ADDITIVE DECOMPOSITION OF MATRICES AND OPTIMIZATION PROBLEMS ON INFINITE TIME INTERVALS

TitleADDITIVE DECOMPOSITION OF MATRICES AND OPTIMIZATION PROBLEMS ON INFINITE TIME INTERVALS
Publication TypeJournal Article
Year of Publication2008
AuthorsLEIZAROWITZ ARIE, LENGER BENJAMIN
JournalDynamic Systems and Applications
Volume17
Start Page283
Pagination20
Date Published2008
ISSN1056-2176
Abstract

We discuss control systems over finite and countable state spaces defined on an infinite time horizon, where, typically, all the associated costs become unbounded as the time grows indefinitely. We consider the limit behavior, as n → ∞, of the expression Pn−1 i=0 v(zi , zi+1) for programs {zi}∞ i=0 in a finite state space X = {xi} N i=1, where v(xi , xj ) is the transition cost from state xi to state xj . To construct optimal programs we will establish and employ an additive decomposition formula which is of the form V = µJ + pηT − ηp T + Θ. In this expression µ is a scalar, J is a matrix that satisfies Jij = 1 for every 1 ≤ i, j ≤ N, p and η are N-dimensional column vectors such that ηi = 1 for all 1 ≤ i ≤ N, and Θ is a matrix satisfying min1≤j≤N Θij = 0 for every 1 ≤ i ≤ N. We will show how to compute µ, p and Θ in time of order O(N5 ). Also, we will discuss infinite horizon optimization problems for certain non-autonomous control systems.

PDFhttps://acadsol.eu/dsa/articles/17/DSA-2007-283-302.pdf
Refereed DesignationRefereed