Optimal Distributed Control Problem for the b-Equation

This paper is concerned with the optimal distributed control problem governed by b-equation. We firstly investigate the existence and uniqueness of weak solution for the controlled system with appropriate initial value and boundary condition. By contrasting with our previous result, the proof without considering viscous coefficient is a big 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 obtain 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.