Can the Airbus A380 safely fly with two engines out on the same wing?
Proving that a multi-engine aircraft is flyable with one or more engines inoperative is part of the certification programme. The Airbus A380 is certified under FAR (Federal Aviation Regulations, USA) part 25 and JAR (Joint Aviation Regulations, Europe) part 25. One of the requirements of FAR/JAR 25 is that the directional control can be maintained when two critical engines (that means under the same wing) fail. 

The following is selectively extracted from FAR part 25 (JAR is similar, there are some differences). Emphasis added by me.   


  Controllability and Maneuverability
  
  §25.143   General.
  
  ....
  
  b) It must be possible to make a smooth transition from one flight
  condition to any other flight condition without exceptional piloting
  skill, alertness, or strength, and without danger of exceeding the
  airplane limit-load factor under any probable operating conditions,
  including—
  
  
  The sudden failure of the critical engine;
  For airplanes with three or more engines, the sudden failure of
  the second critical engine when the airplane is in the en route,
  approach, or landing configuration and is trimmed with the critical
  engine inoperative;
  
  
  ...


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  §25.147   Directional and lateral control.
  
  a)  Directional control; general. It must be possible, with the wings
  level, to yaw into the operative engine and to safely make a
  reasonably sudden change in heading of up to 15 degrees in the
  direction of the critical inoperative engine. This must be shown at
  1.3 VSR1 for heading changes up to 15 degrees (except that the heading change at which the rudder pedal force is 150 pounds need not be
  exceeded), and with—
  
  
  The critical engine inoperative and its propeller in the minimum
  drag position;
  The power required for level flight at 1.3 VSR1, but not more than
  maximum continuous power;
  The most unfavorable center of gravity;
  Landing gear retracted;
  Flaps in the approach position; and
  Maximum landing weight.
  
  
  b) Directional control; airplanes with four or more engines.
  Airplanes with four or more engines must meet the requirements of
  paragraph (a) of this section except that—
  
  
  The two critical engines must be inoperative with their propellers
  (if applicable) in the minimum drag position;
  [Reserved]
  The flaps must be in the most favorable climb position.
  
  
  c) ...
  
  ...
  
  d) ...
  
  e) Lateral control; airplanes with four or more engines. Airplanes
  with four or more engines must be able to make 20° banked turns, with
  and against the inoperative engines, from steady flight at a speed
  equal to 1.3 VSR1, with maximum continuous power, and with the
  airplane in the configuration prescribed by paragraph (b) of this
  section.
  
  f) ...


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  §25.149   Minimum control speed.
  
  ...
  
  g) For airplanes with three or more engines, VMCL-2, the minimum control speed during approach and landing with one critical engine
  inoperative, is the calibrated airspeed at which, when a second
  critical engine is suddenly made inoperative, it is possible to
  maintain control of the airplane with both engines still inoperative,
  and maintain straight flight with an angle of bank of not more than 5
  degrees. VMCL-2 must be established with—
  
  
  ...
  
  ...
  


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  §25.161   Trim.
  
  ...
  
  e) Airplanes with four or more engines. Each airplane with four or
  more engines must also maintain trim in rectilinear flight with the
  most unfavorable center of gravity and at the climb speed,
  configuration, and power required by §25.123(a) for the purpose of
  establishing the en route flight paths with two engines inoperative.