In this paper necessary and sufficient conditions for null-controllability and approximate null-controllability are obtained for the control system w(tt) = w(xx) - q(2)w, w(x)(0, t) = u(t), x > 0, t is an element of (0, T), where q >= 0, T > 0, and u is a control bounded by a hard constant. These problems are considered in the Sobolev spaces. Controls solving these problems are found explicitly. Bang-bang controls solving the approximate null- controllability problem are found by means of the solutions of the Markov power moment problem. Continuous controls solving this problem are constructed with the aid of the Cesaro means of a Fourier series generated by the data of the control system.