Switched Convergence of Second-Order Switched Homogeneous Systems

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.

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, the 𝐻_∞ 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 Section 2, 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 𝑥̇(𝑡)=𝑓_𝜎(𝑡)(𝑥(𝑡)), (1) where 𝑓_𝑖∶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, for 𝑖=1,2, each coordinate of 𝑓_𝑖 is a homogeneous function of grade 𝑚_𝑖; that is, if 𝑓_𝑖=(𝑓¹_𝑖,𝑓²_𝑖) then 𝑓¹_𝑖(𝜆𝑥)=𝜆^𝑚_𝑖𝑓¹_𝑖(𝑥) and 𝑓²_𝑖(𝜆𝑥)=𝜆^𝑚_𝑖𝑓²_𝑖(𝑥), for any 𝑥∈ℝ² and 𝜆∈ℝ.

Throughout the paper, we choose the orientation in ℝ² 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 point 𝑥 in the plane, we will say that the trajectory of 𝑓_𝑖 has positive orientation (clockwise direction) at 𝑥 if det(𝑓_𝑖(𝑥),𝑥)>0. Similarly, we will say that the trajectory of 𝑓_𝑖 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 𝐺_𝑖(𝑥)=det(𝑓_𝑖(𝑥),𝑥) for 𝑖=1,2.

We suppose that the origin is an unstable equilibrium point for each subsystem 𝑓₁ 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 state 𝑥₀∈ℝ² is switched convergent if there exists a switching law 𝜎_𝑥0 such that the solution of the switched system (1) that starts at 𝑥(0)=𝑥₀ converges to the origin, that is, lim_𝑡→+∞𝜙(𝑡;𝑥₀,𝜎_𝑥0)=0, where 𝜙(⋅;𝑥₀,𝜎_𝑥0) is the solution of the switched system (1) for 𝑥(0)=𝑥₀ and the switching law 𝜎_𝑥0.

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 subsystems 𝑓₁ 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 any 𝑥₀∈ℝ²⧵{0} with 𝐹(𝑥₀)=0, it holds that ∇𝐹(𝑥₀)≠0. Similarly, for any 𝑥₀∈ℝ²⧵{0} with 𝐺_𝑖(𝑥₀)=0 for some 𝑖=1,2, it holds that ∇𝐺_𝑖(𝑥₀)≠0.

Now, the following result lets us obtain a consequence of this assumption.

Proposition 3. Let be 𝑔 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 Assumption 2, 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. Let 𝑙₁ and 𝑙₂ be two rays starting from the origin. The set given by 𝑥∈ℝ²∶𝑥=𝜇𝑧₁+1−𝜇𝑧₂,𝑧₁∈𝑙₁,𝑧₂∈𝑙₂,0≤𝜇≤1 (2) 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∶𝐺_𝑖(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 𝐺_𝑖(𝑥)>0 or 𝐺_𝑖(𝑥)<0.

Throughout the paper, we also assume the following.

Assumption 5. {𝑥∈ℝ²∶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 𝑓_𝑖 starting from 𝑥₀ is of clockwise direction (or counterclockwise) and intersects the ray 𝑙₂ at 𝑥_𝑖. Then, (i)if 𝑙₁, 𝑙₂, and the trajectory of 𝑓_𝑖, for 𝑖=1,2, are contained in the same connected component of {𝑥∈ℝ²∶𝐹(𝑥)>0}, then 𝑥₁<𝑥₂𝑥₂<𝑥₁𝑖𝑛𝑐𝑜𝑢𝑛𝑡𝑒𝑟𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒𝑐𝑎𝑠𝑒; (3)(ii)if 𝑙₁, 𝑙₂, and the trajectory of 𝑓_𝑖, for 𝑖=1,2, are contained in the same connected component of {𝑥∈ℝ²∶𝐹(𝑥)<0}, then 𝑥₂<𝑥₁𝑥₁<𝑥₂𝑖𝑛𝑐𝑜𝑢𝑛𝑡𝑒𝑟𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒𝑐𝑎𝑠𝑒. (4)

Before presenting the second lemma, we need to introduce some notations. We will denote by 𝐸₁ 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 each 𝑖=1,2, the set {𝑥∈ℝ²∶𝐺_𝑖(𝑥)=0} is equal to the origin or the union of rays that pass through the origin. If the set {𝑥∈ℝ²∶𝐺_𝑖(𝑥)=0} is equal to {0}, we have that 𝐺_𝑖(𝑥)>0 for all 𝑥∈ℝ²⧵{0} or 𝐺_𝑖(𝑥)<0 for all 𝑥∈ℝ²⧵{0}. If, on the contrary, {𝑥∈ℝ²∶𝐺_𝑖(𝑥)=0} consists of several rays that pass through the origin, ℝ²⧵{𝑥∈ℝ²∶𝐺_𝑖(𝑥)=0} is equal to several cones where it is satisfied that either 𝐺_𝑖(𝑥)>0 or 𝐺_𝑖(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 sets 𝐸1 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‖.
When C(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 for 𝐸₂.

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∶𝐺_𝑖(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: 𝑥∈ℝ²⧵{0}∶𝐹(𝑥)≥0⊂𝑥∈ℝ²∶𝐺₁(𝑥)>0,𝑥∈ℝ²⧵{0}∶𝐹(𝑥)≤0⊂𝑥∈ℝ²∶𝐺₂(𝑥)>0. (5) 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 𝑥∈ℝ²⧵{0}∶𝐹(𝑥)≤0⊂𝑥∈ℝ²∶𝐺₁(𝑥)<0,𝑥∈ℝ²⧵{0}∶𝐹(𝑥)≥0⊂𝑥∈ℝ²∶𝐺₂(𝑥)<0. (6) 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 conditions 𝑥₀∈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, as 𝐸₁ 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 𝜎_𝐼-convergent for any switching law 𝜎_𝐼 of type I.

