**3.2 Existence of a CLF for the mixing flow dynamical system in a slightly perturbed form**

The central aim in the study of the mixing flow dynamical system was associated rather with the fluid mechanics standpoint, namely analyzing the *efficiency of mixing* [10, 11].This is a concept which implies the analysis of deformation efficiencies in length and surface for the material mixed in the basic fluid. The physical phenomena associated are the *multiphase flow* phenomena, and the analysis and numeric simulation for these complex flows is in study. Briefly, in order to obtain a good behavior for the deformation efficiencies, the mathematical context must take into account the following three stages:

*V x*ð Þþ *r*1ð Þ *x β*ð Þ� *x φ*1ð Þ *x ϵ*Σ½ � *x* (21)

�*V x* \_ ð Þþ *<sup>r</sup>*2ð Þ *<sup>x</sup> <sup>β</sup>*ð Þ� *<sup>x</sup> <sup>φ</sup>*2ð Þ *<sup>x</sup> <sup>ϵ</sup>*Σ½ � *<sup>x</sup>* (22)

then the equilibrium point *x\** of (7) is asymptotically stable.

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

The answer to the first question is simply, no.

the further analysis, as we see in the section 3.

**excitable media**

**3.1 Recent results**

relation

*tion gradient F* [10]:

**112**

mention the "Positivstellensazt" as useful appliance [9].

of models, with a very sensitive behavior to initial conditions.

<sup>0</sup><*<sup>J</sup>* <sup>&</sup>lt; <sup>∞</sup>, *<sup>J</sup>* <sup>¼</sup> det *<sup>∂</sup>xi*

non-topological motions like break up or disintegration *are not allowed.*

A few questions immediately come to mind: Do stable polynomial systems always admit a sum-of-squares Lyapunov function? How conservative are we being by limiting ourselves to positive polynomials that admit a sum-of-squares decomposition? Can we determine a priori the degree of the Lyapunov function required?

There is a vast literature on the SOS method to compute Lyapunov functions in various settings and for different kinds of systems [4]. Between them, Parillo given important result on SOS and SDP (Semi Definite Problems) programming in finding Lyapunov functions [9]. As specified above, he introduced an efficient LMI program for finding Lyapunov functions as sum of squares for polynomial systems. The above setting is simplified for finding global Lyapunov functions, but if we are interested to find SOS Lyapunov functions on a compact domain, is important to

In the case of a dynamical system of a simple polynomial form, the stability study and Lyapunov function search can begin with a function test which facilitates

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

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

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

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

> *∂X <sup>j</sup>*

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,

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 *deforma-*

*x* ¼ Φ*t*ð Þ *X* , *X* ¼ Φ*<sup>t</sup>*¼<sup>0</sup>ð Þ *X :* (23)

, *J* ¼ detð Þ *D*Ф*t*ð Þ *x* (24)


Because of its complexity, any perturbation on the mixing flow model changes the behavior of the model in a significant way. Therefore its stability is an important and challenging task. If in [13] it was found a SOS Lyapunov function working both for the initial and the feedback linearized form of the mixing flow dynamical system, this chapter brings the novelty of approaching the mixing flow dynamical system as a *polynomial differential system*, in order to get an optimal search for a Lyapunov function.

When taking into account the feedback control, the transformed model has a significant different repartition of the parameters in the model [12]. The stability analysis was started with the 2d case, with slight perturbations. In what follows, we consider the same slightly perturbed form of the 2d dynamical system as in [13], namely

$$\begin{cases}
\dot{\varkappa\_1} = \mathrm{G}\varkappa\_2 + \varkappa\_1 \\
\dot{\varkappa\_2} = K\mathrm{G}\varkappa\_1 - \varkappa\_2
\end{cases} - 1 < K < 1, G \in \mathrm{R}.\tag{29}$$

Finding a CLF is a not trivial problem. It has been approached by multiple techniques, and Linear Programming, Positivstellensatz (P-sat) [14] and Linear Matrix Inequalities are only few examples. For the present aim the LMI approach

*Qualitative Analysis for Controllable Dynamical Systems: Stability with Control Lyapunov…*

A CLF must satisfy the inequality (32) which can be further evaluated as:

8 ><

*∂V*

choosing a large value of u of the correct sign. Therefore, is essential to establish the

Let us consider the 2d mixing flow dynamical model in its perturbed form (29).

