We consider a wave equation on a semiaxis, namely, w t t (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 and 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.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 4, pp. 525–541, April, 2012.
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Khalina, K.S. Boundary controllability problems for the equation of oscillation of an inhomogeneous string on a semiaxis. Ukr Math J 64, 594–615 (2012). https://doi.org/10.1007/s11253-012-0666-5
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DOI: https://doi.org/10.1007/s11253-012-0666-5