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

  • K. S. Khalina Фiз.-техн. iн-т низьких температур НАН України, Харкiв


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.
How to Cite
KhalinaK. S. “Boundary Controllability Problems for the Equation of Oscillation of an Inhomogeneous String on a Semiaxis”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, no. 4, Apr. 2012, pp. 525-41,
Research articles