Optimal control problem with unilateral constraints for longitudinal vibration of a viscoelastic valve

This paper is dedicated to the study of an optimal distributed control problem with unilateral constraints for a viscoelastic valve vibrating longitudinally and having its motion limited by rigid obstacles at both ends. By the Dubovitskii and Milyutin functional analytical approach, we prove the Pontryagin maximum principle of the investigational system and establish the necessary condition for optimality for the controlled object in fixed final horizon case. In contrast to the finite-dimensional setting, the maximum principle for the infinite-dimensional system does not generally hold as a necessary condition for optimal control. In this paper, we commit ourselves to infinite-dimensional generalizations of the maximum principle and aim at the optimal control theory of partial differential equations. And at the end of the article, we also remark the applicability of the obtained results.