The optimal control problem with necessity condition for a viscous shallow water equation

Abstract

Abstract The optimal control problem for a shallow water equation with a viscous term is analyzed. The existence of optimal control to the control problem is investigated. The necessity condition of optimal control is derived by using the first order Gâteaux derivative of cost functional and adjoint equation. The local uniqueness of the optimal control is established by means of the second order Gâteaux derivative of cost functional. The novelty of this paper is that the necessity condition and local uniqueness of optimal control to the problem are obtained with viscous coefficient ε>0 $\varepsilon>0$.

Keywords

• Shallow water equation
• Optimal control
• Necessity optimal condition
• Local uniqueness