Viability theory: an applied mathematics tool for achieving dynamic systems' sustainability

Sustainability is considered an issue of paramount importance; yet scientists and politicians still seek to understand what it means, practically and conceptually, to be sustainable. This paper's aim is to introduce viability theory, a relatively young branch of continuous mathematics which provides a conceptual framework that is very well suited to sustainability problems.  In particular, viability theory can be used to answer important questions about the sustainability of systems, including those studied in macroeconomics, and can be used to determine sustainable policies for their management.  The principal analytical tool of viability theory is the viability kernel which is the set of all state-space points such that it is possible for evolutions starting from each of those points to remain within the system's predetermined constraints indefinitely. Although, in some circumstances,  kernel determination can be performed analytically,  most practical results in viability theory rely on graphical approximations of viability kernels, which for nonlinear and high-dimensional problems can only be approached numerically. This paper provides an outline of the core concepts of viability theory and an overview of the numerical approaches available for computing approximate viability kernels.  \vikaasa{}, a specialised software application developed by the authors and designed to compute such approximate viability kernels is presented along-side examples of viability theory in action in the spheres of bio-economics and macroeconomics.

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