# Switched Convergence of Second-Order Switched Homogeneous Systems

Authors: Carmen Pérez; Francisco Benítez
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper studies the stabilization of second-order switched homogeneous systems. We present results that solve the problem of stabilizing a switched homogeneous system; that is, we establish necessary and sufficient conditions under which the stabilization is assured. Moreover, given an initial condition, our method determines if there exists a switching law under which the solution converges to the origin and, if there exists this switching law, how it is constructed. Finally, two numerical examples are presented in order to illustrate the results.


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## Body

## 1. Introduction

A switched system is a hybrid system which consists of several subsystems and a rule that orchestrates the switching among them. In the last two decades, there has been increasing interest in stability analysis and control design for switched systems. The motivation is from many aspects. For example, a system with several controllers can be regarded as a switched system. Both discrete- and continuous-time switched systems have been studied from different aspects. Recently, for discrete-time switched systems, theH∞ filtering problem has been studied by including time delay [1–3]. Also, recent works [4–6] have included the stabilization problem for this kind of switched systems from different points of view.The problem of stability has been focused in three main problems [7, 8]: first, find conditions that guarantee that the switched system is asymptotically stable under arbitrary switching; second, define a class of switching laws under which the switched system is stable; third, construct a switching law that makes the switched system asymptotically stable. Recently, the study of the stability has been extended to the case when there exists time delay [9, 10].For the last problem, there are basically two approaches to stabilization for switched systems, one based on detailed analysis of the vector field and the other one based on Lyapunov theory. In papers such as [11, 12] or [13], Lyapunov theory is employed to construct switching laws for a class of switched systems totally composed of unstable subsystems.With respect to the analysis of the vector field, for second-order systems, some necessary and sufficient conditions for stability/stabilizability have been obtained in [14–16] through analysis on the structure of the vector field. In [16], the problem of stabilizing two second-order switched linear systems is solved; that is, a method to define switching laws that decide when a switched system is stabilizable is given. In [15], some results about the stabilization in the nonlinear case are presented.The switched homogeneous systems are a superclass of switched linear systems. The stability of these systems has been studied in [17–19] with the objective of finding conditions under which the switched system is asymptotically stable. Recently, in [20, 21], the problem of stability is also studied. Aleksandrov et al. [20] investigate the stability of these systems from the Lyapunov approach, and Tuna [21] considers discrete-time homogeneous systems under arbitrary switching. These papers solve the problem of stability under arbitrary switching. However, for switched homogeneous systems, the problem of stabilization, that is, the problem of constructing a switching law that stabilizes the system, is still open.In this work, we consider this problem; that is, we search for a switching law that makes the switched homogeneous system asymptotically stable. By using the same idea described in [15, 16], we can generalize the results in order to solve the problem of stabilizing a switched homogeneous system. Hence, we establish the conditions under which the solution of a switched homogeneous system converges to the origin under two types of switching laws. It is important to remark that only the idea in [15] is used, and the results cannot be directly applied to this type of switched systems because the preliminary assumptions in [15] cannot be satisfied for switched homogeneous systems.The remainder of this paper is organized as follows. In Section2, the problem is formulated and the first results are obtained. In Section 3, we present the switching laws employed in order to establish the convergence. The results that assure the convergence of these switched systems are shown in Section 4. Two numerical examples are given in Section 5 in order to illustrate the results. Finally, the conclusions are provided in Section 6.


