  We discuss Floquet engineering of dissipative molecular systems through
periodic driving of an infrared-active vibrational transition, either directly
or via a cavity mode. Following a polaron quantum Langevin equations approach,
we derive correlation functions and stationary quantities showing strongly
modified optical response of the infrared-dressed molecule. The coherent
excitation of molecular vibrational modes, in combination with the modulation
of electronic degrees of freedom due to vibronic coupling can lead to both
enhanced vibronic coherence as well as control over vibrational sideband
amplitudes. The additional coupling to an infrared cavity allows for the
controlled suppression of undesired sidebands, an effect stemming from the
Purcell enhancement of vibrational relaxation rates.
