The well-posedness of the Euler-Bernoulli model is rigorously established, encompassing both the semi-linear case within finite intervals and the linear scenario across infinite intervals. Leveraging the dynamic programming principle, we unveil that the optimal value function stands as the singular viscosity solution to the Hamilton-Jacobi-Bellman equation. Furthermore, we demonstrate the existence of an optimal control strategy across a judiciously defined space of admissible configurations, particularly emphasizing its applicability in scenarios characterized by linear quadratic costs. To assess the efficiency and accuracy, numerical experiment is done, providing a comprehensive evaluation of the performance.