Control Sets of Linear Control Systems on 2-Dimensional Lie Groups. Examples

Control theory provides a robust framework to analyze how dynamical systems can be steered within a given state space using bounded inputs. Through the study of control sets—maximal regions of controllability—one can characterize the extent and limitations of controllability in practical applications. In this manuscript, we investigate control sets for selected models defined on the plane \( \mathbb{R}^2 \) and on the affine group \( Aff_+(2) \). These models are representative of diverse systems in engineering and natural sciences, with concrete applications including mechanical devices, robotic arms, oscillatory systems, and neural circuitry. In this review, we aim to study the control sets for the class of linear control systems (LCS) on two-dimensional Lie groups. A control set with a non-empty interior is relevant in any control system because it identifies the regions in the state space where the challenging property of controllability holds. In simpler terms, if two states are located within the interior of a control set, there are strategies available for the system that can connect these states over a positive time interval. The literature provides several results regarding the existence, uniqueness, and boundedness of these sets. Furthermore, under the so-called Ad-rank condition for the system, a characterization based on the system's positive and negative orbits is available for this kind of control set. However, it is well known that computing these orbits is a difficult task. We begin by reviewing the literature that explicitly presents control sets within our context, offering a comprehensive overview of these control sets. Subsequently, we apply these findings to various application control models.