This paper considers using linear quadratic regulation (LQR) for multi-input control of the Automatic Transmission (AT) upshift inertia phase. The considered control inputs include the transmission input/engine torque, oncoming clutch torque, and traditionally not used off-going clutch torque. Use of the off-going clutch has been motivated by discussed Control Trajectory Optimization (CTO) results demonstrating that employing the off-going clutch during the inertia phase along with the main, oncoming clutch can improve the upshift control performance in terms of the shift duration and/or comfort by trading off the transmission efficiency and control simplicity to some extent. The proposed LQR approach provides setting an optimal trade-off between the conflicting criteria related to driving comfort and clutches thermal energy loss. It ensures tracking a linear-like profile of oncoming clutch slip speed reference, which was found to be nearly optimal based on control trajectory optimization results. A special attention is given on proper implementation of nonlinear energy loss term through LQR cost function cross term and using a clipped optimal control approach to provide that the clutches (described as torque source elements) can only dissipate energy. The LQR approach was applied to a fifth-order powertrain model and different upshift control scenarios ranging from the use of single clutch towards using both clutches and transmission input/engine torque reduction. It is shown that the LQR approach can reproduce Pareto frontiers obtained by multi-objective control parameter optimization demonstrating that apart from being used in closed loop controls, the proposed LQR approach can also be exploited for computationally efficient (off-line) optimization purposes.