We investigate non-Hermitian elastic lattices characterized by non-local feedback control interactions. In one-dimensional lattices, we show that the proportional control interactions produce complex dispersion relations characterized by gain and loss in opposite propagation directions. Depending on the non-local nature of the control interactions, the resulting non-reciprocity occurs in multiple frequency bands characterized by opposite non-reciprocal behavior. The dispersion topology is also investigated with focus on winding numbers and non-Hermitian skin effect, which manifests itself through bulk modes localized at the boundaries of finite lattices. In two-dimensional lattices, non-reciprocity is associated with directional dependent wave amplification. Moreover, the non-Hermitian skin effect manifests as modes localized at the boundaries of finite lattice strips, whose combined effect in two directions leads to the presence of bulk modes localized at the corners of finite two-dimensional lattices. Our results describe fundamental properties of non-Hermitian elastic lattices, and open new possibilities for the design of metamaterials with novel functionalities related to selective wave filtering, amplification and localization. The results also suggest that feedback interactions may be a useful strategy to investigate topological phases of non-Hermitian systems.

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具有非局部反馈控制相互作用的非厄米弹性晶格的动力学和拓扑结构

我们研究了以非局部反馈控制相互作用为特征的非厄米弹性晶格。在一维格子中，我们表明比例控制相互作用产生复杂的色散关系，其特征在于相反传播方向上的增益和损耗。根据控制相互作用的非局部性质，由此产生的非互易性出现在以相反的非互易行为为特征的多个频带中。还研究了色散拓扑，重点是绕组数和非厄米集肤效应，这通过位于有限晶格边界的体模式表现出来。在二维晶格中，非互易性与方向相关波放大有关。而且，非厄米表皮效应表现为模式位于有限晶格条的边界处，其两个方向的组合效应导致存在位于有限二维晶格边角处的体模式。我们的结果描述了非厄米弹性晶格的基本特性，并为设计具有与选择性滤波、放大和定位相关的新功能的超材料开辟了新的可能性。结果还表明，反馈相互作用可能是研究非厄米系统拓扑相的有用策略。并为设计具有与选择性滤波、放大和定位相关的新功能的超材料开辟了新的可能性。结果还表明，反馈相互作用可能是研究非厄米系统拓扑相的有用策略。并为设计具有与选择性滤波、放大和定位相关的新功能的超材料开辟了新的可能性。结果还表明，反馈相互作用可能是研究非厄米系统拓扑相的有用策略。