APPENDIX 2.

Trajectories for which the time of crisis reaches its minimum

Statement:

Let x be a state in K but outside Viab(K). If x0(.) is the trajectory starting from x = (L, P) satisfying

   Equation 23   [23]

and governed by controls u0(.) such that

   Equation 17   [17]

and if x1(.) is another trajectory starting from x and satisfying Eq. 23, then

   Equation 24   [24]

Proof:

The viability constraints on L(t) are L(t) in [Lmin; Lmax] and d L(t)/dt = u(t). So the viability constraints on L(t) impose the following constraints on u(t):

   Equation 25   [25]

Furthermore, u(t) is bounded, u(t) belongs to [-VLmax, VLmax].
The control law u0(.) satisfies these constraints and whatever the control law u1(.) satisfying these constraints, u0(y)≤ u1(y) for all states y in [Lmin; Lmax] × [0; +∞].

Furthermore,

   Equation 26   [26]

and

   Equation 27   [27]

for all states y in [Lmin; Lmax] × [0; + ∞].
Therefore, as L0(0) = L1(0) and P0(0) = P1(0), for all t in [0, T], L0(t) ≤ L1(t), P0(t) ≤ P1(t), and

   Equation 28   [28]

As

   Equation 13   [13]

then

   Equation 24   [24]