## 2. Preliminaries

Consider a switched nonlinear system(1)x˙(t)=fσ(t)(x(t)),
where fi:D→ℝ2, i=1,2, is a vector field of class 𝒞1 in the open and connected set, D, x∈ℝ2, and σ:[0,∞)→{1,2} is a piecewise constant function called switching law indicating the active subsystem at each instant. Such a function σ has a finite number of discontinuities, which we call switching times, on every bounded time interval and takes a constant value on every interval between two consecutive switching times.Moreover, fori=1,2, each coordinate of fi is a homogeneous function of grade mi; that is, if fi=(fi1,fi2) then fi1(λx)=λimfi1(x) and fi2(λx)=λimfi2(x), for any x∈ℝ2 and λ∈ℝ.Throughout the paper, we choose the orientation inℝ2 given by the canonical basis; that is, we will say that a basis in the plane has the same orientation than the canonical basis (positive orientation) if the determinant of the change of basis matrix is strictly positive.Under this orientation, given a pointx in the plane, we will say that the trajectory of fi has positive orientation (clockwise direction) at x if det(fi(x),x)>0. Similarly, we will say that the trajectory of fi has negative orientation (counterclockwise direction) at x if the determinant is negative. Note that if the determinant is null at x, it has no sense to consider the orientation because {f(x),x} is not a basis for ℝ2. Under this assumption, we define Gi(x)=det(fi(x),x) for i=1,2.We suppose that the origin is an unstable equilibrium point for each subsystemf1 and f2. In this work, we will use a generalization of a weaker property than the classic concept of stability. This concept was defined for the nonlinear case in [22].Definition 1 (see [22]).
A statex0∈ℝ2 is switched convergent if there exists a switching law σx0 such that the solution of the switched system (1) that starts at x(0)=x0 converges to the origin, that is, limt→+∞ϕ(t;x0,σx0)=0, where ϕ(·;x0,σx0) is the solution of the switched system (1) for x(0)=x0 and the switching law σx0.The switching laws under which a state is switched convergent will be called convergent switching laws.In order to study the convergence, it is important to consider the set of points where the subsystemsf1 and f2 are parallel. Hence, we define the function F(x)=det(f1(x),f2(x)) and the set Ω={x∈ℝ2:F(x)=0}.In [15, 22], some conditions for F, G1, and G2 are necessary in order to apply the method of stabilization. However, it is easy to prove that, for switched homogeneous systems, these conditions are not satisfied. For this reason, we must study the properties of these functions for switched homogeneous systems and assume a new condition.Assumption 2.
For anyx0∈ℝ2∖{0} with F(x0)=0, it holds that ∇F(x0)≠0. Similarly, for any x0∈ℝ2∖{0} with Gi(x0)=0 for some i=1,2, it holds that ∇Gi(x0)≠0.Now, the following result lets us obtain a consequence of this assumption.Proposition 3.
Let beg a real homogeneous function of grade m and of class 𝒞1 in the open and connected set D⊂ℝ2. If {x∈ℝ2:g(x)=0}≠{0} and ∇g(x0)≠0 for any x0∈ℝ2∖{0} with g(x0)=0, then {x∈ℝ2:g(x)=0}≠{0} is equal to the union of rays that go through the origin.Proof.
