In this dissertation, we study the semilinear two dimensional control system of the form \dot x(t) = A(u(t))x(t), t ϵ\Re, x(0) = x[subscript]0ϵ \Re[superscript]2\\\0, u ϵ U, where A : V → gl(2,\Re) is continuous on an open set V⊂ \Re[superscript]m with V \supset U, U being compact and U = \u : \Re → U locally integrable. The semigroup S corresponds to the solutions with piecewise constant controls, i.e., S = \e[superscript]t[subscript] nB[subscript] n... e[superscript]t[subscript]1[superscript]B[subscript]1 : t[subscript]j≥ 0, B[subscript]j = A(u[subscript]j), u[subscript]jϵ U, j = 1,..., nϵ N\⊂ Gl(2,\Re). S acts in a natural way on \Re[superscript]2\\\0, on the circle, and on the one dimensional projective space P. Under the assumption that the group generated by S in Gl(2,\Re) acts transitively on P, we characterize the sets (control sets) in P, where the system is controllable. We then analyze the Lyapunov spectrum, i.e. the extremal exponential growth rates of the solutions x(t, x[subscript]0, u). If [lambda](x[subscript]0, u) = limsup[limits][subscript]t→[infinity] 1[over] t log ǁ x(t,x[subscript]0, u) ǁ ,then [kappa] = \sup[limits][subscript]uϵ U \sup[limits][subscript]x[subscript]0≠ 0 [lambda](x[subscript]0, u) and [kappa][superscript]* = ϵf[limits][subscript]uϵ U ϵf[limits][subscript]x[subscript]0≠ 0 [lambda](x[subscript]0,u)are the maximal and minimal Lyapunov exponents of the system. This paper gives analytic results for control sets and the extremal Lyapunov exponents in the two dimensional case.