THE FOURIER TRANSFORM METHOD IN CONTROLLABILITY PROBLEMS FOR THE FINITE STRING EQUATION WITH A BOUNDARY CONTROL BOUNDED BY A HARD CONSTANT

Necessary and sufficient conditions for (approximate) null-controllability are obtained for the control system w_tt = w_xx - q²w, w_x(0, t) = u(t), w_x(π, t) = 0, x ∈ (0, π), t ∈ (0, T), where q ≥ 0, T ∈ (0, π], u is a control bounded by a hard constant. These problems are considered in Sobolev spaces. Controls solving them are found explicitly. Continuous controls solving the approximate null-controllability problem are constructed with the aid of the Cesàro means of a Fourier series generated by the data of the control system.