A theory of conservation laws on complex networks was developed and applied in different domains as: vehicular traffic, data networks, irrigation channels, blood flow, vascular stents and supply chains.

The mathematical framework consists in conservation laws (or systems of conservation laws) on a network, modelled as a topological graph. The dynamics on arcs is thus determined while that on nodes needs to be defined. The only conservation through the node is not sufficient to determine a unique solution, thus various additional rules were considered, depending on the physics of the system under consideration.

The theory was developed proving existence of weak entropic solutions on complex networks using a wave-front tracking algorithm and innovative variation estimates. It is interesting to notice that Lipschitz continuous dependence from initial conditions holds only for some type of dynamics at nodes. This result was obtained using a generalized Finsler structure on L1.

Numerical methods were developed to reconstruct solutions on complex networks using Godunov schemes, Kinetic Schemes, Discontinuous Galerkin, Ad-hoc schemes

#### Books and surveys:

- Bressan, S. Canic, M. Garavello, M. Herty, B. Piccoli: Flows on networks: recent results and perspectives, European Mathematical Society Survey, 1 (2014), 47-111. Survey
- Garavello, B. Piccoli: Traffic flow on networks, Applied Math Series vol. 1, American Institute of Mathematical Sciences, Springfield, 2006. Link

#### Recent papers:

- Blandin, X. Litrico, B. Piccoli, A. Bayen: Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation laws, submitted to IEEE Transaction on Automatic Control.
- Herty and B. Piccoli: Numerical methods for the computation of tangent vectors to 2×2 hyperbolic systems of conservation laws, submitted to Communications in Mathematical Sciences.
- Canic, B. Piccoli, J. Qiu, T. Ren: Runge-Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks, preprint arXiv:1403.3750, to appear on Journal of Scientific Computing. pdf
- Manzo, B. Piccoli, L. Rarita: Optimal distribution of traffic flows at junctions in emergency cases, European Journal on Applied Mathematics, 23 (2012), 515-535. pdf
- Rarita, C. D’Apice, B. Piccoli, D. Helbing: Sensitivity analysis of permeability parameters for flows on Barcelona networks, Journal of Differential Equations, 249 (2010), 3110-3131. pdf

#### Other Papers:

- Garavello, B. Piccoli: Conservation laws on complex networks, Annales Institute Henri Poincare (C) Nonlinear Analysis, 26 (2009), 1925-1951. pdf
- Garavello, R. Natalini, B. Piccoli, Terracina A.: Conservation laws with discontinuous ux, Netw. Heterog. Media, 2 (2007), 159-179. link
- Bretti, R. Natalini, B. Piccoli: Numerical approximations of a traffic flow model on networks, Networks and Heterogeneous Media, 1 (2006), 57-84. pdf
- Garavello, B. Piccoli: Source-Destination Flow on a Road Network, Communications Mathematical Sciences, 3 (2005), 261-283. pdf