*<sup>x</sup>*\_<sup>2</sup> <sup>¼</sup> *KGx*<sup>1</sup> � *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>u</sup>* � <sup>1</sup><*<sup>K</sup>* <sup>&</sup>lt;1, *<sup>G</sup>* <sup>∈</sup>*<sup>R</sup>*

*KGx*<sup>1</sup> � *x*<sup>2</sup>

<sup>1</sup>*P*<sup>11</sup> <sup>þ</sup> <sup>2</sup>*x*1*x*2*P*<sup>12</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>2</sup>*P*<sup>22</sup> � �

ð Þ *P*<sup>12</sup> � *P*<sup>11</sup> *x*<sup>1</sup> þ ð Þ *P*<sup>22</sup> � *P*<sup>12</sup> *x*<sup>2</sup> ¼ 0*:*

*<sup>∂</sup><sup>x</sup> <sup>f</sup>*ð Þ¼ *<sup>x</sup>* ð Þ *<sup>P</sup>*<sup>11</sup> <sup>þ</sup> *KGP*<sup>12</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> ð Þ *<sup>P</sup>*0*GP*<sup>11</sup> <sup>þ</sup> *<sup>P</sup>*0*KGP*<sup>22</sup> *<sup>x</sup>* <sup>þ</sup> ð Þ *<sup>P</sup>*<sup>12</sup> � *<sup>P</sup>*<sup>22</sup> *<sup>P</sup>*<sup>2</sup>

We are looking for a positive definite symmetric matrix P, *<sup>P</sup>* <sup>¼</sup> *<sup>P</sup>*<sup>11</sup> *<sup>P</sup>*<sup>12</sup>

*<sup>V</sup>* <sup>¼</sup> <sup>1</sup> 2

� �, *<sup>g</sup>* <sup>¼</sup> �<sup>1</sup>

1*:* � �

*xTPx*, (36)

*<sup>∂</sup><sup>x</sup> g*ð Þ¼ *x* 0 and studying when the

*<sup>∂</sup><sup>x</sup> g*ð Þ¼ *x* 0 implies

*<sup>∂</sup><sup>x</sup> f*ð Þ *x* . Thus, we obtain a

0, (37)

We have a polynomial system of differential equations, with monomials in the right hand sides of the system. In order to put it in a controlled form like (31), we take into account a simple control u at each of the equations of the model (29),

�<sup>∞</sup> *when <sup>∂</sup><sup>V</sup>*

*<sup>∂</sup><sup>x</sup> <sup>f</sup>*ð Þ *<sup>x</sup> when <sup>∂</sup><sup>V</sup>*

*<sup>∂</sup><sup>x</sup> g x*ð Þ 6¼ 0, we can make the inequality (32) hold by

*<sup>∂</sup><sup>x</sup> <sup>g</sup>*ð Þ *<sup>x</sup>* 6¼ <sup>0</sup>

>: *:* (33)

*<sup>∂</sup><sup>x</sup> <sup>g</sup>*ð Þ¼ *<sup>x</sup>* <sup>0</sup>

*<sup>∂</sup><sup>x</sup> <sup>g</sup>*ð Þ¼ *<sup>x</sup>* 0, *<sup>x</sup>* 6¼ <sup>0</sup>*:* (34)

*:* (35)

*<sup>P</sup>*<sup>12</sup> *<sup>P</sup>*<sup>22</sup> � � and

¼

[15, 16] helped to get better track of the model behavior.

*∂V <sup>∂</sup><sup>x</sup> <sup>g</sup>*ð Þ *<sup>x</sup> <sup>u</sup>*

*<sup>∂</sup><sup>x</sup> g x*ð Þ¼ 0 . Thus, we want

*<sup>∂</sup><sup>x</sup> <sup>f</sup>*ð Þ *<sup>x</sup>* <sup>&</sup>lt;0∀*x*∈*R<sup>n</sup> such that <sup>∂</sup><sup>V</sup>*

*x*\_<sup>1</sup> ¼ *Gx*<sup>2</sup> þ *x*<sup>1</sup> � *u*

� �

*inf <sup>u</sup>*∈*<sup>R</sup>*

set of x such that *<sup>∂</sup><sup>V</sup>*

namely

*∂V <sup>∂</sup><sup>x</sup> <sup>f</sup>*ð Þþ *<sup>x</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.96872*

For a fixed x such that *<sup>∂</sup><sup>V</sup>*

*∂V*

�

define a CLF function like

such that (34) holds.

quadratic form like follows:

that means

condition *<sup>∂</sup><sup>V</sup>*

*∂V*

**115**

The vector fields **<sup>f</sup>** and **<sup>g</sup>** are *<sup>f</sup>* <sup>¼</sup> *Gx*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>1</sup>

*<sup>V</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> *<sup>x</sup>*<sup>2</sup>

We have scalar inners in the condition (34). So,*<sup>∂</sup><sup>V</sup>*

From here we take x2 function of x1 and replace in *<sup>∂</sup><sup>V</sup>*

So, we search P by setting the condition *<sup>∂</sup><sup>V</sup>*

*<sup>∂</sup><sup>x</sup> f*ð Þ *x* < 0 is fulfilled.

