On the optimal control problem for the k-FORQ/MCH equation with viscosity

In this paper, we investigate the optimal control problem for the k-FORQ/MCH equation with strong viscosity.We prove the existence and uniqueness of this equation under the initial and boundary conditions by Galerkin method. From these results and the Lion’s theory we deduce the existence of an optimal solution to the control problem governed by the viscous k-FORQ/MCH equation. Using augmented Lagrangian method, we deduce the first-order necessary optimality condition and two second-order sufficient optimality conditions.