This paper investigates the energy harvested from the flutter of a plate in an axial flow by making use of piezoelectric materials. The equations for fully-coupled linear dynamics of the fluid-solid and electrical systems are derived. The continuous limit is then considered, when the characteristic length of the plate's deformations is large compared to the piezoelectric patches' length. The linear stability analysis of the coupled system is addressed from both a local and global point of view. Piezoelectric energy harvesting adds rigidity and damping on the motion of the flexible plate, and destabilization by dissipation is observed for negative energy waves propagating in the medium. This result is confirmed in the global analysis of fluttering modes of a finite-length plate. It is finally observed that waves or modes destabilized by piezoelectric coupling maximize the energy conversion efficiency. ; Comment: 23 pages, 14 figures