An electric power steering apparatus that calculates a first current command value based on at least a steering torque and performs an assist-control of a steering system by driving a motor based on said first current command value, characterized in that said electric power steering apparatus comprises: a first converting section to convert said first current command value to a first rack axial force; a rack position converting section to convert a rotational angle of said motor to a judgment rack position; a rack end approach judging section to judge that a steering approaches to a rack end based on said judgment rack position and to output a rack displacement and a switching signal; a viscoelastic model following control section to generate, with a configuration of a model following control, a second rack axial force including a viscoelastic model as a reference model based on said first rack axial force, said rack displacement and said switching signal; a second converting section to convert said second rack axial force to a second current command value; and an overheat protection control section to calculate an offset, which is subtracted from said first rack axial force or an output of said viscoelastic model of said viscoelastic model following control section, based on vehicle state information; wherein said viscoelastic model following control section comprises: a feed-forward control section to output a third rack axial force by feed-forward controlling based on said first rack axial force; a feed-back control section to output a fourth rack axial force by feed-back controlling based on said rack displacement and said first rack axial force; a first switching section to switch-ON or -OFF an output of said third rack axial force by said switching signal; a second switching section to switch-ON or -OFF an output of said fourth rack axial force by said switching signal; and an adding section to output said second rack axial force by adding outputs of said first and second switching sections (121, 122); wherein said assist-control is performed based on a third current command value obtained by adding said second current command value to said first current command value so as to suppress a rack end hitting. a first converting section to convert said first current command value to a first rack axial force; a rack position converting section to convert a rotational angle of said motor to a judgment rack position; a rack end approach judging section to judge that a steering approaches to a rack end based on said judgment rack position and to output a rack displacement and a switching signal; a viscoelastic model following control section to generate, with a configuration of a model following control, a second rack axial force including a viscoelastic model as a reference model based on said first rack axial force, said rack displacement and said switching signal; a second converting section to convert said second rack axial force to a second current command value; and an overheat protection control section to calculate an offset, which is subtracted from said first rack axial force or an output of said viscoelastic model of said viscoelastic model following control section, based on vehicle state information; wherein said viscoelastic model following control section comprises: a feed-forward control section to output a third rack axial force by feed-forward controlling based on said first rack axial force; a feed-back control section to output a fourth rack axial force by feed-back controlling based on said rack displacement and said first rack axial force; a first switching section to switch-ON or -OFF an output of said third rack axial force by said switching signal; a second switching section to switch-ON or -OFF an output of said fourth rack axial force by said switching signal; and an adding section to output said second rack axial force by adding outputs of said first and second switching sections (121, 122); wherein said assist-control is performed based on a third current command value obtained by adding said second current command value to said first current command value so as to suppress a rack end hitting. a feed-forward control section to output a third rack axial force by feed-forward controlling based on said first rack axial force; a feed-back control section to output a fourth rack axial force by feed-back controlling based on said rack displacement and said first rack axial force; a first switching section to switch-ON or -OFF an output of said third rack axial force by said switching signal; a second switching section to switch-ON or -OFF an output of said fourth rack axial force by said switching signal; and an adding section to output said second rack axial force by adding outputs of said first and second switching sections (121, 122); wherein said assist-control is performed based on a third current command value obtained by adding said second current command value to said first current command value so as to suppress a rack end hitting.