论文标题
一类希尔伯特空间上Sturm-Liouville系统的适应性和可观察性
Well-posedness and observability of Sturm-Liouville systems on a class of Hilbert spaces
论文作者
论文摘要
考虑了在有限间隔的正方形集成函数空间上的Sturm-Liouville运算符类。根据Riesz-spectral特性，这种无限的线性操作员在该空间上的自我相关性和积极性，即任何Sturm-liouville操作员的正（尤其是，尤其是分数）功率的域的希尔伯特空间。在这些空间上，证明任何Sturm-Liouville操作员都是Riesz-Spectral操作员，具有与原始特征值相同的特征值，与重新定制的特征函数相关。这构成了本文的第一个核心结果。还突出显示了与此类Riesz-Spectral Operator生成的C_0-序列相关的属性。另外，作为第二个中心结果，还为这些系统建立了通过点测量算子对近似可观察性的表征。主要的结果应用于扩散转化反应系统，以特别表明动力学操作员是在整数阶的某些Sobolev空间上紧凑的C_0-semigroup的无限发电机，并确定其可观察力。
The class of Sturm-Liouville operators on the space of square integrable functions on a finite interval is considered. According to the Riesz-spectral property, the self-adjointness and the positivity of such unbounded linear operators on that space, a class of Hilbert spaces constructed as the domains of the positive (in particular, fractional) powers of any Sturm-Liouville operator is considered. On these spaces, it is shown that any Sturm-Liouville operator is a Riesz-spectral operator that possesses the same eigenvalues as the original ones, associated to rescaled eigenfunctions. This constitutes the first central result of this paper. Properties related to the C_0-semigroup generated by the opposite of such Riesz-spectral operator are also highlighted. In addition as second central result, a characterization of approximate observability by means of point measurement operators is established for such systems. The main results are applied on a diffusion-convection-reaction system in order notably to show that the dynamics operator is the infinitesimal generator of a compact C_0-semigroup on some Sobolev space of integer order, and to establish its observability.
