The present paper investigates numerically and theoretically the axisymmetric vortex breakdown occurring in a constricted pipe of infinite extension, i.e., the transition from a smooth columnar state to a breakdown state exhibiting a recirculation bubble. Velocity distributions are prescribed at the pipe inlet under the form of Batchelor vortices with uniform axial velocity and variable levels of confinement. A numerical continuation technique is developed to follow the branches of nonlinear steady solutions when varying the swirl parameter. In the most general case, vortex breakdown occurs abruptly owing to a subcritical, global instability of the non-parallel, viscous columnar solution, and results in the coexistence of multiple stable solutions over a finite range of swirl. For highly confined vortices, a second scenario prevails, where the flow transitions smoothly from the columnar to the breakdown state without any instability. The effect of a low-flow rate jet positioned at the pipe wall is then characterized in the perspective of control. Its effectiveness is evaluated in light of several practically meaningful criteria, namely, the ability of the control to optimize either the stability domain or the topology of the columnar state and its ability to alleviate hysteresis. For each criterion, an optimal jet position is determined from nonlinear simulations, the results being in good agreement with that issuing from an asymptotic expansion of the Navier-Stokes equations. Finally, we illustrate the importance of physically motivated control strategies by demonstrating how the wall jet technique can be outdone by an appropriate manipulation of the axial velocity profile prescribed at the pipe inlet.