**Appendix 3**

**Stability and Comparison Systems**

Study of stability using comparison systems [BOR 74, GRU 76].

# A3.1. Vector norms and overvaluing systems

## A3.1.1. Definition of a vector norm

Let E = R^{n} be a vector space and *E*_{1}, *E*_{2},…, *E*_{k} be subspaces of *E*:

Let *x* be a vector of R^{n} for which the projection in subspace *E*_{i} is defined by:

[A3.1]

Note that *p*_{i}(*x*) = *p*(*x*_{i}) is a scalar norm defined over subspace *E*_{i}; it becomes the vector norm:

If *x* and *y* are two vectors of space *E* and ∀*i* = 1, 2,..,*k*, then the following relations are verified:

If *k*−1 subspaces of *E*_{i} are insufficient to define the whole of space *E*, the norm vector is said to be surjective. Furthermore, if every two subspaces *E*_{i} are disjoint, the vector norm is said to be regular:

## A3.1.2. Definition of a system overvalued from a continuous process

We define a process for which the evolution into free state is described by the following equation:

The origin is assumed to be the unique equilibrium point of system [A3.1] and this system ...