  A variety of dynamics in nature and society can be approximately treated as a
driven and damped parametric oscillator. An intensive investigation of this
time-dependent model from an algebraic point of view provides a consistent
method to resolve the classical dynamics and the quantum evolution in order to
understand the time-dependent phenomena that occur not only in the macroscopic
classical scale for the synchronized behaviors but also in the microscopic
quantum scale for a coherent state evolution. By using a Floquet
U-transformation on a general time-dependent quadratic Hamiltonian, we exactly
solve the dynamic behaviors of a driven and damped parametric oscillator to
obtain the optimal solutions by means of invariant parameters of $K$s to
combine with Lewis-Riesenfeld invariant method. This approach can discriminate
the external dynamics from the internal evolution of a wave packet by producing
independent parametric equations that dramatically facilitate the parametric
control on the quantum state evolution in a dissipative system. In order to
show the advantages of this method, several time-dependent models proposed in
the quantum control field are analyzed in details.
