This paper examines the controllability of homogeneous-in-the-state bilinear systems onRⁿ-{0}dx/dt=Ax+uBx (B) where x∈Rⁿ-{0}, u is a real-valued control function and A, B are n×n matrices specified for some real numbers a, b(≠0) and matrix U by B^t+B=2bI, A=Z+aI, Z=(B^t-bI)U+U(B-bI) under a constraint, tr (Z)=0.
For the system we consider the vector fields associated with systems generated by the projection of trajectories from Rⁿ-{0} onto a sphere of n-1 dimension. It is shown that there exist controls such that the induced vector fields become conservative and that if the system (B) is accessible then it is controllable on the sphere because of the compactness of a spherical surface.
Since we can take a course on the sphere and choose a stable mode or an unstable mode of the system (B), every state is attainable to any point on Rⁿ-{0}. This proves that if the system has the accessibility property it is controllable on Rⁿ-{0}.