This page presents the main indicators used in the mathematical theory of viability. You will find their definitions, their role in assessing the viability of a system, and application examples. Each indicator is explained in such a way as to understand how it is calculated, how to interpret it, and in what context it can be used.
Distance to the Viable Set (or Viability Distance)
Definition:
This is the distance between a given state xxx and the set of viable states V.
Formula:
$$
d(x, \mathcal{V}) = \inf_{y \in \mathcal{V}} \| x - y \|
$$
Usefulness:
The greater this distance, the further the state is from the viability conditions; it therefore measures how much a system is "at risk" of exceeding the viable constraints.
Viability Kernel
Definition:
This is the set of initial states from which there exists at least one feasible trajectory satisfying the constraints for all future times.
Formally:
$$
\text{Viab}_F(K) = \left\{ x_0 \in K \,\middle|\, \exists \ x(t), \ \dot{x}(t) \in F(x(t)), \ x(t) \in K, \ \forall t \geq 0 \right\}
$$
Usefulness:
This indicator is crucial: it allows us to identify the "safe zone" of the system in the long term.
Set-Valued Derivative (Contingent Derivative)
Definition:
This is a local indicator that assesses whether the system's dynamics are compatible with the constraints at a given time.
Simplified formula:
$$
F(x) \cap T_K(x) \neq \emptyset
$$
with TK(x) = tangent cone to the set of constraints K at x, and F(x) = system dynamics.
Usefulness:
If the intersection is empty, then the state is not viable in the short term.
