Existence of complete Lyapunov functions with prescribed orbital derivative

Complete Lyapunov functions for a dynamical system, given by an autonomous ordinary differential equation, are scalar-valued functions that are strictly decreasing along orbits outside the chain-recurrent set. In this paper we show that we can prescribe the (negative) values of the derivative along orbits in any compact set, which is contained in the complement of the chain-recurrent set. Further, the complete Lyapunov function is as smooth as the vector field defining the dynamics. This delivers a theoretical foundation for numerical methods to construct complete Lyapunov functions and renders them accessible for further theoretical analysis and development.</p

Microscopic and mesoscopic traffic models

Besides macroscopic traffic flow models, traffic modelling in freeway systems has also been treated with other general approaches, resulting in microscopic and mesoscopic models. Macroscopic models can surely represent large networks efficiently, since they adopt an aggregate representation of the traffic dynamics, but they generally lack the level of detail needed in modelling the individual drivers\u2019 behaviours and choices. Microscopic models are, instead, conceived to explicitly reproduce the drivers\u2019 responses to traffic patterns, reactions to traffic variations, interactions with other vehicles and route choices, i.e. most of the individual behaviours. Consequently, microscopic models are able to provide a lot of information about the features of traffic flow but they require a high computational effort, especially for large road networks. Mesoscopic models fill the gap between microscopic and macroscopic models, by representing the choices of individual drivers at a probabilistic level, but limiting the level of detail on driving behaviours