A Review of Control Sets of Linear Control Systems on Two-Dimensional Lie Groups and Applications

This review article explores the theory of control sets for linear control systems defined on two-dimensional Lie groups, with a focus on the plane (Formula presented.) and the affine group (Formula presented.). We systematically summarize recent advances, emphasizing how the geometric and algebraic structures inherent in low-dimensional Lie groups influence the formation, shape, and properties of control sets—maximal regions where controllability is maintained. Control sets with non-empty interiors are of particular interest as they characterize regions where the system can be steered between states via bounded inputs. The review highlights key results concerning the existence, uniqueness, and boundedness of these sets, including criteria based on the Ad-rank condition and orbit analysis. We also underscore the central role of the symmetry properties of Lie groups, which facilitate the systematic classification and description of control sets, linking the abstract mathematical framework to concrete, physically motivated applications. To illustrate the practical relevance of the theory, we present examples from mechanics, motion planning, and neuroscience, demonstrating how control sets naturally emerge in diverse domains. Overall, this work aims to deepen the understanding of controllability regions in low-dimensional Lie group systems and to foster future research that bridges geometric control theory with applied problems.