Ac motor drives are used in a multitude of industrial and process applicationsrequiring high performances. In high-performance drive systems, the motorspeed should closely follow a specified reference trajectory regardless of anyload disturbances. The motor-control issues are traditionally handled by fixed-gain proportional-integral (PI) and proportional-integral-derivative (PlO)controllers. However, the fixed gain controllers are very sensitive to loaddisturbances. In order to overcome this problem, the fuzzy-logic controller(FLC) has been recently used for motor control purpose. The mathematical toolthis work; we deal with a variety of aspects of these subjects: thedevelopment of different types of contact problems; the detailedformulation of appropriate variational principles; the identification of thespaces in which these formulations make sense; when possible, theestablishment of sufficient conditions for the existence and uniqueness ofsolutions; the proof of convergence theorems and a priori error estimatesfor finite element approximations; the description of numerical schemesfor solving discrete variational inequalities, including the actualnumerical solutions of many representative example problems, and finallythe interpretation of these results with regard to their implications on the. actual physics of the problems considered.This thesis consists of six chapters:Chapter one: addresses historical review on the variationalinequalities and then record some notation, theories that are usedthroughout this thesis.