Optimal control and optimality condition of the Camassa–Holm equation

This paper is devoted to the optimal distributed control problem described by Camassa–Holm equation. We firstly investigate the existence and uniqueness of weak solution for the controlled system with appropriate initial value and boundary condition. By contrast with our previous result, the proof without considering viscous coefficient is a bigger improvement. Secondly, based on the well-posedness result, we find a unique optimal control for the controlled system with the quadratic cost functional. Moreover, by means of the optimal control theory, we establish the sufficient and necessary optimality condition of an optimal control, which is another major novelty of this paper. Finally, we also present the optimality conditions corresponding to two physical meaningful distributive observation cases and give an illustration about how to numerically apply the obtained results.