A320
Air Traffic Control Fully Certified
Communication Breakdown; Situational Awareness
ATC Issue All Types; Conflict Ground Conflict; Less Severe
General None Reported / Taken
Company Policy; Human Factors; Procedure
Aircraft X was holding on Taxiway 1 for an outbound Airbus in taxi lane XX.  Aircraft X taxied to his gate without waiting for the outbound aircraft as instructed.  I made numerous attempts to get Aircraft X to stop; but he had left the frequency to talk with the Company Ramp Tower.  I could have had aircraft pushing off gates YY or ZZ.  I could have had any number of aircraft taxiing on Taxiway 2.  Though this did not happen; it does not diminish the potential for an accident in the future if these types of actions occur again.  After the Tower spoke with the Company Ramp they said they did not tell Aircraft X to taxi; but they did answer affirmative when the pilot of Aircraft X asked if the Airbus in XX was waiting for them.  Recommendation; these scenarios play out almost daily at LAX.  The ramp towers at LAX lost their usefulness after 9/11.  Soon LAX will be burdened with another position; Ground Control 3; at our critically; short-staffed facility.  To provide a modicum of work for this new position; all LAX ramp towers need to close and service must be returned to the tower before someone gets hurt.
LAX Controller voiced concern regarding a ground conflict resulting from the handling provided by 'Ramp Control;' The reporter suggested Ramp Controller operations be terminated for safety concerns.