This paper studies the optimal controller synthesis of an energy-transmission-feedback Linear-Quadratic-Gaussian (LQG) system with imperfect acknowledgments and energy harvesting constraints. The filter, which is provided with several energy options, is capable of receiving imperfect feedback information from the controller. The aim is to jointly select the transmission energy and the controller that would maintain an optimal balance between the control performance and the transmission cost. Firstly, this paper separates the controller synthesis problem from the energy selection problem. Then, the optimal controller synthesis subproblem is characterized by a Riccati equation with continuous packet loss. Simultaneously, the energy selection problem of imperfect feedback is transformed into Markov decision process (MDP) of perfect acknowledgments via the iteration of state information. And the optimal energy selection policy is found by solving a Bellman equation. Ultimately, the effectiveness of the above method is verified through a simulation.