Proof. One of the implications is obvious. Therefore, we will prove that if 𝑥₀ 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 𝑥₀ the point where the solution intersects, for the first time, 𝑙₁. Moreover, there exists {𝑥_𝑛∶𝑛∈ℕ}, the set of points where the solution of the system initialized in 𝑥₀ 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 (𝑥_𝑛)_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 for 𝐸₂∩{𝑥∈ℝ²∶𝐹(𝑥)>0}.

Proof. Firstly, if 𝐸₁∩{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, if 𝑥₀∈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 {𝑥_𝑛∶𝑛∈ℕ} the set of points where the solution intersects, for example, the ray 𝑙₁, from Lemma 7 it is deduced that the sequence (𝑥_𝑛)_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 𝑆₁∩𝑥∈ℝ²∶𝐹(𝑥)<0≠∅,𝑟𝑒𝑠𝑝.𝑆₂∩𝑥∈ℝ²∶𝐹(𝑥)>0≠∅. (7)

Proof. Suppose that 𝑆₁∩{𝑥∈ℝ²∶𝐹(𝑥)<0}≠∅ (the proof is similar with 𝑆₂). Then, from Proposition 11, it is deduced that 𝑥₀∈𝑆₁ is 𝜎_II-convergent under a switching law 𝜎_II 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 law 𝜎 such that 𝑥₀ is 𝜎-convergent and the solution is contained in 𝑆₁ but 𝑆₁∩{𝑥∈ℝ²∶𝐹(𝑥)<0} is empty. Then, 𝑆₁⊆𝑥∈ℝ²∶𝐹(𝑥)≥0. (8) 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 each 𝑥∈𝑙₁∪𝑙₂, 𝐺₁(𝑥)=0 (see Figure 2(a)). (ii)For each 𝑥∈𝑙₁∪𝑙₂, 𝐺₂(𝑥)=0 (see Figure 2(b)). (iii)For each 𝑥∈𝑙₁, 𝐺₁(𝑥)=0 and for each 𝑥∈𝑙₂, 𝐺₂(𝑥)=0 (see Figure 2(c)). (iv)For each 𝑥∈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, that 𝜎(0)=1 and 𝑥₁ is the state where the first switching is produced.
If 𝐺₁(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.
If 𝑆₁ 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.
When 𝐺₁(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 that 𝐺₂(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 in 𝑆₁, 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 subsystem 𝑓₁, 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 subsystem 𝑓₂, 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 into ℛ₁, but this is a contradiction; so we have supposed that the solution converges to the origin in 𝑆₁.

(a)

(b)

(c)

(d)

Figure 2 

Figures for the proof of Theorem 12.

In the previous result, we have proved the convergence when the solution cannot leave a connected component of 𝐸₁ 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. Let 𝑆₁ 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, given 𝑆₂ 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 condition 𝑥₀ 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 𝑆₁∩𝑥∈ℝ²∶det𝐴₁𝑥,𝐴₂𝑥>0≠∅,𝑟𝑒𝑠𝑝.𝑆₂∩𝑥∈ℝ²∶det𝐴₁𝑥,𝐴₂𝑥<0≠∅. (9)

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: 𝑓₁𝑥₁,𝑥₂=𝑥³₁6−𝑥²₁𝑥₂2+𝑥₁𝑥²₂3−𝑥³₂,𝑥²₁𝑥₂6+𝑥³₂3,𝑓₂𝑥₁,𝑥₂=5𝑥³₁2−𝑥²₁𝑥₂2+5𝑥₁𝑥²₂−𝑥³₂,9𝑥³₁2−𝑥²₁𝑥₂2+9𝑥₁𝑥²₂−𝑥³₂. (10) 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: 𝐹𝑥₁,𝑥₂=1𝑥12²¹+2𝑥²₂²9𝑥²₁−33𝑥₁𝑥₂+4𝑥²₂,𝐺₁𝑥₁,𝑥₂1=−2𝑥²₂𝑥²₁+𝑥²₂,𝐺₂𝑥₁,𝑥₂1=−2−3𝑥₁+𝑥₂²𝑥²₁+2𝑥²₂. (11) 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 condition 𝑥₀=(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)).

(a)

(b)

Figure 3 

Solution and rays given by 𝐹(𝑥)=0 in Example 1 (a) and Example 2 (b).

Example 2. Consider a switched nonlinear system given by (1), where the subsystems are defined as follows: 𝑓₁𝑥₁,𝑥₂=𝑥³₁−𝑥²₁𝑥₂2+2𝑥₁𝑥²₂−𝑥³₂,𝑥²₁𝑥₂+2𝑥³₂,𝑓₂𝑥₁,𝑥₂=7𝑥³₁2+𝑥²₁𝑥₂2+7𝑥₁𝑥²₂+𝑥³₂,−25𝑥³₁2−3𝑥²₁𝑥₂2−25𝑥₁𝑥²₂−3𝑥³₂. (12) 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: 𝐹𝑥₁,𝑥₂1=−450𝑥²₂−5𝑥₁𝑥₂−𝑥²₂𝑥²₁+2𝑥²₂²,𝐺₁𝑥₁,𝑥₂1=−2𝑥²₂𝑥²₁+2𝑥²₂,𝐺₂𝑥₁,𝑥₂=12(5𝑥₁+𝑥₂)²𝑥²₁+2𝑥²₂. (13) 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 is 𝑥₀=(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.