Optimal distributed control of a nonlinear viscous dispersive wave equation for unidirectional propagation of a long wave

This paper is concerned with an optimal distributed control problem of a nonlinear viscous dispersive wave equation that approximately describes the unidirectional propagation of long waves. 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 a fixed final horizon case and, subsequently, a remark on the applicability of the obtained results is made for the illustration.