It is deduced from the definition of homogeneous function.Then, under Assumption2, from the previous proposition, it is deduced that if the sets {x∈ℝ2:F(x)=0}, {x∈ℝ2:G1(x)=0} and {x∈ℝ2:G2(x)=0} are different from the origin, they are equal to the union of rays that go through the origin. Hence, if, for example, {x∈ℝ2:F(x)=0} is different from the origin, the complementary of this set is equal to the plane minus some rays that go through the origin. In order to identify these sets, we define the following.Definition 4.
Letl1 and l2 be two rays starting from the origin. The set given by
(2){x∈ℝ2:x=μz1+(1-μ)z2,z1∈l1,z2∈l2,0≤μ≤1}
will be called the cone delimited by l1 and l2 and denoted by C(0,l1,l2).Therefore, with this definition, if{x∈ℝ2:F(x)=0} is different from the origin, the complementary of this set is equal to the union of several cones where it is satisfied that F(x)>0 or F(x)<0. In the same form, we have that if, for example, {x∈ℝ2:Gi(x)=0}≠{0} for some i=1,2, the complementary of this set is equal to the union of several cones where it is satisfied that Gi(x)>0 or Gi(x)<0.Throughout the paper, we also assume the following.Assumption 5.
{
x
∈
ℝ
2
:
F
(
x
)
=
0
}
≠
{
0
}.We can assume this condition without loss of generality because if it is not satisfied, there exists no switched convergent state (see Proposition 2 in [15]). Therefore, in that case, our study has no sense.Under these two assumptions, two lemmas under which the main results will be established can be proved in a similar way to the lemmas presented in [15].Lemma 6.
Consider the switched homogeneous system (1). Let l1 and l2 be two different rays starting from the origin and x0∈l1∖{0}. Suppose that, for each i=1,2, the trajectory of fi starting from x0 is of clockwise direction (or counterclockwise) and intersects the ray l2 at xi. Then,  (i)
ifl1, l2, and the trajectory of fi, for i=1,2, are contained in the same connected component of {x∈ℝ2:F(x)>0}, then
(3)∥x1∥<∥x2∥(∥x2∥<∥x1∥incounterclockwisecase);(ii)
ifl1, l2, and the trajectory of fi, for i=1,2, are contained in the same connected component of {x∈ℝ2:F(x)<0}, then
(4)∥x2∥<∥x1∥(∥x1∥<∥x2∥incounterclockwisecase).Before presenting the second lemma, we need to introduce some notations. We will denote byE1 the region where the first subsystem is of clockwise direction and the second one is of counterclockwise direction. Similarly, we will denote by E2 the region where the first subsystem is of counterclockwise direction and the second one is of clockwise direction.Now, we know that, for eachi=1,2, the set {x∈ℝ2:Gi(x)=0} is equal to the origin or the union of rays that pass through the origin. If the set {x∈ℝ2:Gi(x)=0} is equal to {0}, we have that Gi(x)>0 for all x∈ℝ2∖{0} or Gi(x)<0 for all x∈ℝ2∖{0}. If, on the contrary, {x∈ℝ2:Gi(x)=0} consists of several rays that pass through the origin, ℝ2∖{x∈ℝ2:Gi(x)=0} is equal to several cones where it is satisfied that either Gi(x)>0 or Gi(x)<0, then the sets E1 and E2 are empty, coincide with ℝ∖{0}, or are the union of several cones (see Figure 1).Figure 1
The setsE1 and E2 and the rays {x∈ℝ2:Gi(x)=0}.Lemma 7.
Let (1) be a switched homogeneous system, and suppose that C(0,l1,l2) is a connected component of {x∈ℝ2:F(x)<0}∩E1. One supposes that the trajectory 𝒯1 of f1 goes from x0∈l1 to x1∈l2 (in clockwise direction) and the trajectory 𝒯2 of f2 goes from x1∈l2 to x2∈l1 (in counterclockwise direction); then, it holds that ∥x0∥<∥x2∥.
WhenC(0,l1,l2) is a connected component of {x∈ℝ2:F(x)>0}∩E1, it is obtained that ∥x0∥<∥x2∥.A similar result can be enunciated forE2.


