*2.2. Interval Analysis*

Uncertain parameters are defined by intervals that contain real values of unknown variables in a guaranteed way.

**Definition 1.** *An interval vector* [*x*] *is determined by [45]*

$$\mathbb{P}[\mathbf{x}] = [\underline{\mathbf{x}}, \overline{\mathbf{x}}] = \{a | \underline{\mathbf{x}} \le a \le \overline{\mathbf{x}}, \,\underline{\mathbf{x}} \,\, \overline{\mathbf{x}} \in \mathbb{R}^n\}.$$

**Lemma 1.** *Let x* <sup>∈</sup> <sup>R</sup>*<sup>n</sup> be an interval vector for some x*, *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup> and A* <sup>∈</sup> <sup>R</sup>*m*×*n. Then [29]*

$$A^+\underline{\underline{x}} - A^-\overline{\underline{x}} \le Ax \le A^+\overline{\underline{x}} - A^-\underline{\underline{x}}.$$

**Definition 2.** *The theory of monotone systems states that the solutions to the below system for given x*(0) <sup>≥</sup> <sup>0</sup> *constructed by a matrix A* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* <sup>+</sup> *are non-negative,*

$$\begin{aligned} \mathfrak{x}(k+1) &= A\mathfrak{x}(k) + w(k), \\ \mathfrak{x} \in \mathbb{R}^n, \ w: \mathbb{Z}\_+ \to \mathbb{R}\_+^n, \ k \in \mathbb{Z}\_+, \ k \ge 0. \end{aligned}$$

*and the system is referred to as cooperative or non-negative [46].*
