In this work we deal with the regional observability problem, the purpose here is to reconstruct the initial state for a class of linear time-fractional systems, in a subregion $\omega$ of the evolution domain $\Omega$, using an extension of Hilbert Uniqueness Method (HUM) introduced by Lions. This approach allows us to transform the regional reconstruction problem into a solvability one, which gives the initial state in $\omega$ as a solution of well chosen mapping and leads to an algorithm that enables us to reconstruct it. The efficiency of the proposed algorithm is illustrated with some successful numerical simulations that we give in the last section.

中文翻译：

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线性时间分数系统的区域可观测性

在这项工作中，我们处理了区域可观测性问题，此处的目的是为子区域中的一类线性时间分数系统重建初始状态 $\omega$ 演化域 $\Omega$，使用Lions引入的Hilbert唯一性方法（HUM）的扩展。这种方法使我们能够将区域重建问题转化为可解性，从而获得初始状态。$\omega$作为精心选择的映射的解决方案，并得出了使我们能够对其进行重构的算法。我们在最后一部分给出了一些成功的数值模拟，说明了所提算法的效率。