## 3. Convergent Switching Laws

In order to establish the convergence, two types of switching laws are presented in [15]. In this work, we use the same idea to define the switching laws although, due to the particular properties of the sets {x∈ℝ2:F(x)=0} and {x∈ℝ2:Gi(x)=0}, the definition of these switching laws is different.Firstly, under the switching laws of type I, our goal is that the solution rotates around the origin. For this reason, if the solutions must rotate around the origin in clockwise direction, it is easy to prove that the following must be verified:(5){x∈ℝ2∖{0}:F(x)≥0}⊂{x∈ℝ2:G1(x)>0},{x∈ℝ2∖{0}:F(x)≤0}⊂{x∈ℝ2:G2(x)>0}.
In the same form, it can be proved that if the solution must rotate around the origin in counterclockwise direction, it must be verified that
(6){x∈ℝ2∖{0}:F(x)≤0}⊂{x∈ℝ2:G1(x)<0},{x∈ℝ2∖{0}:F(x)≥0}⊂{x∈ℝ2:G2(x)<0}.
Under these conditions, we define the switching law of type I in the following form.Definition 8.
Let (1) be a switched homogeneous system and suppose that the condition (5) (resp., (6)) is satisfied. Given an initial condition x0∈ℝ2∖{0}, one will say that a switching law σ is of type I if  (i)
σ(t)=1 (resp. σ(t)=2) if ϕ(t;x0,σ)∈{x∈ℝ2:F(x)>0},(ii)
σ(t)=2 (resp. σ(t)=1) if ϕ(t;x0,σ)∈{x∈ℝ2:F(x)<0}, where ϕ is the solution of the switched system (1) under the switching law σ starting from x0.The initial conditionsx0∈ℝ2 for which there exists a switching law of type I are called states of type I, and the switched systems for which there exist switching laws of type I are called switched systems of type I.Now, for the switching laws of type II, asE1 and E2 are equal to ℝ2∖{0} or the union of cones, we define a switching law of type II in the following form.Definition 9.
Let (1) be a switched homogeneous system and x0∈E1 (resp., E2) a state. One says that a switching law is of type II if one switchs whenever the trajectory intersects the boundary of {x∈ℝ2:F(x)<0}∩E1 (resp., {x∈ℝ2:F(x)>0}∩E2).It is important to note that the condition under which the switching law of type II can be defined is that{x∈ℝ2:F(x)<0}∩E1≠∅ or {x∈ℝ2:F(x)>0}∩E2≠∅.


