Optimal distributed controls of a class of nonlinear dispersive equations with cubic nonlinearity

This paper is devoted to the optimal distributed control problem governed by a class of nonlinear dispersive equations with cubic nonlinearity, which contains the famous Novikov equation as special case. We first investigate the existence and uniqueness of the weak solution for the controlled system, and then find an optimal solution for the controlled system with the generalized cost functional. Moreover, by means of the method suggested by A.Ya. Dubovitskii and A.A. Milyutin, we establish the first order necessary optimality condition of optimal control for the controlled system in the fixed final horizon case.