My first thought would be an air quality sensor but I’m not sure how sensitive they would be to food smells. I’ve never used one but there are plenty of examples online.

A fallback would be a timer, if you know when your neighbor usually cooks, you could turn your fan on during those times with some buffer.

Is it common for apartments to have shared extractor ducts? I’ve never heard of a setup like this before, and it sounds like both a pain in the arse and a potential safety issue.

Controlling the kitchen fan is probably the easier bit. Depending on the design of the hood, you might be able to control it with a smart outlet or relay. Turn the hood’s switch to always on and control it with the relay. The difficult bit would be sensing when to turn it on.

Cooking will often produce VOCs, and VOC sensors are easy to obtain, but they are also have other sources that are likely in your own apartment. Maybe try using temperature and humidity sensors int the duct to activate the fan when there is elevated temperature and humidity inside the duct?

The simple way is to use a fan or blower limit switch. The more complicated way would be to use a mass airflow sensor and something like a Raspberry Pi to control the speed of your hood vent.

I wouldn’t bother with that, I’d instead look at it from the other point of view and think about fitting something like a simple vent gravity flap to your hood outlet, that is closed when your hood is off, and opens when your hood is on. I.e seals the vent to prevent smells.

Habe you thought about a mechanical solution, e.g. back pressure flap?

You’ve got two challenges here: automatically controlling the fan, and detecting a suitable trigger for an automation routine.

The very first thing I’d do, though, is see if the landlord can do anything about it before you do anything else. Free air flow between two units may be a fire code violation, or there may be a blockage further up the exhaust duct if the other fan is developing enough pressure to bypass your damper.

As for your question…

Controlling the fan is perhaps the easier of the two. A 4-speed fan switch most likely implements a muktivalue resistor with discrete values-- each ON position allows a different value of current to flow through the switch to the fan motor, thereby controlling the speed. There are wireless fan controllers out there, but they tend to be driven by RF or IR remotes, which CAN be incorporated into a home automation system, but it takes extra steps. I’m not really familiar with the options for operating those controls from a smart assistant, but others here might have better info than me.

Since you want the assistant to turn the fan on low, it might be possible to shim in a mini smart relay that can work with a rocker switch, like a Shelly 1 or similar, wired in series with the rocker switch. Then you can leave the fan on low while you are not using it, and allow the assistant to control it on demand. I don’t know how those switches interact with a multi position switch though, since they are looking for transitions of ON/OFF to energize or deenergize the load terminals. You’d need to research it more. And, of course, you’d be modifying the range hood wiring which might cause you other, nontechnical issues.

As for detecting a trigger condition, this is rather harder. The changes in temperature, humidity or airflow outside your range hood when the neighboring fan is operating will be slight, perhaps within the noise band for the sensor, so my may get false positives, or just not detect it reliably. If you could situate the sensor in the duct itself, then airflow may be a viable option, but again, that may violate code or terms of your lease somehow.

Detecting odors is an altogether different set of challenges. Most of the common gas sensors are looking for specific compounds, like carbon monoxide, carbon dioxide, methane, hydrogen sulfide, etc. Food odors generally only contain trace amounts of those sorts of compounds. Your neighbors would need to consistently cook dishes that offgas one of these detectable compounts, which themselves would need to infiltrate into your air space in sufficient concentration to be reliably detected. I’m not an organic chemist and I am not saying it can’t be done, but I wager a lot would have to be “just so” for your system to work as you describe it.

If it’s not too intrusive for you to do so, you might just consider leaving the fan on low through mealtimes, perhaps using one of the micro relay suggestions in a timer routine on your assistant.

The simplest thing would be a pressure or airflow sensor hard-wired to the switch for the hood itself.

Also consider instead of a 1 way valve installing a damper that only opens when your range good is turned on.

Shove a thermocouple up the duct. When neighbor is cooking it will likely affect the temperature in the duct noticably. Use that as your fan trigger.

Sounds hard If you can’t hook into the neighbours stove or hood. Maybe detect if the hood of the neighbors is turned on through a windspeed sensor (these things that rotate). TBH this sound like you should just fix the mechanism/back flow valve that restricts the airflow coming in if your hood is turned off.

Could you add a fan at the exhaust end to create an ever present, gentle, negative pressure to encourage the air to flow that way?

Acinfiniti makes duct fans that have a temperature sensor and when it detects the change in temperature it turns the fan on. I took one apart and used the sensor, pcb and fan in a different project without problems.