A control apparatus for a vehicular drive system (12; 105) provided with a step-variable transmission portion (20; 110) which is adapted to be shifted to a selected one of a plurality of speed positions having respective speed ratio values, with engaging and releasing actions of a plurality of coupling devices, and a drive power source portion (39; 103) operatively connected to an input shaft of the step-variable transmission portion in a power transmittable manner, the control apparatus being configured to implement a speed synchronizing control upon a shift-down action of the step-variable transmission portion in a coasting run of a vehicle (10; 100), wherein a rotating speed (ωi) of the input shaft of the step-variable transmission portion is raised with an input shaft torque (Ti) transmitted from the drive power source portion to the input shaft when both of a releasing-side coupling device and an engaging-side coupling device are placed in a released state in which an engaging torque (Tcb) is not generated, from a pre-shift-down synchronizing speed to a post-shift-down synchronizing speed, characterized in that said control apparatus comprises a torque control portion which is configured to be operated during the speed synchronizing control implemented after initiation of an inertia phase of the shift-down action of the step-variable transmission portion (20;110) in the coasting run of the vehicle (10; 100) upon the shift-down action of the step-variable transmission portion (20; 110) in the coasting run of the vehicle (10; 100); to set a target value (Ti*) of the input shaft torque transmitted from the drive power source portion (39; 103) to the input shaft on the basis of a temperature (Toil) of a lubricant in the step-variable transmission portion such that the target value of the input shaft torque is larger when the temperature of the lubricant is comparatively low than when the temperature is comparatively high and is larger than a dragging torque (Tloss) generated in the step-variable transmission portion, after the initiation of the inertia phase; and to increase the input shaft torque so as to coincide with the target value of the input shaft torque, after the initiation of the inertia phase. to be operated during the speed synchronizing control implemented after initiation of an inertia phase of the shift-down action of the step-variable transmission portion (20;110) in the coasting run of the vehicle (10; 100) upon the shift-down action of the step-variable transmission portion (20; 110) in the coasting run of the vehicle (10; 100); to set a target value (Ti*) of the input shaft torque transmitted from the drive power source portion (39; 103) to the input shaft on the basis of a temperature (Toil) of a lubricant in the step-variable transmission portion such that the target value of the input shaft torque is larger when the temperature of the lubricant is comparatively low than when the temperature is comparatively high and is larger than a dragging torque (Tloss) generated in the step-variable transmission portion, after the initiation of the inertia phase; and to increase the input shaft torque so as to coincide with the target value of the input shaft torque, after the initiation of the inertia phase.