For this model, it was found a strong result. A Lyapunov function V was found *both* for the initial model (29) and for its feedback linearized form. Namely, it was constructed the sum-of-squares function

$$V: \boldsymbol{R}^2 \to \boldsymbol{R}, V(\boldsymbol{\omega}) = \boldsymbol{\varkappa}\_1^2 + \frac{1}{|\boldsymbol{K}|} \boldsymbol{\varkappa}\_2^2, \forall \boldsymbol{\varkappa} = (\boldsymbol{\varkappa}\_1, \boldsymbol{\varkappa}\_2) \in \boldsymbol{R}^2. \tag{30}$$

The conditions (8)–(11) are fulfilled for suitable conditions on the parameters, for both models, so V is a Lyapunov function, and the origin is an asymptotically stable equilibrium point.

In what follows we consider the problem of finding a *control Lyapunov function* (CLF) for the model (29). We use a simplified form of the control system (5), namely we consider a control system of the form

$$
\dot{\mathfrak{x}} = f(\mathfrak{x}) + \mathfrak{g}(\mathfrak{x})u \tag{31}
$$

where f and g are smooth vector fields, *x t*ð Þ<sup>∈</sup> *Rn*, *u t*ð Þ<sup>∈</sup> *<sup>R</sup>*, *<sup>f</sup>*ð Þ¼ <sup>0</sup> 0.

**Definition 2**. A function V is a control Lyapunov function (CLF) for this system if *<sup>V</sup>* : *<sup>R</sup><sup>n</sup>* ! *<sup>R</sup>* is a smooth, radially bounded, and positive definite function such that

$$\inf\_{\mathbf{x}\in\mathcal{R}}\left\{\frac{\partial V}{\partial\mathbf{x}}f(\mathbf{x})+\frac{\partial V}{\partial\mathbf{x}}\mathbf{g}(\mathbf{x})\mathbf{u}\right\}<\mathbf{0},\forall\mathbf{x}\neq\mathbf{0}.\tag{32}$$

Existence of such a V implies that (29) is globally asymptotically stabilizable at origin.

We have to notice that the condition (32) for the CLF is in fact equivalent with that in (14) introduced by Sontag [6]. If such a CLF is given, it is shown [6] that a feedback low u = k(x) with k(0) = 0, can be constructed from the CLF producing a closed loop globally asymptotically stable. Hence, the problem of globally asymptotically stabilizing (31) is reduced to finding a CLF for the system.

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

Finding a CLF is a not trivial problem. It has been approached by multiple techniques, and Linear Programming, Positivstellensatz (P-sat) [14] and Linear Matrix Inequalities are only few examples. For the present aim the LMI approach [15, 16] helped to get better track of the model behavior.

A CLF must satisfy the inequality (32) which can be further evaluated as:

$$\inf\_{\mathbf{x}\in\mathcal{R}}\left\{\frac{\partial V}{\partial\mathbf{x}}f(\mathbf{x})+\frac{\partial V}{\partial\mathbf{x}}\mathbf{g}(\mathbf{x})\mathbf{u}\right\}=\begin{cases}-\infty & \text{when}\frac{\partial V}{\partial\mathbf{x}}\mathbf{g}(\mathbf{x})\neq\mathbf{0} \\ \frac{\partial V}{\partial\mathbf{x}}f(\mathbf{x}) & \text{when}\ \frac{\partial V}{\partial\mathbf{x}}\mathbf{g}(\mathbf{x})=\mathbf{0} \end{cases}.\tag{33}$$

For a fixed x such that *<sup>∂</sup><sup>V</sup> <sup>∂</sup><sup>x</sup> g x*ð Þ 6¼ 0, we can make the inequality (32) hold by choosing a large value of u of the correct sign. Therefore, is essential to establish the set of x such that *<sup>∂</sup><sup>V</sup> <sup>∂</sup><sup>x</sup> g x*ð Þ¼ 0 . Thus, we want

$$\frac{\partial V}{\partial \mathbf{x}} f(\mathbf{x}) < 0 \forall \mathbf{x} \in \mathbb{R}^n \text{ such that } \frac{\partial V}{\partial \mathbf{x}} \mathbf{g}(\mathbf{x}) = \mathbf{0}, \mathbf{x} \neq \mathbf{0}. \tag{34}$$

