  In a previous work, we constructed a positive-definite total energy
functional for the axially symmetric linear perturbative theory of Kerr black
hole spacetimes. That work is based on the dimensional reduction of dynamical
axisymmetric spacetimes into 2+1 Einstein-wave map system. In the construction
of the positive-definite energy, various dynamical terms, at the boundary of
the orbit space, critically occur.
  In this work, after setting up the initial value problem in harmonic
coordinates, we prove that the positive energy for the axially symmetric linear
perturbative theory of Kerr black holes is strictly conserved in time, by
establishing that all the boundary terms dynamically vanish for all times. This
result implies a form of dynamical linear stability of Kerr black holes
