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 integratorswhich 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 themost relevant qualitative properties of the solution for the matrix Riccati differential equations as well as for the remaining equations. The analytical approximations allowus to study the problem in terms of the parameters involved. Some numerical examples are also consideredwhich show that exponential methods are, in general, superior to standard methods. © 2012 Elsevier B.V. All rights reserved.