Upper and lower kinematical bounds for the expectation values of arbit
rary observables of driven quantum systems in mixed states are derived
, and criteria for their attainability established. The results are ap
plied to the problem of maximizing the energy of a laser-driven four-l
evel Morse oscillator model for HF, as well as a four-level harmonic-o
scillator model. [S1050-2947(98)01510-8].