Proposition 1. The dynamics (3) leaves invariant the 3D-space D

(m — ka)

П = (s, x,xd) s + x + ———————— xA

( m — a)

Proof. The invariance of R+ is guaranteed by the following properties:

s = 0 ^ s = kax > 0,

x = 0 =>- x = 0,

xd = (m — a) x > 0.

Consider the quantity M = s + x + ^. One can easily check from equations (3) that one has M = 0, leading to the invariance of the set П. |

Let s be the number s = ц 1 (m) or + to .

Proposition 2. The trajectories of dynamics (3) converge asymptotically toward an equilibrium point

s*, 0, ——p — (s0 + — t0 — s*) m — ka

with s* < min(so + xo, s).

Proof. The invariance of the set О given in Proposition 1 shows that all the state variables remain bounded. From equation xd = (m — a)x with m > a, and the fact that xd is bounded, one deduces that x(-) has to converge toward 0, and xd(•) is non increasing and converges toward x* such that x* Є [0, (so + xo)(m — a) / (m — ka)]. Then, from the invariant defined by the set О, s(-) has also to converges to some s* < so + xo. If s* is such that s* > s, then from equation x = (y(s) — m)x, one immediately see that x{-) cannot converge toward 0. |