This paper presents a bio-mimetic approach to the design and control of a fish-like robot with compliant parts. One of the key contributions of this work is the use of continuous structures instead of discrete assemblies. In this framework, the motion of the robot is accomplished by copying the kinematics of a biological fish swimming in a sub-carangiform mode. The flexible part referred to as the tail is modeled as a cantilever beam with non-uniform cross-section actuated by a time varying moment. The geometrical and inertial properties of the tail are known. The method of assumed mode is used to derive the equations of motion of the tail; a relationship between the applied torque and the lateral line deflections is calculated. The expression of the torque mimicking the midline kinematics of a biological fish is then computed. A prototype implementing the proposed approach is built. Experiments are performed on a given tail in air and in water. The calculated and experimental midline deflections are then compared.