Viability Theory is one of the few mathematical theories relying on set-valued analysis instead of the usual single-valued one of classical analysis. Its development, motivated specially by economic and biological sciences, started in the early 1980s and since then has found numerous applications. Mathematics research in Viability Theory and its applications are undertaken by academic researchers and engineers of LASTRE.
In a nutshell, Viability Theory investigates evolutions:
· constrained to adapt to an environment;
· evolving under several kinds of uncertainty;
· using for this purpose controls, subsets of controls, and, in the case of networks, connectionist matrices;
· regulated by feedback laws (static or dynamic) that are then "computed" according to given principles, such as the inertia principle, intertemporal optimization, etc.;
· co-evolving with their environment;
· corrected by introducing adequate controls (viability multipliers) when viability or capturability is at stakes.
The mathematical techniques of viability theory rely on:
·differential inclusions (with memory);
· calculus of connections (and many-to-many applications Mutation analysis);
· control theory and differential games;
· set-valued numerical analysis;
· equations governing the morphological evolution of forms and allowing to analyze the co-evolution of the variables and constraints;
· impulse differential inclusions (hybrid continuous / discrete time);
· optimization (measuring a priori transition costs of control);
· algebra and tensor calculus to study the evolution of networks and minimize the connectionist complexity.
Investigations in GAIA-TROP will concern the evaluation of uncertain evolutionary systems by building qualitative and quantitative indicators taking into account prescribed constraints and objectives. The regulation map (retroaction, feedbacks) providing the controls or the commands piloting the evolutions so as to fulfill desired conditions can also be built whenever such evolutions exist. Interdisciplinary knowledge and agreement on constraints and objectives will be discussed and established. However, the dynamics describing agents’ behaviors or biological and ecological “processus” may be much difficult to represent, their disclosure through a synthesis and sharing of the existing knowledge will be actually a preliminary task to be tackled by the GAIA-TROP interdisciplinary research group.
The aim of our approach consists in a decision support tool dedicated to agriculture in an insular context in order to better understand the mechanisms that underly agro-systems and to validate the decisions so as to ensure, if possible, viability in the presence of global changes and other types of uncertainty.