Boundary controllability problems for the equation of oscillation of an inhomogeneous string on a semiaxis - Khalina, K. S. (Ukrains’kyi Matematychnyi Zhurnal, 2012)
We consider a wave equation on a semiaxis, namely, w_tt(x,t) = w_xx(x,t) — q(x)w(x,t), x > 0. The equation is controlled by one of the following two boundary conditions: w(0,t) = u_0(t) and w_x(0,t) = u_1(t), t (0,T), where u_0, u_1 are controls. In both cases, the potential q satisfies the condition q C[0, ∞), the controls belong to the class L^∞ and the time T > 0 is fixed. These control systems are considered in Sobolev spaces. Using the operators adjoint to the transformation operators for the Sturm - Liouville problem, we obtain necessary and sufficient conditions for the null-controllability and approximate null-controllability of these systems. The controls that solve these problems are found in explicit form.