A control device that controls a vehicle transmission device including an input drivingly coupled to an internal combustion engine, an output drivingly coupled to wheels, a shift input drivingly coupled to the input via a fluid coupling having a lock-up clutch, and a speed change mechanism disposed on a power transmission path connecting the shift input and the output, wherein the speed change mechanism is capable of performing both continuously variable shifting that continuously changes a speed ratio, and stepped shifting that changes a speed ratio in a stepwise manner, the control device including: an electronic control unit that is configured to perform, when the stepped shifting is performed, rotation maintained engagement control that controls an engagement pressure of the lock-up clutch such that a rotational speed of the input follows a predetermined target rotational speed, regardless of a change in rotational speed of the shift input.