The recent advancements of mobile edge computing (MEC) technologies and unmanned aerial vehicles (UAVs) have provided resilient and flexible computation services for ground users beyond the coverage of terrestrial service. In this paper, we focus on a UAV-assisted MEC system in which the UAV equipped with MEC servers is used to assist user devices in computing their tasks. To minimize the weighted average energy consumption and delay in the UAV-assisted MEC system, a LQR-Lagrange-based DDPG (LLDDPG) algorithm, which jointly optimizes the user task offloading and the UAV trajectory design, is proposed. To be specific, the LLDDPG algorithm consists of three subproblems. The DDPG algorithm is used to address the issue of UAV desired trajectory planning, and subsequently, the LQR-based algorithm is employed to achieve the real-time tracking control of UAV desired trajectory. Finally, the Lagrange duality method is proposed to solve the optimization problem of computational resource allocation. Simulation results indicate that the proposed LLDDPG algorithm can effectively improve the system resource management and realize the real-time UAV trajectory design.