Is it possible to focus the sun in such way? 
Imagine the sliding part of the mirror is controlled by computer and opens on intervals.
Is it possible to increase the power of the beam by making it bounce between the mirrors thus going through the lens and then releasing it resulting in beam with more power ?
 A: 
Is it possible to focus the sun in such way?

Yes, as others have pointed out, all of the ideas in your sketch are already used in existing designs - perhaps excepting the shutter (which actually performs no useful purpose so far as I can see). 
As Chris White commented - "this exact design (with the shutter permanently open) is a Schmidt-Cassegrain telescope, probably the most popular high-end consumer scope these days."


Is it possible to increase the power of the beam by making it bounce between the mirrors

No, focusing a beam of light, or reflecting it, does not increase the power. The amount of power is the amount of light energy entering the system per second. That is limited by the diameter of the entry pupil. Energy is conserved.
A: Fun fact: You cannot focus the beams from the sun create a hotter area than the surface of the sun itself. This would break the second law of thermodynamics. 6000 °C is hot enough for many spectacular destructive applications, though, a youtube search on "melting steel with sun light" offers hours of entertainment.
If you want to make something that potentially focuses more energy than the surface of the sun, head for a solar pumped laser.
