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The Viability Theory

Viability theory is one of the few mathematical theories specially motivated by economic and biological sciences. Its development started in the early 1980s as a radical break with the static mathematical approaches of decision making. It has since found applications in automatic control.

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 (mutational and morphological viability);
- 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 of the inertial (measuring a priori transition costs of control);
- algebra and tensor dynamics to study the evolution of networks and minimize the connectionist complexity.

Mathematics research in viability theory and its applications are undertaken by academic researchers and engineers of VIMADES and LASTRE .

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