**4. Conclusions**

with the notation

we have

when

*<sup>P</sup>*<sup>0</sup> <sup>¼</sup> *<sup>P</sup>*<sup>12</sup> � *<sup>P</sup>*<sup>11</sup> *P*<sup>12</sup> � *P*<sup>22</sup>

*<sup>∂</sup><sup>x</sup> <sup>f</sup>*ð Þ *<sup>x</sup>* <sup>&</sup>lt;<sup>0</sup>

Taking into account that K,G are also real parameters, the inequality (38)

*<sup>P</sup>* <sup>¼</sup> <sup>1</sup> <sup>1</sup>

1*=*<sup>2</sup> 1 � �

which is a feasible condition since from the model we have �1<*K* <1, *G* ∈ *R*. Now we can verify if the test value (39) of P can produce a feasible CLF function V. LMI approach has very reliable numeric tools [14, 15]. We choose for the present

where A is the system matrix and P is the matrix in the definition (36) of CLF.

2*P*<sup>11</sup> þ 2*P*12*KG P*11*G* þ *KGP*<sup>22</sup>

!

*P*<sup>11</sup> þ *P*22*KG* 2*P*12*G* � 2*P*<sup>22</sup>

*<sup>A</sup>* <sup>¼</sup> <sup>1</sup> *KG G* �1 � �

Replacing with the test values from (39) for P, the condition "*ATP* <sup>þ</sup> *PA* negative semidefinite" implies the following conditions for the parameters

*<sup>G</sup>* <sup>&</sup>lt; � <sup>1</sup>

<sup>2</sup> <sup>∙</sup> <sup>1</sup> <sup>1</sup> <sup>þ</sup> *<sup>K</sup> :*

Thus we can conclude that, in feasible conditions for the parameters, the model (35) can admit a CLF V and thus the model can be globally stabilizable at the origin.

*=*2

implies few tests. Therefore for the moment we choose the matrix P like

aim the basic matrix inequality, namely the Lyapunov inequality:

With this choice we find the condition

In our model (35), the matrix A is given by:

*<sup>A</sup>TP* <sup>þ</sup> *PA* <sup>¼</sup>

These conditions are feasible, for a negative G.

and thus we obtain

**116**

The sign of the form (37) is of course dominated by the coefficient of x2

*∂V*

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

*:*

*P*<sup>11</sup> þ *KGP*<sup>12</sup> < 0*:* (38)

*KG* < � 2, (40)

*ATP* <sup>þ</sup> *PA* <sup>≤</sup><sup>0</sup> (41)

*KG* < � 2, (44)

*:* (39)

, (42)

*:* (43)

. Thus,

In the present chapter, the aim of stabilizing the two-dimensional dynamical systems arising from excitable media is taken into account.

Stabilizing a dynamical system from excitable media is not an easy task, especially because of the sensitive dependence of these models on initial conditions. The mixing flow models are not an exception, since the repartition of their parameters has a great influence on the model trajectory behavior.

The Lyapunov function is a strong appliance in studying stability. Still, constructing it is not easy. This is why approaching this for controllable systems make easier the task of searching the stability.

Searching for a Lyapunov function becomes easier if we take into account the iterative algorithm in the "sum of squares" programming. For the mixing flow example of this chapter, the test values (39) for the matrix P shown that in feasible conditions for the parameters, we can construct a CLF for the model (35) and thus the model is globally stabilizable in the origin.

Constructing the CLF functions for controllable systems provide interesting results, and this is shown by the above 2d mixing flow model. The 2d mixing flow model is controllable in some slightly perturbed forms [12, 13]. Therefore, in this chapter the standpoint is that we started with the simple control u taken on both equations of the dynamical system. The control is not implied in the further calculus on finding the CLF V, which made easier the above analysis. A next aim is to find CLF functions for other perturbations cases of the mixing flow model, on one hand, and also to approach iterative LMI algorithms for finding the matrix P. The test values (39) for the matrix P give rise to an easier further statement of a SDP problem for the mixing flow dynamical system.

Thus, this chapter adds a new standpoint in the mathematical context of the mixing flow dynamical system, by considering it as a polynomial system of differential equations. For a model whose mathematical apparatus is rather associated with mechanics and fluid mechanics issues, approaching the stability by searching Lyapunov functions with computational algebraic appliances is new. This approach could open a new way in the study of stabilizing of the mixing flow dynamical model - which is a far from equilibrium model, and also in the qualitative analysis of the differential systems associated to transport phenomena.
