An optimal distributed control problem of the viscous generalized Camassa-Holm equation

In this work, an optimal distributed control problem of the viscous generalized Camassa-Holm equation is considered. By the Dubovitskii and Milyutin functional analytical approach, we prove the Pontryagin maximum principle of the investigational system. The necessary condition for optimality is established for the controlled object in the fixed final horizon case and, subsequently, a remark on how to apply the obtained results is made as an illustration.