Are there economic studies surrounding the efficacy of gun control?
To answer the headline question directly: yes, there are. 

While economists are among the central participants, this is really an interdisciplinary policy-oriented question. The Science of Gun Policy is a fairly recent and comprehensive review of the research overall, published by the Rand Corporation. It's over 400 pages long and I haven't read it. But if you glance through it you will get a sense of the many complex angles that this question has been examined from. Relevant outcomes are not limited to violent crime per se.

Table 21.1 on pages 332-333 summarize the evidence for which policy measures may affect which outcomes, according to the current evidence. Regarding violent crime specifically, there is moderate evidence ("two or more studies—at least one of which was not compromised by serious methodological weaknesses... and contradictory evidence was not found in other studies with equivalent or stronger methods.") to show: 


waiting periods reduce total homicides
background checks by dealers reduce firearm homicides (evidence for private-seller background checks is less conclusive)
firearm prohibitions for individuals subject to restraining orders decrease intimate partner homicides.


I have not looked through the relevant studies for these particular findings to see which ones are specifically by economists, but in any event, these are probably the most conclusive findings you will find. Someone else might be able to give you a more in-depth look at the economics research per se.