Donnell's model of a shallow (and thin) circular cylindrical shell is formulated by a system of three partial differential equations, only one of which contains explicit time dependence. It constitutes one of the most important linear shell models, yet problems associated with its boundary stabilization and control have not been carefully studied. In this paper, we set up the functional-analytic framework, derive dissipative boundary conditions, and determine the infinitesimal generator of the semigroup of evolution. Using a frequency domain method along with energy multipliers, we establish the result of uniform exponential decay of energy under geometric conditions identical to those of the case of a thin Kirchhoff plate. Our approach, incorporating energy multipliers in the frequency domain with a contrapositive argument, appears to be new. It has the beneficial effect of avoiding the necessity to estimate lower order terms when the shell radius is not large. We also consider the case in which the domains contain angular corners; special treatment is required to handle the additional energy contributed by the twisting moments at corner points. Under the assumption of sufficient regularity, uniform exponential decay of energy is also established for such domains.