Weak solution of the Novikov equation and optimal control

The aim of this paper is to discuss optimal distributed control problem governed by the Novikov equation. We first investigate the existence and uniqueness of weak solution for the controlled system. In contrast to the existing research result, the proof of solution mapping is local Lipschitz continuous that is one major novelty. Moreover, using the well-posedness result, we obtain a unique optimal control and optimal solution for the controlled system. At last, we also demonstrate the sufficient and necessary optimality condition of an optimal control by means of the optimal control theory, not limited to the necessary condition, which is another significant improvement of this paper.