Control sets of linear control systems on ${\mathbb{R}}^2$. The real case

In this paper, we study the dynamical behavior of a linear control system on ${\mathbb{R}}^2$ when the associated matrix has real eigenvalues. Different from the complex case, we show that the position of the control zero relative to the control range can have a strong interference in such dynamics if the matrix is not invertible. In the invertible case, we explicitly construct the unique control set with a nonempty interior.