# Transformation Operators in the Problems of Controllability for the Degenerate Wave Equation with Variable Coefficients

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We study a control system \( {w}_{tt}=\frac{1}{\rho }{\left(k{w}_x\right)}_x+\gamma w,w\left(0,t\right)=u(t),x\in \left(0,l\right),t\in \left(0,T\right), \) in special modified spaces of the Sobolev type. Here, *ρ, k,* and *γ* are given functions on [0*, l*) , *u* 𝜖 *L*^{∞}(0*,T*) is a control, and *T >* 0 is a constant. The functions *ρ* and *k* are positive on [0*, l*) and may tend to zero or to infinity as *x* → *l.* The growth of distributions from these spaces is determined by the growth of *ρ* and *k* as *x* → *l.* Applying the method of transformation operators, we establish necessary and sufficient conditions for the *L*^{∞}-controllability and approximate *L*^{∞}-controllability at a given time and for free time.

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