Due to the crucial role in the development and utilization of ocean resources, research on autonomous underwater vehicles (AUVs) has received widespread attention from both academia and industry in recent years. Existing AUVs are typically expensive and have poor control accuracy, which significantly limits their development and use. Motivated by this, a low-cost, long-endurance, and high-precision AUV prototype "Sea Explorer-I" is independently developed by our laboratory. In this paper, a thorough research is conducted for "Sea-Explorer I", including its structural design, mathematical modeling, and output feedback linear quadratic tracking (LQT) control problems. The principal contributions of this paper are as follows: For "Sea-Explorer I", high-precision flow field simulations are conducted using computational fluid dynamics (CFD). This approach integrates dynamic and static grids with superimposed translational and rotational motions, enabling the calculation of the hydrodynamic coefficients required for mathematical modeling. Moreover, from the perspective of stochastic control, this paper describes the optimal trajectory tracking control problem of an AUV as an LQT problem, then the optimal LQT control strategy is derived, and the necessary and sufficient stabilization conditions are also presented. Finally, the effectiveness of the obtained results is validated by the simulation results.