Classical control theory deals with controlled ODEs and PDEs. An interesting setting, where both ODE and PDE control appear naturally, is that of multi-agent systems and crowd dynamics. Recently, control problems involving measure-valued solutions attracted a lot of attention as a tool to deal with such systems. The measure can serve as the probabilistic representation of the position of a single agent, the mean-field type limit for a large group configuration or a multi-scale representation of the agents’ population. First we describe the connections between the control of microscopic and mean-field macroscopic equations for crowds. In particular, we show the difficulties in passing to the limit for control strategies and possible multi-scale approaches. Then we turn the attention to problems directly formulated in terms of controlled dynamics of a Radon measure. We show some recent results for optimal control problems as well as a new set of evolution equations called measure differential equations.

Benedetto Piccoli is Distinguished Professor of Mathematics at Rutgers University–Camden. He got his Bachelor from University of Padua, Master and PhD from the International School of Advanced Studies of Trieste (Italy). After serving as Research Director at CNR in Rome, he joined Rutgers–Camden as the inaugural holder of the Joseph and Loretta Lopez endowed chair in Mathematics. He served as Program Director and Director of the Center for Computational and Integrative Biology and currently serves as Associate Provost for Research. Under his direction the PhD program in computational biology became the second largest in Delaware Valley. His research interests span various areas of applied mathematics and has extended to  interdisciplinary collaborations with engineers, economists, physicians, biologists and other. His research interests include social dynamics, control theory, traffic flowed on networks, crowd dynamics, math finance and systems biology. He has authored more than 260 research papers and 6 books (h-index 42, citations ca. 7000 in Google Scholar). He is the recipient of the 2009 Fubini Prize and was in the 2012 inaugural cohort of American Mathematical Society Fellows. He is the founder and Editor in Chief of the journal Networks and Heterogeneous Media.