Corollary 1

In an open system, being the ith tank an input one, the Gershgorin circle corresponding to the ith row looks like the disk in Figure 6.

Independently of the previous results, it is easy to observe that all the solutions corresponding

For this purpose, when analyzing eigenvalues with Reð Þ λ<sup>i</sup> < 0, there are two cases to be

In the first case, the corresponding ODE solutions are a linear combination of the functions

<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

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 �

According to the position of the Gershgorin disks for an MP-matrix (see Figure 8), the ODE

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

In this case the ODE solutions are a linear combination of the following functions:

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 ,

Considering all these results, it is obvious that the stability of the ODE system solutions will

In the previous section, some particular cases with λ<sup>i</sup> ¼ 0 and/or λ<sup>i</sup> ¼ a þ bi ∉ℜ (with a < 0, b 6¼ 0) were considered. A first question to analyze is if there exists an MP that satisfies any of these conditions. For this purpose, let us consider the closed MP of Figure 9, in which ODE system can be written as , and the corresponding MP-matrix is

, and <sup>c</sup> <sup>¼</sup> <sup>Φ</sup>

V3

<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>

Square Matrices Associated to Mixing Problems ODE Systems

http://dx.doi.org/10.5772/intechopen.74437

<sup>2</sup>;…; t

<sup>q</sup> � �, where the number

. If Φ and Vi are chosen such that a ¼ 1,

<sup>n</sup>exp ð Þ �at cos ð Þ! bt <sup>t</sup>!þ<sup>∞</sup>

0 and

55

to the eigenvalues with Reð Þ λ<sup>i</sup> < 0 tend to vanish when t ! þ∞.

<sup>n</sup>exp ð Þ! �λit <sup>t</sup>!þ<sup>∞</sup>

and GMð Þ λ<sup>i</sup> as in the other case. It is easy to prove that t

0, ∀n ¼ 0, 1,…, q, since a < 0 .

solutions corresponding to an eigenvalue λi, with Reð Þ¼ λ<sup>i</sup> 0, can be analyzed.

must be λ<sup>i</sup> ¼ 0, since the Gershgorin disks look like those in Figure 8.

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>q</sup>exp ð Þ �at cos ð Þ bt ; <sup>t</sup>

2exp ð Þ �λit ; …; <sup>t</sup>

considered: λ<sup>i</sup> ∈ ℜ and λi∉ℜ.

exp ð Þ �λit ; texp ð Þ �λit ; t

λ<sup>i</sup> < 0, it follows that t

<sup>n</sup>exp ð Þ �at sin ð Þ! bt <sup>t</sup>!þ<sup>∞</sup>

exp ð Þ �0t ; texp ð Þ �0t ; t

�a 0 a b �b 0 0 c �c 1

0 B@

and the polynomial becomes a constant.

depend exclusively on AMð Þ0 and GMð Þ0 .

7. Several questions and a conjecture

CA, being <sup>a</sup> <sup>¼</sup> <sup>Φ</sup>

V<sup>1</sup> , <sup>b</sup> <sup>¼</sup> <sup>Φ</sup> V2

sin ð Þ bt ;…; t

t
