Scale of viability and minimal time of crisis (Q1380729)

The authors consider a differential inclusion with a given constant (a closed set \(K\)). The problem consists to associate to any initial condition \(x_0\in K\) the minimal time such that a trajectory starting from \(x_0\) stays outside \(K\). This is called by the authors the ``time of crisis'' and is defined by: \[ \text{Min}\{\lambda[t/x(t)\not\in K]\text{ where }x(\cdot)\text{ describes all solutions to }x(t)\in F(x(t)),\;x(0)= x_0\} \] (where \(\lambda\) is the real Lebesgue measure). This ``time of crisis'' can be viewed as a value-function which associates to any \(x_0\) its minimal time spent outside \(K\). The main result of the present paper consists in proving that to determine the ``time of crisis'' is equivalent to solve a viability problem in a space variable with one dimension more. More precisely, the epigraph of the ``time of crisis'' is the viability kernel of some set for suitable differential inclusion. This approach consisting to reduce the determination of a value-function to the determination of a viability kernel is fruitful and has been already carried out for various optimal control problems. This allows to build algorithms. Here an algorithm is provided, its convergence is proved, and a simple economic example is studied, up to the numerical computation. The authors also give an equivalent formulation of their main result in terms of viscosity solution to a suitable PDE. The interest of this ``time of crisis'' lies in the ability to deal with ``relaxed'' constraints and to give a measure of the ``violation'' of constraints. The authors have done a very interesting paper introducing this notion and studying it using tools of viability theory.