DEFINITENESS OF QUADRATIC FUNCTIONALS
DEFINITENESS OF QUADRATIC FUNCTIONALS
复制标题
二次函数的定性
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
W. Kratz
中科院分区：
文献类型：
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作者：
W. Kratz
摘要：本文讨论二次泛函，它是变分法中的二次变分。我们考虑一般的线性齐次边界条件。这种泛函的确定性导致了一个线性哈密顿微分系统，由运动方程和欧拉方程组成。本文的主要新方面是我们不要求系统的可控性或正态性。这使得有必要引入所谓微分系统的联合基的“广义”焦点的新概念。基于这个概念我们推导出二次函数定性的充要条件。特别是，非负性的结果对于可控系统来说甚至是新的。除了这些非负定性和正定性的特征之外，另一个主要结果在于证明广义焦点总是孤立的。
Abstract: This paper deals with quadratic functionals, which occur as second variations in the calculus of variations. We consider general, linear homogeneous boundary conditions. The definiteness of such functionals leads to a linear Hamiltonian differential system, consisting of the equation of motion and the Euler equation. The main new aspect in this paper is the fact that we do not require controllability or normality of the system. This makes it necessary to introduce a new notion of "generalized" focal points of so-called conjoined bases of the differential system. Based on this concept we derive necessary and sufficient conditions for the definiteness of quadratic functionals. Particularly, the result on nonnegativity is even new for controllable systems. Besides these characterizations of nonnegative and positive definiteness another main result consists in the proof that the generalized focal points are always isolated.