History and Context of the Mathematical Theory of Viability

1. Origins of the Theory

The mathematical theory of viability was initiated in the 1980s by the French mathematician Jean-Pierre Aubin. It arose from a fundamental need: to understand and formalize the capacity of a dynamic system to respect constraints over time, without necessarily seeking to optimize a specific objective.

Unlike classical optimization approaches, which aim for a precise result, viability focuses on the long-term sustainability of the system, its ability to survive in a constrained and often uncertain environment.

2. Scientific and Philosophical Context

At the end of the 20th century, scientific challenges evolved: modeling sustainability, managing natural resources, preserving biodiversity, stabilizing ecosystems… These are all issues that go beyond the scope of pure optimization and require tools to ensure compatibility between trajectories and constraints.

The theory of viability is part of this dynamic. It draws inspiration from several fields:

Mathematical analysis (differential inclusions, convex geometry),

Control theory,

Mathematical biology and ecology,

Sustainable and social economics.

It proposes a new way of thinking: rather than seeking the best solution, seeking viable solutions.

3. Formalization and Early Work

Jean-Pierre Aubin's early work defined the fundamental concepts:

Viable set: a set of states from which there exists at least one trajectory that respects the constraints for any future time.

Differential inclusion: a mathematical tool used to model complex dynamics with multiple speeds or choices.

Capture basin: the set of initial states that can reach a target while remaining viable.

These ideas were formalized in his seminal work, Viability Theory, published in 1991, which remains a key reference in the field.

4. Contemporary Developments

Since its inception, viability theory has expanded into numerous fields:

Robotics and autonomous systems: ensuring the safety of a robot in a changing environment.

Economics and resource management: balancing production and conservation.

Resilience models: understanding a system's capacity to cope with disturbances.

It has also given rise to interdisciplinary projects led by the CNRS (French National Centre for Scientific Research) or the European Union, and to tools such as ViabLab, a virtual laboratory dedicated to experimenting with viable models.

5. Current Challenges and Relevance

Today, the mathematical theory of viability is particularly relevant in a world facing planetary boundaries: climate change, social crises, and technological uncertainties.

It provides a rigorous framework for reasoning about the sustainability of human, economic, ecological, and technological trajectories. Its approach is particularly well-suited to systems that are:

Complex,

Interdependent,

Constrained by operating rules,

Evolving in an uncertain environment.

Key takeaway:

Viability theory does not seek to maximize an objective, but rather to guarantee the survival of a system within a defined framework. It is a science of sustainable possibility, which is finding concrete applications today in addressing the major challenges of the 21st century.