Exponential stability of globally projected dynamic systems

In this paper, we further analyze and prove the stability and convergence of the dynamic system proposed by Friesz et al.(1994), whose equilibria solve the associated variational inequality problems. Two sufficient conditions are provided to ensure the asymptotic stability of this system with a monotone and asymmetric mapping by means of an energy function. Meanwhile this system with a monotone and gradient mapping is also proved to be asymptotically stable using another energy function. Furthermore, the exponential stability of this system is also shown under strongly monotone condition. Some obtained results improve the existing ones and the given conditions can be easily checked in practice. Since this dynamic system has wide applications, the obtained results are significant in both theory and applications.