This paper explores stability issues for operational route guidance control strategies for vehicular traffic networks equipped with advanced information systems, and develops a general procedure for the stability analysis of the associated dynamic traffic assignment (DTA) problems. The route guidance control strategies are modeled as dynamical systems, and the associated solution procedure enables computational tractability for real-time deployment. An important study insight is that the Lyapunov functions for the route guidance control models are their corresponding objective functions under DTA. This overcomes the key difficulty of constructing meaningful Lyapunov functions for DTA problems.