Convergence results and optimal control problems via gap functions for n-player generalized multiobjective games with applications

Abstract

The aim of this paper is to study some new results on the convergence of solutions for controlled systems driven by generalized multiobjective games, optimal control problems where the systems are governed by generalized multiobjective games and controlled systems of traffic networks. First, we recall the controlled systems of generalized multiobjective games proposed by Hung and Keller (Math. Nachr. 296 (2023), 3676–3698). Second, we introduce gap functions and a key Assumption 3.6 using nonlinear scalarization functions for these games. Results on the lower convergence and convergence of the solutions for such problems using the key Assumption 3.6 are established. Third, we revisit optimal control problems governed by generalized multiobjective games. We investigate necessary and sufficient conditions for the convergence of solutions to optimal control problems. Finally, as a real-world application, we consider the special case of controlled systems of traffic networks. The necessary and sufficient conditions for the convergence of solutions for these problems are also obtained. Many examples are given for the illustration of our results.