## 4. Switched Convergence of Switched Homogeneous Systems

In this section, we solve the problem of the convergence of switched homogeneous systems by using the previous switching laws.In [15], the hypothesis of the result that assures the convergence under switching laws of type I is not straightforward to check. However, for switched homogeneous systems, in order to solve the convergence, we only check the conditions given in the previous section.Theorem 10.
Consider the switched homogeneous system (1). Suppose that (5) or (6) is satisfied. Let x0∈ℝ2∖{0} be a state. It holds that x0 is σ-convergent for any switching law σ if and only if x0 is σI-convergent for any switching law σI of type I.Proof.
One of the implications is obvious. Therefore, we will prove that ifx0 is convergent for any switching law, it is also convergent for a switching law of type I. Firstly, as (5) or (6) is satisfied, the solution for the switching law of type I rotates around the origin. Thus, if we choose a ray l1 contained in {x∈ℝ2:F(x)=0}, there exists x¯0 the point where the solution intersects, for the first time, l1. Moreover, there exists {xn:n∈ℕ}, the set of points where the solution of the system initialized in x¯0 under a switching law of type I intersects the ray l1. Therefore, if we proceed as in the proof of Theorem 1 in [15], we can prove that the sequence (xn)n converges to the origin and, thus, the switching law of type I is convergent.In [15], only a sufficient condition for the convergence of switching laws of type II is presented. However, in this case, more results are deduced from the study of this type of switching laws. Firstly, we have the following proposition.Proposition 11.
Consider the switched homogeneous system (1). If E1∩{x∈ℝ2:F(x)<0}≠∅, then x0 is switched convergent for any x0∈E1∩{x∈ℝ2:F(x)<0}.A similar result can be obtained forE2∩{x∈ℝ2:F(x)>0}.Proof.
Firstly, ifE1∩{x∈ℝ2:F(x)<0}≠∅, we have that this set is equal to a cone. Therefore, there exist l1 and l2 such that E1∩{x∈ℝ2:F(x)<0}=C(0,l1,l2).
Moreover, ifx0∈E1∩{x∈ℝ2:F(x)<0}, then it is possible to define for x0 a switching law of type II. Hence, the solution under this switching law intersects the rays l1 and l2. If we denote by {xn:n∈ℕ} the set of points where the solution intersects, for example, the ray l1, from Lemma 7 it is deduced that the sequence (xn)n converges to the origin. As the solution is bounded, we have that the solution also converges to the origin.Now, it is possible to obtain, in this case, a necessary and sufficient condition for the switched convergence.Theorem 12.
Consider the switched homogeneous system (1). Suppose that E1 (resp., E2) is not empty. Let S1 (resp., S2) be a connected component of E1 (resp., E2), x0∈S1 (resp., S2) and σ a switching law such that the solution under the switching law σ starting from x0 is contained in S1. Then, x0 is σ-convergent if and only if
(7)S1∩{x∈ℝ2:F(x)<0}≠∅,(resp.S2∩{x∈ℝ2:F(x)>0}≠∅).Proof.
Suppose thatS1∩{x∈ℝ2:F(x)<0}≠∅ (the proof is similar with S2). Then, from Proposition 11, it is deduced that x0∈S1 is σII-convergent under a switching law σII of type II. Moreover, by definition of the switching law of type II, the solution under this switching law is contained in S1.
Suppose now that there exists a switching lawσ such that x0 is σ-convergent and the solution is contained in S1 but S1∩{x∈ℝ2:F(x)<0} is empty. Then,
(8)S1⊆{x∈ℝ2:F(x)≥0}.
Therefore, S1 is a cone so it is contained in {x∈ℝ2:F(x)≥0}, which is equal to the union of cones. Thus, S1=C(0,l1,l2), and in l1 and l2, it is satisfied that G1(x)=0 or G2(x)=0. Now, we can distinguish the following cases.  (i)
For eachx∈l1∪l2, G1(x)=0 (see Figure 2(a)).(ii)
For eachx∈l1∪l2, G2(x)=0 (see Figure 2(b)).(iii)
For eachx∈l1, G1(x)=0 and for each x∈l2, G2(x)=0 (see Figure 2(c)).(iv)
For eachx∈l1, G2(x)=0 and for each x∈l2, G1(x)=0 (see Figure 2(d)).
Before studying each case, we can suppose, without loss of generality, thatσ(0)=1 and x1 is the state where the first switching is produced.
IfG1(x)=0 if x∈l1∪l2, let y1 and y2 be the states where the trajectory of f2 that goes through x1 intersects l1 and l2, respectively (see Figure 2(a)). Hence, we consider the closed curve consisting of the union of this trajectory, the arc [y1,0] of l1, and the arc [0,y2] of l2. Then, by the Jordan curve theorem, there exist two regions that are open, connected, and disjoint and whose union is the complementary set of the union of the curves. Let ℛ1 denote the bounded region whose boundaries are the considered curves. When, according to the switching law σ, the second switching is produced in the trajectory of f2, we switch to the first subsystem and the solution leaves ℛ1 so F(x)≥0.
