O-rings, and compressed air.
O-rings, and compressed air?? How I implement those things with an arduino 
SamyTech:
O-rings, and compressed air?? How I implement those things with an arduino
You would implement those O-rings on tube sections so air won't leak between the tubes. You'd have to seal the top and bottom tubes too. Then if you pump air in it will push on all the tubes making them slide up on the O-rings.
My first thought was use rockets and a gyro steering system. It might rise a couple 1000 feet.
take two tubes. one fits into the other and slides.
put a cap on the smaller one so it gives you a platform and put your lights on that.
get a gear motor that will fit inside of the inner tube
mount that gear motor to a threaded rod.
cut a hole in the smaller tube high enough so that the gear motor and threaded rod will fit inside of the tube.
put your nut in that hole.
secure the motor to the base.
when the motor spins the inner tube lifts and your project is on top
you can buy a linear actuator and do the same thing if you are not mechanically inclined
since this is your project, start making sketches of how things might fit.
post them and we can help you figure it out.
Or put a spiral spring inside of your tubes with a latch that Arduino causes to be released and the lighthouse pops up.
It will need a stop like a string tied to center top and center bottom so it rises only so far.
If it has to go down and up, have a motor winch-wind the string to pull the top down.
With a spring you don't need O-rings or air pump.
SamyTech:
I cannot use a linear actuator because the rod would protrude above the headlight, and I want a system that does not protrude from inside the headlight
I think you have a misconception of how a linear actuator works.
The shaft of the actuator "TELESCOPES" out of the end of the unit like a hydraulic ram moves in and out of a cylinder.
The rod will not protrude above the top of the lighthouse.
How much space have you got under the platform?
Tom.. 
I make a diagram explaining why I think a threaded rod would protude above the highest point of the lighthouse:
You might look at this scheme to make a telescoping mast, and put it inside your lighthouse. You can add as many sections as you want.
Use the Arduino and a H-bridge to control a small DC motor to act as a winch to pull or release the cord and make the lighthouse rise.
Paul
Thank you so mach for the showing me the system of the telescope, I will use it, now I am making the designs in Fusion 360. When I finish the design I will post the results. I hope I would finish the designs today or tomorrow.
That drawing helped explain what you’re trying to do....
Words are excellent, and we had no problem understanding WHAT you were trying to do, but we didn’t know what you were working with..
GoForSmoke:
Or put a spiral spring inside of your tubes with a latch that Arduino causes to be released and the lighthouse pops up.
Like this?
SamyTech:
I have to make a lighthouse of 300mm maximum at the beginning and with a push botton I have to activate it and make it to rase vertically, I don't now what mechanism would be usefull for this project. Any idea ??
Two designs that might be considered are a scissor lift mechanism and a telescoping pole setup that is operated by a cable. As to using air in a telescoping tube setup, instead of sealing with O-rings, you might make an internal bladder out of thin plastic sheeting. If the tower is thin, you might push it up with metal cable. There are probably more complex ideas available
aarg:
Like this?
Except with nested tubes you would never see the spring.
Would it take more motor -or leverage- to compress the spring? Yes but I'd love to see the prof when the thing pops up!
The rocket notion however, hey "rise straight up"? I'm givin' it all I can Cap'n! She'll blow if I give it any more!
Back in the day, most of my rockets turned into confetti somewhere near apogee. The OKC bombing put an end to that.
The easy way I think it would be the telescope.
I was also thinking of a scissor lift kind of mechanism. At least no risk of air leaks, and much easier to not only go up, but also go down, and both in a highly controlled manner. You can probably buy scissor lift mechanisms off the shelf - look at those plastic grabbers you can find in the $1-store, or whatever your location's equivalent. That's the same thing.
Scissor lift.... +1 to zoomcat.