2017
Том 69
№ 9

All Issues

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

Khalina K. S.

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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.

English version (Springer): Ukrainian Mathematical Journal 64 (2012), no. 4, pp 594-615.

Citation Example: Khalina K. S. Boundary controllability problems for the equation of oscillation of an inhomogeneous string on a semiaxis // Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 525-541.

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