**3. Existence of control Lyapunov function for dynamical systems from excitable media**

#### **3.1 Recent results**

The concerning for the dynamical models arising from excitable media is not new. The dynamical systems modeling the mixing flow have an important place, because of the complexity of the model. It is in fact about far from equilibrium class of models, with a very sensitive behavior to initial conditions.

Let us recall the statistical idea of a flow, generally represented by the application

$$\mathbf{x} = \Phi\_t(\mathbf{X}),\\\mathbf{X} = \Phi\_{t=0}(\mathbf{X}).\tag{23}$$

That means, X is mapped in x after a time t. In continuum mechanics, the relation (23) is named *flow*, it is a diffeomorphism of class C<sup>k</sup> and must satisfy the relation

$$0 < J < \infty,\\
J = \det\left(\frac{\partial \mathbf{x}\_i}{\partial X\_j}\right),\\
J = \det(D\Phi\_t(\mathbf{x})) \tag{24}$$

where D denotes the derivation operation with respect to the reference configuration, in this case with **X**. If the Jacobean J is unitary, it is said we have an isochoric flow. The relation (24) implies two particles, **X**<sup>1</sup> and **X**<sup>2</sup> which occupy the same position **x** at a given moment, or a particle which splits in two parts. That means, non-topological motions like break up or disintegration *are not allowed.*

The mixing flow is a special type of flow, implying a basic fluid (water) in which a biological material is moving (mixing) in different conditions and with different velocities. Therefore, the *stretching and folding* are strongly related phenomena. With respect to X there is defined the basic measure of deformation, the *deformation gradient F* [10]:

*Qualitative Analysis for Controllable Dynamical Systems: Stability with Control Lyapunov… DOI: http://dx.doi.org/10.5772/intechopen.96872*

$$\mathbf{F} = \left(\nabla\_X \Phi\_t(\mathbf{X})\right)^T,\\ F\_{\vec{\eta}} = \left(\frac{\partial \mathbf{x}\_i}{\partial X\_j}\right). \tag{25}$$

For a material filament and correspondingly for a material surface, in a mixing flow, there are defined another two basic deformation measures, the *length deformation λ* and *surface deformation η*. In this context, a specific analysis for deformations of infinitesimal elements is the so-called "*good mixing concept*", related to the boundaries of the quantities λ and η. The class of flows with a special form of F is of very large interest in the literature, as it contains the so-called "constant stretch history motion" (CSHM flows). Details can be found in [10].

When studying the mixing flow phenomena, one starts from the widespread kinematic 2d mixing flow

$$\begin{cases}
\dot{\varkappa\_1} = G\varkappa\_2 \\
\dot{\varkappa\_2} = K G\varkappa\_1,
\end{cases} \quad -1 < K < 1,\\
G \in \mathcal{R}.\tag{26}$$

Although this is a linear model, when associating the corresponding initial condition.

$$\mathbf{x}\_1(\mathbf{0}) = \mathbf{x}\_1(t=\mathbf{0}) = X\_1; \mathbf{x}\_2(\mathbf{0}) = \mathbf{x}\_2(t=\mathbf{0}) = X\_2\tag{27}$$

it is obtained a complex solution for the Cauchy problem (26)-(27) [11]. From geometric standpoint, the streamlines of the above model satisfy the relation *x*2 <sup>2</sup> � *<sup>K</sup>* <sup>∙</sup> *<sup>x</sup>*<sup>2</sup> <sup>1</sup> ¼ *const:* and this is corresponding to some ellipses with the axes rate 1 ∣*K*∣ � �<sup>1</sup>*=*<sup>2</sup> if K is negative, and to some hyperbolas with the angle *<sup>β</sup>* <sup>¼</sup> arctan <sup>1</sup> ∣*K*∣ � �<sup>1</sup>*=*<sup>2</sup> between the extension axis and x2, if K is positive [10].

To this broad isochoric flow, we can associate easily the corresponding 3d dynamical system [11]:

$$\begin{cases} \dot{\boldsymbol{\dot{x}}}\_1 = \boldsymbol{G} \bullet \boldsymbol{\varkappa}\_2 \\\\ \dot{\boldsymbol{\dot{x}}}\_2 = \boldsymbol{K} \bullet \boldsymbol{G} \bullet \boldsymbol{\varkappa}\_1, \ -1 < \boldsymbol{K} < \mathbf{1}, \boldsymbol{c} = \text{const.} \\\\ \dot{\boldsymbol{\dot{x}}}\_3 = \boldsymbol{c} \end{cases} \tag{28}$$

with the third component for the moving velocity of the system.

In the 3d case, the non-periodic model exhibits a complicate behavior. A lot of comparative computational analysis proved the great influence of the parameters on the model behavior, leading to far from equilibrium models [11]. The perturbed model was also taken into account, and it was found out that its sensitivity with respect to the parameters is significant, both in 2d and 3d case [11, 12].
