Saddle point criteria for multi-dimensional control optimisation problem involving first-order PDE constraints

In this paper, the multi-dimensional control optimisation problem involving first-order PDE constraints is considered. The absolute value penalty function method is used to solve the considered problem. Further, the saddle point of the Lagrange functional defined for the aforesaid control optimisation problem is discussed. Thereafter, the exactness property of the absolute value penalty function method is studied with the help of saddle-point theory. Then, the relationship between a minimiser of the penalised multi-dimensional control optimisation problem and a saddle point of the aforesaid control problem is established under the convexity assumption. Moreover, we have verified the theoretical results with applications.