IfS1 is delimited by the rays given by G2(x)=0, we consider the trajectory of f1 that goes through x0 and intersects l1 and l2 at y1 and y2, respectively (see Figure 2(b)). Again, we define the closed curve consisting of this trajectory, the arc [y2,0] of l2, and the arc [0,y1] of l1. Then, by the Jordan curve theorem, there exist two regions that are open, connected, and disjoint and whose union is the complementary set of the union of the curves. Let ℛ1 denote the bounded region whose boundaries are the considered curves. When, according to the switching law σ, the second switching is produced in the trajectory of f2, we switch to the first subsystem and the solution leaves ℛ1 so F(x)≥0.
WhenG1(x)=0 if x∈l1 and G2(x)=0 if x∈l2, we choose the trajectory of f1 starting from x0 that intersects l2 at y2 and the trajectory of f2 starting from x0 that intersects l1 at y1 (see Figure 2(c)). Now, the closed curve considered is the union of the trajectories of f1 and f2 and the arcs [0,y1] and [y2,0] of l1 and l2, respectively. By the Jordan curve theorem, there exist two regions that are open, connected, and disjoint and whose union is the complementary set of the union of the curves. Let ℛ1 denote the bounded region whose boundaries are the considered curves. Again, when the next switching is produced, the solution goes into the unbounded region, so F(x)≥0.
Finally, suppose thatG2(x)=0 if x∈l1 and G1(x)=0 if x∈l2. Then, let y1 be the point where the trajectory of -f1 starting from x0 intersects l1, and let y2 be the point where the trajectory of -f2 starting from x1 intersects l2 (see Figure 2(d)). Consider the closed curve equal to the union of the trajectories of f1 and f2 and the arcs [0,y1] and [0,y2] of l1 and l2, respectively. By the Jordan Curve Theorem, we obtain a bounded region ℛ1. In this case, it is clear that when the next switching is produced at x1, the solution goes into the unbounded region.
Hence, in all cases, we have obtained a closed curve and two regions (bounded and unbounded), and the solution, in the next switching, goes into the unbounded region. As, by hypothesis, the solution converges to the origin and is contained inS1, this solution must go into the bounded region ℛ1 and, thus, must intersect the closed curve. In particular, it must intersect the arc that goes from y1 to y2 and is given by the trajectories of f1 and/or f2. We will prove that this is not possible.
If the solution is given by the subsystemf1, it is clear that it cannot intersect the arc of trajectory given by f1 so they are solutions of the same subsystem. And it cannot intersect the arc of trajectory of f2 so, in that case, we could find a state x¯∈S1 such that F(x¯)<0.
In the same form, if the solution is given by the subsystemf2, it is clear that it cannot intersect the arc of trajectory of f2 so they are solutions of the same subsystem, and it cannot intersect the trajectory of f1 so, in that case, we could find a state x¯∈S1 such that F(x¯)<0.
Therefore, as the solution cannot intersect the closed curve, it cannot go intoℛ1, but this is a contradiction; so we have supposed that the solution converges to the origin in S1.Figures for the proof of Theorem12.
(a)
(b)
(c)
(d)In the previous result, we have proved the convergence when the solution cannot leave a connected component ofE1 or E2. Our goal is proving the convergence without assuming that the solution is in a cone. For this reason, the following definition is needed.Definition 13.
LetS1 be a connected component of E1. One will say that S1 is a sealed cone if it is a cone equal to C(0,l1,l2) such that G1(x)=0 if x∈l2 and G2(x)=0 if x∈l1.
Similarly, givenS2 a connected component of E2, one will say that S2 is a sealed cone if it is a cone equal to C(0,l1,l2) such that G1(x)=0 if x∈l1 and G2(x)=0 if x∈l2.Given an initial conditionx0 in the interior of a sealed cone S1, if we choose the trajectory of f1, we know that this trajectory is of clockwise direction and cannot intersect l2; so in this ray, it holds that G1(x)=0. In the same manner, if we choose the trajectory of f2, this trajectory cannot intersect l1 so, in this ray, it holds that G2(x)=0. Therefore, the following result can be deduced.Proposition 14.
Consider the switched homogeneous system (1). Suppose that E1 (resp., E2) is not empty. If S1 (resp., S2) is a sealed cone, for any x0∈S1 (resp., S2) and any switching law σ, the solution of the switched system (1) is in S1 (resp., S2).This proposition proves that if the solution starts in a sealed cone, it cannot leave this cone. Hence, by using the previous theorem and this proposition, we have the following result.Corollary 15.
Consider the switched homogeneous system (1). Suppose that E1 (resp., E2) is not empty. If S1 (resp., S2) is a sealed cone of E1 (resp., E2) and x0∈S1 (resp., S2), then x0 is σ-convergent for some switching law σ if and only if
(9)S1∩{x∈ℝ2:det(A1x,A2x)>0}≠∅,(resp.S2∩{x∈ℝ2:det(A1x,A2x)<0}≠∅).


