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 equation 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 and 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.
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