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The paper deals with the optimal control of the electrical drive systems with linear dependent on speed load torque. The minimum of the Joule losses in the transient period is imposed. Different types of motors are considered. The approach refers to the current control. Two types of problems depending on the final conditions are considered. The optimization is done in the sense of minimization of the losses in the transient period. Taking into account the great values of the current in the transient period, the dominant losses are the copper ones and consequently, only these losses are considered. One considers that the flux is constant and therefore, only one control current (torque producing current component in the case of the vector control) is used. The basic results for the optimal control of an electrical drive system in the above mentioned conditions are presented [ABSTRACT FROM AUTHOR]