## 5. Numerical Examples

Now, in order to illustrate the results, two numerical examples are presented.Example 1.
Consider a switched nonlinear system given by (1), where the subsystems are defined as follows:
(10)f1(x1,x2)=(x136-x12x22+x1x223-x23,x12x26+x233),f2(x1,x2)=(5x132-x12x22+5x1x22-x23,9x132-x12x22+9x1x22-x23).
Firstly, this switched system is homogeneous because each coordinate is homogeneous of grade 3. In order to apply the results in the paper, we study the functions F, G1, and G2:
(11)F(x1,x2)=112(x12+2x22)2(9x12-33x1x2+4x22),G1(x1,x2)=-12x22(x12+x22),G2(x1,x2)=-12(-3x1+x2)2(x12+2x22).
From the expressions of F, G1, and G2, it is deduced that Assumptions 2 and 5 are satisfied. Furthermore, we have that {x∈ℝ2:G1(x)=0} is equal to the ray given by x2=0 and {x∈ℝ2:G2(x)=0} is equal to the ray given by x2=3x1. Moreover, the trajectories of both subsystems are of counterclockwise direction for any initial condition. Therefore, E1=E2=∅ and the switching laws of type II cannot be defined.
Also,{x∈ℝ2:F(x)=0} is equal to two rays that go through the origin. Moreover, it is easy to prove that (6) holds; thus, we can define switching laws of type I and, by Theorem 10, the switching laws determine the convergence of the system.
Let see what happens for the initial conditionx0=(1,1). First, F(x0)<0; thus, according to the definition of switching law of type I, we choose the first subsystem and switch when the solution intersects the rays in {x∈ℝ2:F(x)=0}. In this case, the solution converges to the origin (see Figure 3(a)).Solution and rays given byF(x)=0 in Example 1 (a) and Example 2 (b).
(a)
(b)Example 2.
Consider a switched nonlinear system given by (1), where the subsystems are defined as follows:
(12)f1(x1,x2)=(x13-x12x22+2x1x22-x23,x12x2+2x23),f2(x1,x2)=(7x132+x12x22+7x1x22+x23,-25x132-3x12x22-25x1x22-3x23).
Firstly, this switched system is homogeneous because each coordinate is homogeneous of grade 3. In order to apply the results in the paper, we again study the functions F, G1, and G2:
(13)F(x1,x2)=-14(50x22-5x1x2-x22)(x12+2x22)2,G1(x1,x2)=-12x22(x12+2x22),G2(x1,x2)=12(5x1+x2)2(x12+2x22).
From the expressions of F, G1, and G2, it is deduced that Assumptions 2 and 5 are satisfied. Furthermore, we have that {x∈ℝ2:G1(x)=0} is equal to the ray given by x2=0 and {x∈ℝ2:G2(x)=0} is equal to the ray given by x2=-5x1. Moreover, the trajectory of f1 is of counterclockwise direction and the trajectory of f2 is of clockwise direction. Therefore, E2 is the complementary of these rays, that is, equal to the union of two cones.
Furthermore, as the set{x∈ℝ2:F(x)=0} is equal to two rays and E2∩{x∈ℝ2:F(x)>0} is not empty, the switching laws of type II can be defined and are convergent by Theorem 12. It is important to note that the cones in E2 are not sealed; thus, we cannot apply in this case Corollary 15.
In particular, if the initial condition isx0=(1,1), then x0∈E2 but F(x0)<0. Thus, we first choose the subsystem f1 until going into {x∈ℝ2:F(x)>0}. Then, we define a switching law of type II, and, by Proposition 11, the switching law of type II is convergent (see Figure 3(b)).


## 6. Conclusions

In this work, we have solved the problem of stabilization of a second-order switched homogeneous system. Although the idea in [16] for the linear case is used and the results for nonlinear case in [15] are mentioned, it has been necessary to study this kind of switched systems properly so it is not possible to apply directly these results. Furthermore, the study of this kind of switched systems generalizes the linear case by reducing the number of cases to only two switching laws.As for future work, these results can be applied to study the stabilization of switched nonlinear systems where the conditions in [15] cannot be satisfied, for example, in switched polynomial systems. Hence, it could be possible to solve completely the problem of convergence of second-order switched nonlinear systems. Furthermore, projections may be employed to apply this research to the convergence of higher order switched homogeneous systems. Finally, the method could be employed in real process that can be modelled by homogeneous systems.



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