A max-plus based fundamental solution for a class of discrete time linear regulator problems

Efficient Riccati equation based techniques for the approximate solution of discrete time linear regulator problems are restricted in their application to problems with quadratic terminal payoffs. Where non-quadratic terminal payoffs are required, these techniques fail due to the attendant non-quadratic value functions involved. In order to compute these non-quadratic value functions, it is often necessary to appeal directly to dynamic programming in the form of grid- or element-based iterations for the value function. These iterations suffer from poor scalability with respect to problem dimension and time horizon. In this paper, a new max-plus based method is developed for the approximate solution of discrete time linear regulator problems with non-quadratic payoffs. This new method is underpinned by the development of new fundamental solutions to such linear regulator problems, via max-plus duality. In comparison with a typical grid-based approach, a substantial reduction in computational effort is observed in applying this new max-plus method. A number of simple examples are presented that illustrate this and other observations.