Appendix C: Max-Plus Convergence
1999
C.1 Introduction This appendix presents basic facts concerning the space in which the information state takes values such as algebraic structure (in which max is linear), interpretation in terms of max-plus measures, weak convergence, compactness criteria, limit identification, and metrization of the weak topology. The concepts and results discussed are max-plus analogues of probabilistic concepts and results; so there is an optimization-probability duality. Indeed, from the point of view of the theory of large deviations, this is very natural, since in that theory probability is linked with optimization. In preparing this appendix, material was drawn from several sources, given in the bibliography. We have attempted to specialize the results from the literature so that they are directly relevant for our purpose of studying the dynamical properties of possibly singular information states for use in developing the nonlinear H∞ theory, as presented in this book. The principal references used were [AQV94], [Aki95], [Aki96], [AW83], [DE97], [Jia95], [LM95], [Qua90], [Qua92], [MS92]; see also the recent papers [FM], [HM98b].