Several phenomena in modern science and technology can be described by means of non-linear differential equations such as hyperbolic system with source term, kinetic equations and advection reaction diffusion models. These models play a fundamental role in classical physics, and in industrial applications, for example in plasma physics, granular gases, semiconductor design, or metheorolgy and geophysical problems. More recently these tools of mathematical science have been extended to interacting particle systems in order to describe soft-science problems such as: financial markets, opinon dynamics, traffic flow, crowd evacuation, cancer cells growth, biological newtwork formation.
In each of these contexts one of the fundamental aspect is to determine the conditions, and the forcing terms which highly influence the dynamics, and can be controlled.
Our project wants to analyze these themes by developing efficient numerical methods for control problems where the constrains are described by the evolution of kinetic, and hyperbolic models.
The research outlines we will develop are:
1) Control of kinetic models
(G. Albi, G. Dimarco, L. Pareschi, G. Puppo, M. Semplice, G. Visconti, M. Zanella).
2) Hamilton-Jacobi equation and applications to optimal control
(S. Cacace, E. Carlini, M. Falcone, R. Ferretti, G. Paolucci, L. Saluzzi).
3) IMEX method for differential equations
(G. Albi, G. Dimarco, L. Pareschi, S. Boscarino, G. Russo).
4) High-Order schemes for conservation and balance laws
(E. Abbate, S. Boscarino, G. Russo, G. Puppo, M. Semplice, A. Thomman, G. Visconti).
5) Mean-field optimal control and mean-field games, and uncertainty quantification
(G. Albi, S. Cacace, E. Carlini, G. Dimarco, L. Pareschi, M. Zanella).