Generalized Wasserstein distance and its application to transport equations with source

In this article, we generalize the Wasserstein distance to measures with different masses. We study the properties of such distance. In particular, we show that it metrizes weak convergence for tight sequences. We use this generalized Wasserstein distance to study a transport equation with source, in which both the vector field and the source depend on the measure itself. We prove existence and uniqueness of the solution to the Cauchy problem when the vector field and the source are Lipschitzian with respect to the generalized Wasserstein distance

Infinite time regular synthesis

In this paper we provide a new sufficiency theorem for regular syntheses. The concept of regular synthesis is discussed in where a sufficiency theorem for nite time syntheses is proved. There are interesting examples of optimal syntheses that are very regular but whose trajectories have time domains not necessarily bounded This research is motivated by the fact that one of the main tools toward the construction of optimal syntheses is the proof of a strong sufficiency theorem. The regularity assumptions of the main theorem in are verified by every piecewise smooth feedback control generating extremal trajectories that reach the target in finite time with a finite number of switchings indeed even by more complicate syntheses like the Fuller one presenting trajectories with an infinite number of switchings. In the case of this paper the situation is even more complicate, since we admit both trajectories with finite and infinite time. It is important to notice that in spite of its complexity this situation is encountered in many simple cases like linear quadratic problems see the example of the last section and We use weak differentiability assumptions on the synthesis and weak continuity assumptions on the associated value function. However, in this paper we need the value function to be continuous at the origin see Remark for more details. The general case of synthesis generated by general piecewise smooth feedback deserves a further careful investigatio

Existence and approximation of probability measure solutions to models of collective behaviors

In this paper we consider first order differential models of collective behaviors of groups of agents based on the mass conservation equation. Models are formulated taking the spatial distribution of the agents as the main unknown, expressed in terms of a probability measure evolving in time. We develop an existence and approximation theory of the solutions to such models and we show that some recently proposed models of crowd and swarm dynamics fit our theoretic paradigm.Comment: 31 pages, 1 figur

On the continuum approximation of the on-and-off signal control on dynamic traffic networks

In the modeling of traffic networks, a signalized junction is typically treated using a binary variable to model the on-and-off nature of signal operation. While accurate, the use of binary variables can cause problems when studying large networks with many intersections. Instead, the signal control can be approximated through a continuum approach where the on-and-off control variable is replaced by a continuous priority parameter. Advantages of such approximation include elimination of the need for binary variables, lower time resolution requirements, and more flexibility and robustness in a decision environment. It also resolves the issue of discontinuous travel time functions arising from the context of dynamic traffic assignment. Despite these advantages in application, it is not clear from a theoretical point of view how accurate is such continuum approach; i.e., to what extent is this a valid approximation for the on-and-off case. The goal of this paper is to answer these basic research questions and provide further guidance for the application of such continuum signal model. In particular, by employing the Lighthill-Whitham-Richards model (Lighthill and Whitham, 1955; Richards, 1956) on a traffic network, we investigate the convergence of the on-and-off signal model to the continuum model in regimes of diminishing signal cycles. We also provide numerical analyses on the continuum approximation error when the signal cycles are not infinitesimal. As we explain, such convergence results and error estimates depend on the type of fundamental diagram assumed and whether or not vehicle spillback occurs to the signalized intersection in question. Finally, a traffic signal optimization problem is presented and solved which illustrates the unique advantages of applying the continuum signal model instead of the on-and-off model