existence of $M^+$ are fundamental to guarantee the solvability of classical minimization problems given by linear-quadratic control processes.
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