MATHEMATICAL CONTROL AND doi:10.3934/mcrf.2018008 RELATED FIELDS Volume 8, Number 1, March 2018 pp. 177–193 OPTIMAL CONTROL OF URBAN AIR POLLUTION RELATED TO TRAFFIC FLOW IN ROAD NETWORKS Lino J. Alvarez-V´ azquez * Depto. Matem´atica Aplicada II, Universidade de Vigo, E.I. Telecomunicaci´ on 36310 Vigo, Spain N´ estor Garc´ ıa-Chan Depto. F´ ısica, Universidad de Guadalajara, C.U. Ciencias Exactas e Ingenier´ ıas 44420 Guadalajara, Mexico Aurea Mart´ ınez Depto. Matem´atica Aplicada II, Universidade de Vigo, E.I. Telecomunicaci´ on 36310 Vigo, Spain Miguel E. V´ azquez-M´ endez Depto. Matem´atica Aplicada, Universidade de Santiago de Compostela E. Polit´ ecnica Superior, 27002 Lugo, Spain Abstract. Air pollution is one of the most important environmental problems nowadays. In large metropolitan areas, the main source of pollution is vehicular traffic. Consequently, the search for traffic measures that help to improve pollution levels has become a hot topic today. In this article, combining a 1D model to simulate the traffic flow over a road network with a 2D model for pollutant dispersion, we present a tool to search for traffic operations that are optimal in terms of pollution. The utility of this tool is illustrated by formulating the problem of the expansion of a road network as a problem of optimal control of partial differential equations. We propose a complete algorithm to solve the problem, and present some numerical results obtained in a realistic situation posed in the Guadalajara Metropolitan Area (GMA), Mexico. 1. Introduction. Air pollution is one of the most troubling environmental prob- lems in the world today, where the untimely death of more than three million people per year has already been noted for this reason [17]. In big cities, one of the main causes of this pollution is the vehicular traffic. Governments have become aware of the problem, and we are currently witnessing the incessant search for measures to address the situation: for instance, promoting the use of public transport and electric vehicles, restricting the number of cars that can enter the center of the cities (entry restricted, for example, by even/odd number plate), applying the prohibition of entry to those cars that most pollute, etc. 2010 Mathematics Subject Classification. Primary: 49M25, 35Q93; Secondary: 90C56. Key words and phrases. Optimal control, partial differential equations, air pollution, traffic flow, road network. * Corresponding author. 177