Interesting question as most will ask the opposite if ps4 controller can be use on ps5.
Anyway based on a quick google search, nobody have definite answer on this yet.
Aug 3 - Sony is confirming today that its existing PS4 controller will not work with PS5 games. “We believe that PS5 games should take advantage of the new capabilities and features we’re bringing to the platform, including the features of DualSense wireless controller,” says Sony in a statement confirming the company’s plans.
Sony says that existing DualShock 4 controllers and officially licensed third-party PS4 controllers “will work with supported PS4 games.” Sony doesn’t list its supported games, but the company does confirm that officially licensed racing wheels, arcade sticks, and flight sticks will work with PS5 games and supported PS4 games. Sony’s existing PS Move Motion Controllers and the PlayStation VR Aim Controller will also work with supported PS VR games on the PS5.
This move does mean that PS5 owners will need to buy additional controllers to play PS5 games, much like how PS3 controllers didn’t work on the PS4.