Abstract: This paper concerns the exponential utility maximization problem for semi-Markov decision process with Borel state and action spaces, and nonnegative rewards. The optimal criterion is maximize the expectation of exponential utility of the total rewards in infinite horizon. Under the regular and compactness-continuity conditions, we establish the corresponding optimality equation, and prove the existence of an exponential utility optimal stationary policy by an invariant embedding technique. Moreover, we provide an iterative algorithm for calculating the value function as well as the optimal policies. Finally, we illustrate the computational aspects of an optimal policy with an example.