This paper addresses black-box system optimization using Bayesian system equation modeling based on a Gaussian process model. Unlike the conventional Bayesian optimization framework, which typically updates predictive statistical models for black-box objective functions during the optimization process, the proposed framework updates the statistical model of the black-box system equation rather than the objective function. This allows the previously obtained system model to be directly reused if the objective function needs to be modified for re-optimization. The proposed method incorporates an adjustment mechanism into the acquisition system function derived from the predictive model, adapting to a predefined system evaluation limit as the termination criterion, making it effective in high-cost evaluation scenarios.