 In this paper it is considered an electric drive with static torque with constant component and speed proportional component. Using the classic calculus of variations is determined the extremal control and trajectory and the overheating that ensures maximum exploitation of the system resources represented by the achievement of a maximum variation of speed in the acceleration processes  analytic and numerical model, extremal trajectory, extremal control, optimal control, overheating I. INTRODUCTION N the case of electric drives working in continuous duty type, it is necessary to perform the start%up process and in the case of those electric drives working in continuous duty type with periodical change of speed, it is necessary to perform changes of speed. To estimate the heating process at the drive system acceleration, as performance number can be adopted the maximum exploitation of the system resources. Using the classic calculus of variations can be solved this optimization problem. II. PROBLEM FORMULATION Considered an electric drive with static torque with constant component, speed and square speed proportional component [1], [2], [3] Boteanu N. Niculae is currently an Associate Professor in Faculty of Engineering in Electromechanical Environment and Industrial Informatics, Electromechanical Engineering Department,, University of Craiova, ROMANIA, e%mail address nboteanu@em.ucv.ro Marius%Constantin O.S. Popescu is currently an Associate Professor in Faculty of Engineering in Electromechanical Environment and Industrial Informatics, Electromechanical Engineering Department, University of Craiova, ROMANIA, e%mail address popescu_ctin2006@yahoo.com, Manolea Gheorghe is Professor in Faculty of Engineering in Electromechanical Environment and Industrial Informatics, Electromechanical Engineering Department,, University of Craiova, ROMANIA, e%mail address ghmanolea@em.ucv.ro Petri9or Anca is Engineers in Faculty of Engineering in Electromechanical Environment and Industrial Informatics, Electromechanical Engineering Department, University of Craiova, ROMANIA, e%mail address apetrisor@em.ucv.ro 2 0 1 2 s M =M + k ω + k ω or (1) 2 0 1 2 s F =F + k v + k v . Neglecting the electromagnetic inertia in respect of the mechanics inertia, supposing a constant inertia moment, the electric drive will be described by the general movement equation s s dω dv M=M +J , or F=F +J dt dt (2) and by the dependence between speed and acceleration. ω= ωdt ∫  or v= vdt ∫  . (3) To expand the interpretations and the conclusions, with and for the restraint of the value intervals, will be introduced relative coordinates. In this sense, considering as a reference for time the mechanical constant of time N N Jω T= M (4) and for electricity, couple and speed, their nominal values will be obtained the relative values N N N N N s 0 0 0 N N N N 2 2 1 N 1 N 2 N 2 N N N N N s t i M F ω v , , , , T I M F ω v M F M F , , M F M F kω kv kω kv , M F M F 1 2 s    i ν k k τ = = = = = = = = = = = = = = (5) and for relative acceleration there will be the relation N N v /T v /T ν ω = = ω    . (6) In the hypothesis of proportionality between the electromagnetic couple and the burden power, the equations (1), (2) and (3) in the relative coordinates it becomes [18] , = + + + + = ∫  2 0 1 2 0 1 2 s   kν kν  k k =0 ν νdτ (7) Dynamic of Electrical Drive Systems with Heating Consideration Boteanu Niculae, Popescu Marius%Constantin, Manolea Gheorghe, Anca Petri9or I INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Issue 3, Volume 3, 2009 299