First, the problem of stability of an equilibrium figure
F
∗
F_*
for an abstract system is reduced to the sign of the difference between the energy of the perturbed motion at initial time, and that of
F
∗
F_*
. All control conditions are only sufficient conditions to ensure nonlinear stability.
Second, employing the local character of the nonlinear stability, some nonlinear instability theorems are proven by a direct method.
Third, the definition of loss of control from initial data for motions
F
F
is introduced. A class of equilibrium figures
F
∗
F_*
is constructed such that:
F
∗
F_*
is nonlinearly stable; the motions, corresponding to initial data sufficiently far from
F
∗
F_*
, cannot be controlled by their initial data for all time. A lower bound is computed for the norms of initial data above which the loss of control from initial data occurs.