A new approach for solving the multidimensional control optimization problems with first-order partial differential equations constraints

The purpose of this paper is to provide the linearization technique to solve the multidimensional control optimization problem (MCOP) involving first-order partial differential equation (PDEs) constraints. Firstly, we use the modified objective function approach for simplifying the aforesaid extremum problem (MCOP) and show that the solution sets of the original control optimization problem and its modified control optimization problem (MCOP) ${}_{\Omega }$ are equivalent under convexity assumptions. Further, we use the absolute value exact penalty function method to transform (MCOP) ${}_{\Omega }$ into a penalized control problem (MCOP) ${}_{\Omega \varrho }$. Then, we establish the equivalence between a minimizer of the modified penalized optimization problem (MCOP) ${}_{\Omega \varrho }$ and a saddle point of the Lagrangian defined for the modified optimization problem (MCOP) ${}_{\Omega }$ under appropriate convexity hypotheses. Moreover, the results established in the paper are illustrated by some examples of MCOPs involving first-order PDEs constraints.