What about utilizing a DHCP address for the time being to see if the network disconnections still occur.
Beyond potential addressing issues the only other thing I can think is there may be something wrong with the switch port, patch cable, or cable run that you are using.
An easy way to check all of these at once and either confirm it is the controller or rule it out would be to get another network device of any kind, set it to the same static address and plug it into the same patch cable.
If the network disconnections occur on this other device it is either:
-Network switch
-Patch cable/cable run
-Network addressing
If the network disconnections stop then I would say the OpenSprinkler controller is possibly bad.