Brockett's condition implies exponential composition-operator stabilization

Let \overset{\raisebox{0ex}{\scalebox{0.45}{\bullet}}}{x} =f(x,u) be the control system under consideration. Brockett's condition means that the vector field $f$ is open at the origin. A composition operator $T_h$ has a stationary symbol when its symbol $h$ is stationary, and continuity at the origin means that $h$ is continuous there. Composition-operator stabilization conjecture. If the vector field $f$ satisfies Brockett's condition, then the system is locally exponentially stabilizable by a composition operator $T_h$ with a stationary symbol that is continuous at the origin. This conjecture proposes a sufficiency result complementary to the established necessity results: openness of $f$ should yield exponential stabilization in the composition-operator framework, while the claim itself remains open.