The objective of this research is to provide a physics-based understanding of the global stability characteristics of laminar separation bubbles (LSBs) that are actively controlled by periodic two-dimensional forcing through a blowing and suction slot. Particular emphasis is on shedding light on the origin of the three-dimensionality in the controlled LSBs that could ultimately lead to laminar-turbulent transition. Toward this end, a highly efficient global stability solver based on the linearized Navier-Stokes equations has been developed. The new solver is capable of investigating both convective and absolute (temporally growing) primary and secondary instability mechanisms. Using secondary stability analysis, a critical forcing amplitude was found below which the secondary absolute instability mechanism becomes active for a limited range of spanwise wavenumbers. Of special interest is the frequency of the temporally most unstable mode with exactly half the forcing frequency. For forcing amplitudes larger than the critical forcing amplitude, it was found that the controlled absolutely stable LSBs are secondarily unstable (with respect to a spatial convective instability) to a wide range of frequencies and spanwise wavenumbers. When including vortical free-stream disturbances obtained from an isotropic free-stream turbulence model in the secondary stability analysis, the most viable path to transition was identified for controlled convectively unstable LSBs. In addition, Direct Numerical Simulations (DNS) were carried for selected forcing amplitudes. Whereas the DNS results further corroborated the findings obtained from the secondary stability analysis, they provided additional insight into the underlying flow physics of the nonlinear stages of the transition process for forced absolutely unstable LSBs. An unexpected finding from the DNS is that for forcing amplitudes close but smaller than the previously found critical amplitude, free-stream turbulence (with sufficient intensity) has a stabilizing effect on the most temporally unstable mode, and can therefore change the transition scenario driven by an absolute instability to one dominated by a convective instability mechanism.