@article{CampanaCiaramellaBorzi2021,
  author    = {Campana, Francesca Cal{\`a} and Ciaramella, Gabriele and Borz{\`i}, Alfio},
  title     = {Nash Equilibria and Bargaining Solutions of Differential Bilinear Games},
  series = {Dynamic Games and Applications},
  volume    = {11},
  journal   = {Dynamic Games and Applications},
  number    = {1},
  doi       = {10.1007/s13235-020-00351-2},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-283897},
  pages     = {1-28},
  year      = {2021},
  abstract  = {This paper is devoted to a theoretical and numerical investigation of Nash equilibria and Nash bargaining problems governed by bilinear (input-affine) differential models. These systems with a bilinear state-control structure arise in many applications in, e.g., biology, economics, physics, where competition between different species, agents, and forces needs to be modelled. For this purpose, the concept of Nash equilibria (NE) appears appropriate, and the building blocks of the resulting differential Nash games are different control functions associated with different players that pursue different non-cooperative objectives. In this framework, existence of Nash equilibria is proved and computed with a semi-smooth Newton scheme combined with a relaxation method. Further, a related Nash bargaining (NB) problem is discussed. This aims at determining an improvement of all players' objectives with respect to the Nash equilibria. Results of numerical experiments successfully demonstrate the effectiveness of the proposed NE and NB computational framework.},
  language  = {en}
}
@article{BartschBorziFanellietal.2021,
  author    = {Bartsch, Jan and Borz{\`i}, Alfio and Fanelli, Francesco and Roy, Souvik},
  title     = {A numerical investigation of Brockett's ensemble optimal control problems},
  series = {Numerische Mathematik},
  volume    = {149},
  journal   = {Numerische Mathematik},
  number    = {1},
  doi       = {10.1007/s00211-021-01223-6},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-265352},
  pages     = {1-42},
  year      = {2021},
  abstract  = {This paper is devoted to the numerical analysis of non-smooth ensemble optimal control problems governed by the Liouville (continuity) equation that have been originally proposed by R.W. Brockett with the purpose of determining an efficient and robust control strategy for dynamical systems. A numerical methodology for solving these problems is presented that is based on a non-smooth Lagrange optimization framework where the optimal controls are characterized as solutions to the related optimality systems. For this purpose, approximation and solution schemes are developed and analysed. Specifically, for the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov-Tadmor method, a Runge-Kutta scheme, and a Strang splitting method are discussed. The resulting optimality system is solved by a projected semi-smooth Krylov-Newton method. Results of numerical experiments are presented that successfully validate the proposed framework.},
  language  = {en}
}