Magnus integrators for solving linear-quadratic differential games

We consider Magnus integrators to solve linear-quadratic N-player differential games. These problems require to solve, backward in time, non-autonomous matrix Riccati differential equations which are coupled with the linear differential equations for the dynamic state of the game, to be integrated forward in time. We analyze different Magnus integrators which can provide either analytical or numerical approximations to the equations. They can be considered as time-averaging methods and frequently are used as exponential integrators. We show that they preserve some of the most relevant qualitative properties of the solution for the matrix Riccati differential equations as well as for the remaining equations. The analytical approximations allow us to study the problem in terms of the parameters involved. Some numerical examples are also considered which show that exponential methods are, in general, superior to standard methods.The authors acknowledge the support of the Generalitat Valenciana through the project GV/2009/032. The work of SB has also been partially supported by Ministerio de Ciencia e Innovacion (Spain) under the coordinated project MTM2010-18246-C03 (co-financed by the ERDF of the European Union) and the work of EP has also been partially supported by Ministerio de Ciencia e Innvacion of Spain, by the project MTM2009-08587.Blanes Zamora, S.; Ponsoda Miralles, E. (2012). Magnus integrators for solving linear-quadratic differential games. Journal of Computational and Applied Mathematics. 236(14):3394-3408. https://doi.org/10.1016/j.cam.2012.03.008339434082361