ArcSDE stores authorization/license info in a table called SDE_server_config. Use the template below to update this table when a new authorization is needed:

UPDATE [MyGDB].[dbo].[SDE_server_config]
SET [char_prop_value] = 'arcsdeserver,101,ecp000000000,31-dec-2016,JA4FAKE3RA9DJLMAP105'
WHERE prop_name = 'AUTH_KEY'

The keycode above is fake. A real code can be found in the keycodes file after you authorize ArcGIS Server. This file is typically in \\Program Files\ESRI\License<release>\sysgen on Windows or /arcgis/server/framework/runtime/.wine/drive_c/Program Files/ESRI/License<release>/sysgen on Linux.