In this article, we study linear control systems on a 4-dimensional solvable Lie group. Our motivation stems from the model introduced in Baspinar et al. (J Math Neurosci 10:11, 2020), which presents a precise geometric framework in which the primary visual cortex \( V1 \) is interpreted as a fiber bundle over the retinal plane \( M \) (identified with \( \mathbb {R}^{2} \) ), with orientation \( \theta \in S^{1} \) , spatial frequency \( \omega \in \mathbb {R}^{+} \) , and phase \( \phi \in S^{1} \) as intrinsic parameters. For each fixed frequency \( \omega \) , this model defines a Lie group \( G(\omega ) = \mathbb {R}^{2} \times S^{1} \times S^{1} \) , which we adopt in this work as the state space group \( G \) of our linear control system. We also present new results concerning controllability and characterize the control sets associated with this class of systems.
