In this article, we investigate a class of quaternionic functional equations of the two-sided Sylvester type, which arise in areas such as control theory, robotics, signal processing, and image analysis. Although quaternionic matrix equations have been extensively studied, two-sided Sylvester systems remain particularly challenging due to the noncommutativity of quaternion multiplication and the increased structural complexity they entail. We derive general solutions to these systems by establishing the necessary and sufficient conditions for solvability, unifying and by extending previous theoretical results. We propose an efficient algorithm to compute the general solution in both full-rank and rank-deficient cases, using Moore-Penrose inverses and projection operators The practical interest of our method is demonstrated through applications in perturbation theory, image processing, and robust control. We present numerical examples to validate the proposed approach, including a case study involving a multi-joint robotic manipulator. These results highlight both the theoretical relevance and computational advantages of the proposed method.

In applied and computational mathematics, quaternions are fundamental in representing three-dimensional rotations. However, specific types of quaternionic linear matrix equations remain few explored. This study introduces new quaternionic linear matrix equations and their necessary and sufficient conditions for solvability. We employ a methodology involving lemmas and ranks of coefficient matrices to develop a novel algorithm. This algorithm is validated through numerical examples, showing its applications in advanced fields. In control theory, these equations are used for analyzing control systems, particularly for spacecraft attitude control in aerospace engineering and for control of arms in robotics. In quantum computing, quaternionic equations model quantum gates and transformations, which are important for algorithms and error correction, contributing to the development of fault-tolerant quantum computers. In signal processing, these equations enhance multidimensional signal filtering and noise reduction, with applications in color image processing and radar signal analysis. We extend our study to include cases of $\eta$-Hermitian and i-Hermitian solutions. Our work represents an advancement in applied mathematics, providing computational methods for solving quaternionic matrix equations and expanding their practical applications.