From the above it looks like the brown wires are connected to the relay contacts and these are switching the loads. On the other side of the relays you have terminal 13 which is ground and terminal 14 which energises the coil from whatever control system you are currently using.
If you are wanting to decouple the relays from the existing control system then you could use something like the following where the ESP8266 microcontroller is flashed with ESPHome or Tasmota.
The relays in the above have 3 terminals which are usually “common”, “normally open”, and “normally closed” so choose the option that matches your existing setup.
Also, it looks like your Finder relay each has two contacts so you may need the 4 channel version of the above rather than the 2 channel version.
If the relays are no longer working then you could just replace them with the same or similar 10 Amp relay with a 12 volt coil.
Whether they are bad and need to be replaced or not you can accomplish ‘smart’ control by using a low power relay or solid state switching device to control the relay coil activation. Leave the high power switching to the existing relays and just switch the low power 12 volt control.
Are these now controlled by a mechanical switch or by some other automation system? That could be a consideration in your final design.
Here is a 4 channel smart wifi relay board which works with Alexa or Google fro Amazon:
Thank @Jonah1970 and @davel4wa for your replay
let me try to add more context
First goal should be replace both with sometime smaller
Second goal (if achievable) keep the existing control (I didn’t make it, but as far I understand after “reverse engineer” the behavior. It is an alarm system, arm and disarm a zone trigger Relay1 or Relay2 and control the roller shutter. Relay1 controls roller shutter UP, Relay2 controls roller shutter DOWN. There are 2 brown cables for each relay because they control the roller shutter on the first and second floor) and add “smart control”. So, I can still trigger the relays as I do today or using the “smart” brain with HomeAssistant
This is one of the main pains with the existing config, I can’t control the first or second floors independently also if the existing wires can do that
After some search I found some 12v relay in a box in my garage (from some old projects), but I have no idea if that can be ok for my goal, as I have “only” 2 wires 36/39 or 38/36 and I don’t have an additional one to connect the INT1 or INT2
When you say ‘roller shutter’ it does not sound like a home type window shade but like a heavy security roll up ‘door.’ If that is the case, it likely requires a fair amount of power. The relays on the relay module you show has 10A relays rated for line voltage so that should work.
Are the 1st and 2nd floor shutters opened and closed simultaneously or independently? When the alarm system is armed/disarmed do the shutters activate automatically?
the guy that made the installation “improperly” re-used the capability of the alarm system to pull up a 12V exit for an amount of time, and mapped this behavior to a zone.
when the zoneA is on, the 2 contact are on and provide 13v to the finder releA, then the releA remain on for an period of time that allow the roller blind to go up.
when the zoneB is on, the 2 contact are on and provide 13v to the finder releB, then the releB remain on for an period of time that allow the roller blind to go down
then I need to manually disarm the zones to be able to insert it again an roll up and down the blind
probably not the best one, but this is how it works and why I decided to have a look to figured out if there is a better way to do it