Q:

Are there any existing technologies, or research, that investigate the sustainable production of food for manned exploration trips? 

With NASA planning to eventually get people to Mars, and with privately funded attempts such as Mars-One planning to do the same, and relatively soon, there is the issue of having to feed anyone you send for the 6 or so months journey.  
(Obviously the issue extends to when you reach the planet but this question is focused on in flight food production as this deals more with the realm of long distance, manned space exploration)
What technologies currently exist or what research is being done focusing on sustainable food production for these long, manned, exploration trips? 

A:

The largest scale experiment to test out such ideas was the Biosphere 2.  The experiment was probably more akin to setting up a habitat on Mars than growing in a spacecraft, but it is a good starting point for all such technologies.  
Basically, the facility was set up so that 8 people could live there for an indefinite period of time, growing their own food, and maintaining the equipment and environment.  In practice, it is quite similar to the type of environment that might exist on a Mars colony. 
The program was plagued with issues, which I will point you to the Wikipedia page to learn more about, but it was largely sustainable, and later versions actually worked quite well. 
Furthermore, there has been several programs to grow plants on board the International Space Station, and while they aren't a significant part of the diet yet, they have been eaten.  They are working to improve these programs constantly. 

A:

China conducted a 105-day controlled environment project with a crew of three scientists in 2014.  In the facility (called Yuegong-1) five kinds of cereals, fifteen kinds of vegetables, and one kind of fruit were grown.  Yellow mealworms were grown as well for protein.  The mission completed successfully.  China hopes to utilize this facility as a testbed for the controlled environment that will be in place in their completed space station (Tiangong).