In this paper necessary and sufficient conditions of L ∞ -controllability and approximate L ∞ -controllability are obtained for the control system wttis a control; and q ≥ 0, α > 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 .

In the paper, problems of controllability, approximate controllability, reachability and approximate reachability are studied for the control systemIt is proved that each end state of this system is approximately reachable in a given time T , and each its initial state is approximately controllable in a given time T . A necessary and sufficient condition for reachability in a given time T is obtained in terms of solvability a Markov power moment problem. It is also shown that there is no initial state that is null-controllable in a given time T . The results are illustrated by examples. and the seering conditionHere, for m = 0, 1, 2, H m 0 = ϕ ∈ L 2 (0, +∞) | ∀k = 0, m ϕ (k) ∈ L 2 (0, +∞)

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