Transformation operators in controllability problems for the degenerate wave equation with variable coefficients

Abstract

We study the control system $w_{tt} = \cfrac1{\rho } (kw_x) x + \gamma w,\; w(0, t) = u(t),\; x \in (0, l), t \in (0, T)$, in special modified spaces of the Sobolev type. Here, $\rho , k,$ and \gamma are given functions on $[0, l)$; $u \in L^{\infty} (0, T)$ is a control, and $T > 0$ is a constant. The functions $\rho$ and $k$ are positive on $[0, l)$ and may tend to zero or to infinity as $x \rightarrow l$. The growth of distributions from these spaces is determined by the growth of $\rho$ and $k$ as $x \rightarrow l$. Applying the method of transformation operators, we establish necessary and sufficient conditions for the $L^{\infty}$ -controllability and approximate $L^{\infty}$ -controllability at a given time and at a free time.