Title: Regional Gradient Observability for Fractional Differential Equations with Caputo Time-Fractional Derivatives Show Chinese title

Title: 区域梯度可观测性对于具有Caputo时间分数阶导数的分数阶微分方程

Abstract: We investigate the regional gradient observability of fractional sub-diffusion equations involving the Caputo derivative. The problem consists of describing a method to find and recover the initial gradient vector in the desired region, which is contained in the spacial domain. After giving necessary notions and definitions, we prove some useful characterizations for exact and approximate regional gradient observability. An example of a fractional system that is not (globally) gradient observable but it is regionally gradient observable is given, showing the importance of regional analysis. Our characterization of the notion of regional gradient observability is given for two types of strategic sensors. The recovery of the initial gradient is carried out using an expansion of the Hilbert Uniqueness Method. Two illustrative examples are given to show the application of the developed approach. The numerical simulations confirm that the proposed algorithm is effective in terms of the reconstruction error. Show Chinese abstract

Abstract: 我们研究涉及Caputo导数的分数次扩散方程的区域梯度可观测性。 该问题包括描述一种方法，以在包含在空间域中的目标区域内找到和恢复初始梯度向量。 在给出必要的概念和定义后，我们证明了精确和近似区域梯度可观测性的某些有用特征。 给出了一个分数系统示例，该系统不是（全局）梯度可观测的，但它却是区域梯度可观测的，这表明了区域分析的重要性。 我们为两种类型的策略传感器提供了区域梯度可观测性概念的特征。 初始梯度的恢复是通过Hilbert唯一性方法的展开来进行的。 给出了两个说明性例子，以展示所开发方法的应用。 数值模拟证实了所提出的算法在重建误差方面是有效的。