Boundary controllability problems for the equation of oscillation of an inhomogeneous string on a semiaxis

Abstract

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 $w_x(0,t) = u_1(t), t \in (0,T)$, where $u_0, u_1$ are controls. In both cases, the potential q satisfies the condition $q \in C[0, \infty)$, the controls belong to the class $L^{\infty}$ 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.