Online optimal control of a multicarrier home energy system, including a fuel cell (FC) with combined heat and power functionality, a furnace, and a battery as an energy storage system, is presented in this paper. An appropriately defined objective function (OF) is formulated to address the operation costs of the home energy system by considering the electrical time-of-use price and time-varying electrical and thermal demand. To determine the optimal control strategy of the FC, the defined OF is solved analytically to reach a closed-form solution for the optimal operation of the FC, which results in the global optimum compared with the near-optimal point in previous works of FC scheduling. To attain the whole system optimal operation, a rule-based solution for the optimal operation of the battery is proposed according to its characteristics. The proposed method is applied to a realistic case study including real load demand and experimentally verified data of the FC. The results are compared with previous approaches, in which optimization techniques have been used to achieve near-optimal scheduling of FC with/without a battery. It is shown that the proposed control strategy results in more exact scheduling and can be applied successfully in real applications.