  We present a new theorem describing stable solutions for a driven quantum
system. The theorem, coined `inertial theorem', is applicable for fast driving,
provided the acceleration rate is small. The theorem states that in the
inertial limit eigenoperators of the propagator remain invariant throughout the
dynamics, accumulating dynamical and geometric phases. The proof of the theorem
utilizes the structure of Liouville space and a closed Lie algebra of
operators. We demonstrate applications of the theorem by studying three
explicit solutions of a harmonic oscillator, a two-level and three-level system
models. These examples demonstrate that the inertial solution is superior to
that obtained with the adiabatic approximation. Inertial protocols can be
combined to generate a new family of solutions. The inertial theorem is then
employed to extend the validity of the Markovian Master equation to strongly
driven open quantum systems. In addition, we explore the consequence of new
geometric phases associated with the driving parameters.
