ILLUSION OR REALITY? THE MATHEMATICS OF ATTAINABLE GOALS AND IRREDUCIBLE UNCERTAINTIES

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Abstract
In complex, high-dimensional systems, it is usually far from obvious what states are attainable within the constraints on admissible actions. Similarly, when only parts of the system are physically measurable, a vital practical as well as philosophical question arises as to how much inherent uncertainty remains in determining the true state of the system. In system-theoretic jargon, these are problems of reachability and constructibility. This paper presents an overview of the current mathematical state-of-the-art as it relates to the reachability/constructibility question. Particular emphasis is given to those results which seem most useful for dealing with practical system problems. After an introductory section to motivate the subject, a survey of the principal mathematical results for linear systems is given, along with a discussion of multidimensional nonlinear problems. The paper concludes with several actual problems from ecology, urban systems, water resource systems, and transportation networks where reachability and/or constructibility questions play an important role in the analysis.

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