Linear control systems on matrix Lie groups have an important place in terms of their application to real-life problems [1]. By the definition in the context of Lie groups a linear control system is determined by the pair Σ = (G, D), where the state space is a real finite dimensional Lie group G with the Lie algebra L(G) and the dynamic D is given by the family of differential equations on G

                                                                                       g(t)=X(g(t))+Σi=1ui(t)Y (g(t))

where X is an element of normalizer of L(G) and called drift vector field, and the control vectors Y 1, Y 2, . . . , Y n belong to L(G), where L(G) denotes the Lie algebra of left-invariant vector fields. The control function u = (u1, . . . , un) is in the class of piece-wise constant functions from [0,∞) to Rn. The main purpose of geometric control theory is to investigate whether it is possible to reach any other state from a given specific state in a positive time via admissible trajectories. For example, from a given initial condition x0, can a new condition x1 be reached by transferring x0 via the admissible control u in a positive time? Considering this for disease and epidemic models, is it possible to find a medical strategy to transform an initial level of disease, at another final level of health, in a positive time [2]?  In this presentation, we work on the controllability properties of some kind of control theory problems.