Achieving precise control over self-propelled undulatory swimmers requires a deep understanding of their intricate dynamics. This paper presents a method for addressing optimal control problems in this context by leveraging surrogate models. We develop a Navier-Stokes solver using a volume penalization method to simulate the fluid-structure interaction inherent in swimming dynamics. An offline phase generates training data through open-loop simulations across a defined range of control inputs, enabling the training of a surrogate model. This model significantly reduces computational costs, particularly in optimization and control contexts. Utilizing these surrogate models, we compute control strategies to address two key challenges: precise velocity tracking and optimizing swimmer efficiency. Firstly, we employ model predictive control (MPC) to enable velocity tracking against a reference signal, allowing swift adjustments of the swimmer's frequency and amplitude. Secondly, we tackle the minimization of the swimmer's cost of transport, resulting in a solution akin to a burst-and-coast strategy. Despite achieving energy performance comparable to continuous swimming cases, mismatches between the surrogate model and the high fidelity simulation significantly impact the quality of the obtained solution. This work sheds light on the potential of surrogate models in optimizing self-propelled swimming behavior and underscores the importance of addressing model mismatches for more accurate control strategies in the future.