Optimal control of vibration of an Euler-Bernoulli beam conveying fluid

This paper investigates the optimal boundary control of vibrations in an Euler-Bernoulli beam that transports fluid at a variable velocity. The beam equation incorporates an integral term, which serves to model the viscoelastic damping effect. By utilizing the functional analytical approach first proposed by Dubovitskiĭ and Milyutin, we establish a Pontryagin-type first-order necessary optimality condition for the system under scrutiny within the framework of a fixed finite time horizon. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed boundary control strategy and to validate the established theoretical results.