Optimality conditions for a class of multi-objective control problems

This paper is devoted to studying a class of infinite-dimensional multi-objective control problems. First, an abstract linear system is driven by a leader control and two follower controls. For each leader control, a pair of follower controls is searched for as a Nash equilibrium of the game problem, while the aim of leader controls is to solve a constrained infinite-dimensional optimization problem. The first-order necessary condition for solutions to this optimization problem is given through a finite codimensionality condition. As applications, some multi-objective control problems for wave equations and quasi-linear parabolic equations are studied, respectively.