A back propagation through time algorithm is used to optimize control variable trajectories of an automated automotive transmission for the engagement task. The optimization criterion is to maximize the end vehicle speed, and minimize the dissipated clutch energy and the shuffle mode vibration content. The optimization gives optimal profiles of the engine speed and clutch slip speed state variables, which are then used as reference variables of an optimal feedback controller. Since the engagement is carried out over a finite time horizon, a time-variant form of the linear quadratic regulator (LQR) is designed. The design is based on a transformation of the nonlinear Riccati matrix equation into a linear differential equations in the controller gains using the properties of Kronecker product. The time-variant LQR controller is examined by computer simulation and compared with the constant-gain controller.