In the game description it is not mentioned if Raiden IV supports controllers? Can somebody please confirm that I can play this game not only with a keyboard.

Does it also support joysticks?

I'm not sure if you are referring to a flight stick or a flight stick (unintentional rhyming), either way I haven't tested them. It would also depend on the company that made the stick, and the firmware the stick uses.

I was referring to fightsticks, such as the ones used in arcade fighting games, like Street Fighter 2, or arcade space shooters, like Gradius, R-Type, and Raiden 2.

For example: http://slagcoin.com/joystick/brand_controllers.html

No, no. Those like this one are usually called an arcade stick or an arcade controller as a whole. Joystick here is traditionally based on digital switches.
While flightsticks are usually shaped more like fighter jet control grip and are specialized in relatively precisional analogue measurements.

There are arcade SHMUPS which utilize flightstick capabilities, though. Demon Star for example. It has even an FFB support.