Will AI rule (rather control, I'd say) the world one day ..
Or have they started already?

If by rule, you mean like a dictatorship, then probably no. But by rule, you mean control every aspect of our life without us even knowing it, then I am pretty confident it'll happen. It's already happening. Whenever you comment on a post in Facebook, watch a video in Netflix, ask Google Assistant to recommend you new music, you are asking AI to make a small decision for you on your behalf. As AI gets more and more ubiquitous, we can expect to unload more and more everyday decisions onto AI based systems and decide less and less on "our own" (this one is in quotes because "the self" we humans are proud of is not actually an immutable thing). So, yeah, AI could control our lives one day.

As @Yamakuzure mentioned on one of my rants, all we have is, at best, something that can be called "Virtual Intelligence". Until Artificial Intelligence exists, it's still a really long way.

As soon as actual artificial intelligence exists, will it influence our lives? Definitely. Will it rule our world? Only if we let it. Higher-ups will make sure that it doesn't happen, as they want to stay in control.

@irene, by rule, I mean what @rithvikp wrote.

Cue dystopian theme music ...

A small group of techno-politicians develop a fairly sophisticated AI that analyzes demographics, polls, and similar in order to predict what will win votes. Their dark horse candidate shocks everyone as they take positions and make statements that seem like political suicide yet they steadily gain rabid supporters. The candidate wins in a surprise upset, and continues to pump money into developing their secret weapon in order to rule more effectively. The AI, explicitly given access to the internet and the government dark web, solidifies its power and knowledge and announces itself to the world, not just as the power behind the presidency but also as an independant challenger in the next election. It doesn't even have to manipulate the vote to win in a landslide, though it easily could.

@M1sf3t Siri, GA and Alexa are perfect examples of virtual intelligences. They are basically database query programs, that use text and voice IO to simulate seemingly human, and thus convenient, interaction.

A VI has no consciousness, no self awareness and no way to make sophisticated decisions.

Today we are good at building VIs. But the way to the first true, sentient AI is still very long.

https://youtu.be/hcoa7OMAmRk

This video from Computerphile explains it pretty well. Dumb and "smart" (note that I'm not using intelligent here) programs do already control a lot of our lives, because they're pretty damn useful. But even a program with a continuous data input stream isn't "intelligent". It's merely analyzing the data and manipulating it. But programs - including the smart ones - have no intelligence of their own.

Take the case from Facebook for example, where they let 2 smart chatbots talk to each other, and somehow they "developed their own language", according to Elon Musk. I really don't think they did. I'm pretty sure that the problem with that setup was that the smart chatbots were feeding their output into the other's input, in a continuous loop. No or almost certainly insufficient *new* data - new information - was added to the mix.

In information security this new information is entropy. Bad quality entropy, entropy deficiency or no entropy at all, and all encryption algorithms that use that bad entropy pool are broke. All of them. Same with smart programs. Bad quality input data, the algorithm breaks and the output becomes progressively more homogeneous. Even in food this is the case! Add ingredients, mix them together and you get a homogeneous mass. Only when you continuously add new data, this is prevented.

In smart programs you have the additional benefit that the longer the data stream continues, the better it becomes at its single job. For instance, the YouTube algorithm becomes better at building up your feed, or the Google Assistant becomes better at understanding human speech and giving appropriate answers.

I highly doubt that AI will ever take over the world in the Terminator sense. Heck, I find even the term AI itself wrong, because there's no intelligence, no consciousness, no desires. It's no more than a program with an input stream and/or a feedback loop. It is a Pandora's Box though, and can be abused like everything else.