On the optimal control problem for the Novikov equation with strong viscosity

• The existence of a unique weak solution to the viscous Novikov equation is obtained.
• The existence of an optimal solution to the optimal control problem is shown.
• The first-order necessary optimality condition is deduced.
• Two second-order sufficient optimality conditions are established.

In this paper, we consider an optimal control problem for the Novikov equation with strong viscosity. Using the Faedo–Galerkin method we derive the existence of a unique weak solution to this equation. Applying Lions' theory, we obtain the existence of an optimal solution to the control problem for this equation. We also deduce the first-order necessary optimality condition. Moreover we establish two second-order sufficient optimality conditions, which require coercivity of the augmented Lagrangian functional on the whole space or on a suitable subspace.