In this paper we study exact boundary controllability for a linear wave
equation with strong and weak interior degeneration of the coefficients
in the principle part of the elliptic operator. The objective is to
provide a well-posedness analysis of the corresponding system and derive
conditions for its controllability through boundary actions. Passing to
a relaxed version of the original problem, we discuss existence and
uniqueness of solutions, and using the HUM method we derive conditions
on the rate of degeneracy for both exact boundary controllability and
the lack thereof.