LP well-posed controlled systems for bounded quasi-equilibrium problems and their application to traffic networks

Abstract

The aim of this paper is to establish the Levitin–Polyak well-posedness (LP well-posedness, for short) for two new classes of controlled systems of the bounded quasi-equilibrium problems, and two new classes of associated optimal control problems. First, we introduce two classes of the bounded quasi-equilibrium problems, and show, under suitable conditions, the equivalence between the LP well-posedness and existence of solution to these problems. Results on metric characterization of the LP well-posedness and LP well-posedness in the generalized sense for such problems in terms of the behavior of the approximate solution sets are provided. Second, we establish two classes of optimal control problems for systems described by the generalized bounded quasi-equilibrium problems. We also studied the LP well-posedness and LP well-posedness in the generalized sense for these problems. Finally, as a real-world application, we study the special case of controlled systems of the bounded traffic network problems.