In this paper necessary and sufficient conditions of L-infinity-controllability and approximate L-infinity-controllability are obtained for the control system w(tt) = w(xx) - p(x)w, w(0, t) = u(t), x > 0, t is an element of (0, T). Here p(x) = q(2) + r(x), vertical bar r(x)vertical bar <= alpha e(-x), x > 0; u is a control; and q >= 0, alpha > 0, T > 0 are constants. These problems are considered in the Sobolev spaces. Using the transformation operator of the Sturm-Liouville problem on the positive half-axis, we see that the control system with an exponentially perturbed potential q(2) replicates the controllability properties of the system with the constant potential q(2). Conditions of controllability for the system with the potential p are obtained from the conditions for the system with the constant potential q(2).