Corollary 5

In an open system with input and non-input tanks, all the eigenvalues satisfy the condition Reð Þ λ<sup>i</sup> ≤ 0.

Independently of the previous results, it is easy to observe that all the solutions corresponding to the eigenvalues with Reð Þ λ<sup>i</sup> < 0 tend to vanish when t ! þ∞.

For this purpose, when analyzing eigenvalues with Reð Þ λ<sup>i</sup> < 0, there are two cases to be considered: λ<sup>i</sup> ∈ ℜ and λi∉ℜ.

In the first case, the corresponding ODE solutions are a linear combination of the functions exp ð Þ �λit ; texp ð Þ �λit ; t 2exp ð Þ �λit ; …; <sup>t</sup> <sup>q</sup>exp ð Þ �λit � �, where the number <sup>q</sup> depends on the algebraic and geometric multiplicity of λ<sup>i</sup> (i.e., AMð Þ λ<sup>i</sup> and GMð Þ λ<sup>i</sup> ). Taking into account that λ<sup>i</sup> < 0, it follows that t <sup>n</sup>exp ð Þ! �λit <sup>t</sup>!þ<sup>∞</sup> 0, ∀n ¼ 0, 1,…, q.

In the second case—which really happens, as it will be observed later—we have λ<sup>i</sup> ¼ a þ bi ∉ℜ (with <sup>a</sup> <sup>&</sup>lt; <sup>0</sup>, b ¼6 0). The ODE solutions are a linear combination of exp ð Þ �at cos ð Þ bt ; exp ð Þ �at � sin ð Þ bt ;…; t <sup>q</sup>exp ð Þ �at cos ð Þ bt ; <sup>t</sup> <sup>q</sup>exp ð Þ �at sin ð Þg bt , where the number <sup>q</sup> depends on AMð Þ <sup>λ</sup><sup>i</sup> and GMð Þ λ<sup>i</sup> as in the other case. It is easy to prove that t <sup>n</sup>exp ð Þ �at cos ð Þ! bt <sup>t</sup>!þ<sup>∞</sup> 0 and t <sup>n</sup>exp ð Þ �at sin ð Þ! bt <sup>t</sup>!þ<sup>∞</sup> 0, ∀n ¼ 0, 1,…, q, since a < 0 .

According to the position of the Gershgorin disks for an MP-matrix (see Figure 8), the ODE solutions corresponding to an eigenvalue λi, with Reð Þ¼ λ<sup>i</sup> 0, can be analyzed.

For this purpose it is important to observe that if an eigenvalue λ<sup>i</sup> satisfies Reð Þ¼ λ<sup>i</sup> 0, then it must be λ<sup>i</sup> ¼ 0, since the Gershgorin disks look like those in Figure 8.

In this case the ODE solutions are a linear combination of the following functions: exp ð Þ �0t ; texp ð Þ �0t ; t 2exp ð Þ �0<sup>t</sup> ;…; <sup>t</sup> <sup>q</sup>exp ð Þ �0<sup>t</sup> � � <sup>¼</sup> <sup>1</sup>; <sup>t</sup>; <sup>t</sup> <sup>2</sup>;…; t <sup>q</sup> � �, where the number q depends on AMð Þ0 and GMð Þ0 . In other words, the corresponding solutions are polynomial, and so, they will not tend to vanish nor remain bounded when t ! þ∞, unless AMð Þ¼ 0 MGð Þ0 , and the polynomial becomes a constant.

Considering all these results, it is obvious that the stability of the ODE system solutions will depend exclusively on AMð Þ0 and GMð Þ0 .
