Basic Principles of the Mathematical Theory of Viability

Viability theory offers a new way of approaching dynamic systems. Rather than seeking to optimize a single objective, it seeks to determine the conditions under which a system can evolve sustainably without exceeding a set of constraints.

1. The Concept of Viability

A system is viable if it can evolve over time while respecting state constraints. These constraints can be:

• Ecological: a population must not fall below a critical threshold.
• Economic: a debt level must not exceed a ceiling.
• Physical: a robot must not leave a safe zone.

In other words, a system is viable if there exists at least one temporal trajectory that remains within these constraints at all times.

2. Constrained Dynamical Systems

The theory applies to dynamical systems mathematically formalized by differential equations or differential inclusions, of the form:

x˙(t)∈F(x(t))

where:

• x(t) is the state of the system at time t,
• F(x) is a multivalued vector field (allowing for several possible directions),
• The trajectory x(t) must remain within a set of constraints K⊆Rn

3. The Viable Set

The viable set is the set of initial states from which it is possible to remain within the constraints forever.

We denote this set:

ViabF(K)

It is defined by:

ViabF(K) = {x0 ∈ K | ∃x(⋅) such that x(0) = x0 and x(t) ∈ K ∀t ≥ 0}

4. Viable Trajectories

A viable trajectory is a dynamic solution that respects:

The system's evolution laws (via F),

The imposed constraints (via K).

There can be several viable trajectories for the same initial state. The goal is therefore not to choose the best one, but to understand all the possible trajectories that do not violate the constraints.

5. Associated Tools and Concepts

Here are some fundamental related concepts:

• Differential inclusions: generalize differential equations to allow for non-unique evolutions.
• Nagumo tangency conditions: allow us to characterize viable sets.
• Capture basin (viable catchment basin): the set of states from which a target can be reached without violating constraints.
• Viability kernel: the smallest set containing all viable trajectories.

6. Why is this important?

Because in reality, there isn't always an ideal solution. But it's essential to know what is possible without jeopardizing the system.

Because this approach applies equally well to ecosystems, autonomous robots, and social or economic systems.

Because it provides a robust mathematical framework for reasoning in terms of sustainability, resilience, and adaptive governance.

In summary

Viability theory is based on a simple but powerful idea: not to seek the best solution, but to avoid unacceptable solutions. It mathematically defines the set of safe states in which a system can continue to function without breaking the rules.