Finite Element Analysis of Nash Equilibrium of Bi-objective Optimal Control Problem Governed by Stokes Equation with $L^2$-norm State-Constraints
This paper investigates the Nash equilibrium of a bi-objective optimal control problem governed by the Stokes equations. A multi-objective Nash strategy is formulated, and fundamental theoretical results are established, including the existence, uniqueness, and analytical characterization of the equilibrium. A finite element framework is developed to approximate the coupled optimal control system, and the corresponding optimality conditions for both the continuous and discrete formulations are rigorously derived and analyzed. Furthermore, \textit{a priori} finite element error estimates are obtained for the discrete problem, ensuring convergence and stability of the proposed method. The theoretical results are corroborated by numerical experiments, which demonstrate the accuracy and computational efficiency of the finite element approach.