Linear control systems on a 4D solvable Lie group used to model primary visual cortex $V1$

In this article, we study linear control systems on a 4-dimensional solvable Lie group. Our motivation stems from the model introduced in \cite{baspinar}, 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 $θ\in S^{1}$, spatial frequency $ω\in \mathbb{R}^{+}$, and phase $φ\in S^{1}$ as intrinsic parameters. For each fixed frequency $ω$, this model defines a Lie group $G(ω) = \mathbb{R}^{2} \times S^{1} \times S^{1}$, 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.