Controllability of Linear Control Systems on Solvable Lie Groups with Hyperbolic Drift

We study the controllability of linear control systems on connected solvable Lie groups with hyperbolic drift. Under a suitable splitting assumption on the control directions, we show that the controllability problem can be reduced to the controllability of induced systems on the positive and negative hyperbolic components of the group. We then establish sufficient conditions ensuring controllability of these induced systems based on the ad-rank condition and suitable drift-reachability assumptions. As a consequence, we obtain a controllability criterion for the original system. The results provide a hyperbolic counterpart to existing controllability results for linear systems on solvable Lie groups and are illustrated by explicit examples.