In this thesis, we study the controllability problem for two systems of partial differential equations. We will first consider the wave equation with variable coefficients and potential in one dimension, $u_{tt} - (a(x)u_x)_x + pu = 0$, with control function $v(t)$ acting on the boundary. We consider a class of functions corresponding to a special weight function that contains the variable coefficient $a(x)$. From here, we derive a global Carleman estimate for this system, and establish the controllability property. We then later extend the class of admissible functions $a(x)$ for which the controllability property holds true. We then study the controllability problem for the degenerate heat equation in one dimension. For $0\leq \alpha <1$, on $(0,1) \times (0,T)$, we consider $w_t - (x^{\alpha}w_x)_x = f$. This equation is degenerate because the diffusion coefficient $x^{\alpha}$ is positive in the interior of the domain and vanishes at the boundary. We consider this problem under the Robin boundary conditions. Again, we derive a Carleman estimate for this system, taking into account the new boundary terms that arise from the Robin conditions.
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