Let us consider the 2d mixing flow dynamical model in its perturbed form (29). We have a polynomial system of differential equations, with monomials in the right hand sides of the system. In order to put it in a controlled form like (31), we take into account a simple control u at each of the equations of the model (29), namely

$$\begin{cases}
\dot{\mathbf{x}}\_1 = \mathbf{G}\mathbf{x}\_2 + \mathbf{x}\_1 - u \\
\dot{\mathbf{x}}\_2 = \mathbf{K}\mathbf{G}\mathbf{x}\_1 - \mathbf{x}\_2 + u
\end{cases} - \mathbf{1} < K < \mathbf{1}, \mathbf{G} \in \mathbb{R}.\tag{35}$$

The vector fields **<sup>f</sup>** and **<sup>g</sup>** are *<sup>f</sup>* <sup>¼</sup> *Gx*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>1</sup> *KGx*<sup>1</sup> � *x*<sup>2</sup> � �, *<sup>g</sup>* <sup>¼</sup> �<sup>1</sup> 1*:* � �

We are looking for a positive definite symmetric matrix P, *<sup>P</sup>* <sup>¼</sup> *<sup>P</sup>*<sup>11</sup> *<sup>P</sup>*<sup>12</sup> *<sup>P</sup>*<sup>12</sup> *<sup>P</sup>*<sup>22</sup> � � and define a CLF function like

$$V = \frac{1}{2} \mathfrak{x}^T P \mathfrak{x},\tag{36}$$

that means

• modeling the global swirling streamlines;

Lyapunov function.

namely

• local modeling of the concentrated vorticity structure;

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

Because of its complexity, any perturbation on the mixing flow model changes the behavior of the model in a significant way. Therefore its stability is an important and challenging task. If in [13] it was found a SOS Lyapunov function working both for the initial and the feedback linearized form of the mixing flow dynamical system, this chapter brings the novelty of approaching the mixing flow dynamical system as a *polynomial differential system*, in order to get an optimal search for a

When taking into account the feedback control, the transformed model has a significant different repartition of the parameters in the model [12]. The stability analysis was started with the 2d case, with slight perturbations. In what follows, we consider the same slightly perturbed form of the 2d dynamical system as in [13],

For this model, it was found a strong result. A Lyapunov function V was found *both* for the initial model (29) and for its feedback linearized form. Namely, it was

> <sup>1</sup> þ 1 j j *<sup>K</sup> <sup>x</sup>*<sup>2</sup>

The conditions (8)–(11) are fulfilled for suitable conditions on the parameters, for both models, so V is a Lyapunov function, and the origin is an asymptotically

In what follows we consider the problem of finding a *control Lyapunov function* (CLF) for the model (29). We use a simplified form of the control system (5),

where f and g are smooth vector fields, *x t*ð Þ<sup>∈</sup> *Rn*, *u t*ð Þ<sup>∈</sup> *<sup>R</sup>*, *<sup>f</sup>*ð Þ¼ <sup>0</sup> 0. **Definition 2**. A function V is a control Lyapunov function (CLF) for this system if *<sup>V</sup>* : *Rn* ! *<sup>R</sup>* is a smooth, radially bounded, and positive definite function

> *∂V <sup>∂</sup><sup>x</sup> <sup>g</sup>*ð Þ *<sup>x</sup> <sup>u</sup>*

Existence of such a V implies that (29) is globally asymptotically stabilizable at

We have to notice that the condition (32) for the CLF is in fact equivalent with that in (14) introduced by Sontag [6]. If such a CLF is given, it is shown [6] that a feedback low u = k(x) with k(0) = 0, can be constructed from the CLF producing a closed loop globally asymptotically stable. Hence, the problem of globally asymp-

� �

� 1<*K* < 1, *G* ∈*R*

2, <sup>∀</sup>*<sup>x</sup>* <sup>¼</sup> ð Þ *<sup>x</sup>*1, *<sup>x</sup>*<sup>2</sup> <sup>∈</sup> *<sup>R</sup>*<sup>2</sup>

*x*\_ ¼ *f*ð Þþ *x g*ð Þ *x u* (31)

<0, ∀*x* 6¼ 0*:* (32)

*:* (29)

*:* (30)

*x*\_<sup>1</sup> ¼ *Gx*<sup>2</sup> þ *x*<sup>1</sup> *x*\_<sup>2</sup> ¼ *KGx*<sup>1</sup> � *x*<sup>2</sup>

(

*<sup>V</sup>* : *<sup>R</sup>*<sup>2</sup> ! *<sup>R</sup>*,*V*ð Þ¼ *<sup>x</sup> <sup>x</sup>*<sup>2</sup>

namely we consider a control system of the form

*inf <sup>u</sup>*∈*<sup>R</sup>*

*∂V <sup>∂</sup><sup>x</sup> <sup>f</sup>*ð Þþ *<sup>x</sup>*

totically stabilizing (31) is reduced to finding a CLF for the system.

constructed the sum-of-squares function

stable equilibrium point.

such that

origin.

