## 1 Introduction  
Locally resonant metamaterials have revolutionized the field of material science by enabling the manipulation of mechanical waves through unique structural designs that are not possible with conventional materials. Such metamaterials utilize an intricate arrangement of embedded resonators to selectively amplify or attenuate waves, yielding capabilities like enhanced vibration isolation, targeted wave trapping, and precise steering. The cross-disciplinary value of these materials is evident in their wide-ranging applications, from improving acoustic insulation and energy harvesting to managing seismic waves and developing advanced sensors. The concept of metamaterials is not exclusive to structural dynamics; its origins can be traced back to research in optics by [PERSON] et al. [1], and it has since become a subject of extensive study in various fields, including acoustics. Moreover, the concept of mechanical locally resonant metamaterials was first introduced by [PERSON] et al. [2], who demonstrated an elastic locally resonant bandgap phenomenon, akin to a mass-spring oscillator. Since then, various types of mechanical locally resonant metamaterials have been extensively investigated in the literature. The field of resonator couplings and dispersion has seen substantial progress in recent years. For instance, [PERSON] et al. [3] innovated a superconducting architecture utilizing a ring resonator for multiqubit connectivity, enhancing the efficiency of quantum processors. In the realm of optics and spectroscopy, [PERSON] et al. [4] developed a novel experimental setup to measure the dispersion of organic exciton polaritons, revealing the quantized interactions between light and matter in organic materials. Additionally, [PERSON] et al. [5] demonstrated the coherent internally coupled distant magnonic resonators via superconducting circuits for integrated magnonic networks that can operate coherently at quantum-compatible scales. From optics and materials science to structural vibration and energy harvesting, these advancements bridge diverse fields to pioneer new applications and efficiencies. [PERSON] et al. [6] proposed a modified metamaterial beam that combined vibration suppression and energy harvesting functions in internally coupled resonators in the low-frequency range. In their design, local resonators were alternately coupled, with piezoelectric elements attached for energy conversion. [PERSON] and [PERSON] [7] also contributed by introducing a novel metamaterial with a multiresonator mass-in-mass lattice system, where the internal coupling was achieved through a linear spring, leading to the formation of two additional bandgaps over conventional designs. The research trajectory in metastructure system formulation has been significantly advanced by [PERSON] et al. [8], who developed a robust framework that culminates in transfer functions, allowing for nuanced manipulation and control of system responses. [PERSON] et al. 's mathematical framework, leveraging Laplace transformations, further simplifies the analysis of metastructures, especially in damping low-frequency vibrations, thus enhancing the practical applicability of these complex systems in engineering solutions. [PERSON] et al. [9] developed the mathematical framework using Laplace transformations for analyzing locally resonant metastructures, simplifying examination of their responses, and deriving a closed-form expression for bandgap frequency range, validated through dispersion analysis and experimental tests. Traditional methods focus on dispersion analysis and limit the scope of analysis to wave propagation without offering insights into control strategies. This work develops a mathematical framework to derive a close-form formulation for analyzing both conventional and internally coupled resonators in metastructures, integrating control system theory and the transfer function method to provide enhanced control mechanisms and bandgap tuning methods through resonator stiffness adjustments.