Controllability Problems for the Non-Homogeneous String that is Fixed at the Right End Point and has the Dirichlet Boundary Control at the Left End Point

In the paper, the necessary and sufficient conditions of null-controllability and approximate null-controllability are obtained for the control system w(tt)(x, t) = w(xx)(x, t) - q(x)w(x, t), w(0, t) = u(t), w(d, t) = 0, x is an element of (0, d), d > 0 t is an element of (0, T), 0 < T <= d, where q(x) is an element of C-1[0, d], q(x) >= 0, q'(+)(0) = q'_(d) - 0, u is a control vertical bar u(t)vertical bar <= 1 on (0, T). The problems for the control system are considered in the modified Sobolev spaces. The control that solves these problems is found explicitly. The bang-bang controls solving the approximate null-controllability problem are constructed as the solutions of the Markov trigonometric moment problem.