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International Journal of Computer Games Technology
Volume 2009, Article ID 162450, 12 pages
http://dx.doi.org/10.1155/2009/162450
Research Article

A Shortest-Path Lyapunov Approach for Forward Decision Processes

1Center for Computing Research, National Polytechnic Institute, Avenue Juan de Dios Batiz s/n, Edificio CIC, Col. Nueva Industrial Vallejo, 07738 Mexico City, Mexico
2Center for Applied Science and High Technology Research, National Polytechnic Institute Legaria 69 Col. Irrigación, 11500 Mexico City, Mexico

Received 1 May 2008; Accepted 7 September 2008

Academic Editor: Abdennour El Rhalibi

Copyright © 2009 Julio B. Clempner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In previous work, attention was restricted to tracking the net using a backward method that knows the target point beforehand (Bellmans's equation), this work tracks the state-space in a forward direction, and a natural form of termination is ensured by an equilibrium point 𝑝 βˆ— ( 𝑀 ( 𝑝 βˆ— ) = 𝑆 < ∞ a n d 𝑝 βˆ— β€’ = βˆ… ) . We consider dynamical systems governed by ordinary difference equations described by Petri nets. The trajectory over the net is calculated forward using a discrete Lyapunov-like function, considered as a distance function. Because a Lyapunov-like function is a solution to a difference equation, it is constructed to respect the constraints imposed by the system (a Euclidean metric does not consider these factors). As a result, we prove natural generalizations of the standard outcomes for the deterministic shortest-path problem and shortest-path game theory.