**114**

• introducing the elements of chaotic turbulence.

$$V = \frac{1}{2} \left( \varkappa\_1^2 P\_{11} + 2\varkappa\_1 \varkappa\_2 P\_{12} + \varkappa\_2^2 P\_{22} \right)$$

such that (34) holds.

So, we search P by setting the condition *<sup>∂</sup><sup>V</sup> <sup>∂</sup><sup>x</sup> g*ð Þ¼ *x* 0 and studying when the condition *<sup>∂</sup><sup>V</sup> <sup>∂</sup><sup>x</sup> f*ð Þ *x* < 0 is fulfilled.

We have scalar inners in the condition (34). So,*<sup>∂</sup><sup>V</sup> <sup>∂</sup><sup>x</sup> g*ð Þ¼ *x* 0 implies

$$(P\_{12} - P\_{11})\mathbf{x}\_1 + (P\_{22} - P\_{12})\mathbf{x}\_2 = \mathbf{0}.$$

From here we take x2 function of x1 and replace in *<sup>∂</sup><sup>V</sup> <sup>∂</sup><sup>x</sup> f*ð Þ *x* . Thus, we obtain a quadratic form like follows:

$$\frac{\partial V}{\partial \mathbf{x}} f(\mathbf{x}) = (P\_{11} + K \mathbf{G} P\_{12}) \mathbf{x}^2 + (P\_0 \mathbf{G} P\_{11} + P\_0 K \mathbf{G} P\_{22}) \mathbf{x} + (P\_{12} - P\_{22}) P\_0^2,\tag{37}$$

with the notation

$$P\_0 = \frac{P\_{12} - P\_{11}}{P\_{12} - P\_{22}}.$$

The sign of the form (37) is of course dominated by the coefficient of x2 . Thus, we have

$$\frac{\partial V}{\partial \mathbf{x}} f(\mathbf{x}) < 0$$

when

$$P\_{11} + KGP\_{12} < 0.\tag{38}$$

**4. Conclusions**

**Author details**

University of Craiova, Romania

provided the original work is properly cited.

Adela Ionescu

**117**

In the present chapter, the aim of stabilizing the two-dimensional dynamical

*Qualitative Analysis for Controllable Dynamical Systems: Stability with Control Lyapunov…*

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

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

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

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 Lyapunov function is a strong appliance in studying stability. Still, constructing it is not easy. This is why approaching this for controllable systems

systems arising from excitable media is taken into account.

has a great influence on the model trajectory behavior.

make easier the task of searching the stability.

*DOI: http://dx.doi.org/10.5772/intechopen.96872*

the model is globally stabilizable in the origin.

problem for the mixing flow dynamical system.

the differential systems associated to transport phenomena.

\*Address all correspondence to: adelajaneta2015@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Taking into account that K,G are also real parameters, the inequality (38) implies few tests. Therefore for the moment we choose the matrix P like

$$P = \begin{pmatrix} \mathbf{1} & \mathbf{1}\sharp \\ \mathbf{1}\sharp & \mathbf{1} \end{pmatrix}. \tag{39}$$

With this choice we find the condition

$$KG < -2,\tag{40}$$

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 aim the basic matrix inequality, namely the Lyapunov inequality:

$$A^T P + PA \le 0\tag{41}$$

where A is the system matrix and P is the matrix in the definition (36) of CLF. In our model (35), the matrix A is given by:

$$A = \begin{pmatrix} \mathbf{1} & KG \\ G & -\mathbf{1} \end{pmatrix},\tag{42}$$

and thus we obtain

$$A^T P + PA = \begin{pmatrix} 2P\_{11} + 2P\_{12}KG & P\_{11}G + KGP\_{22} \\\\ P\_{11} + P\_{22}KG & 2P\_{12}G - 2P\_{22} \end{pmatrix} . \tag{43}$$

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

$$KG < -2,\tag{44}$$

$$G < -\frac{1}{2} \bullet \frac{1}{1+K}.$$

These conditions are feasible, for a negative G.

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