# Exponential Stability of Switched Positive Homogeneous Systems

Authors: Dadong Tian; Shutang Liu
Journal: Complexity
 (2017)

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## Abstract

This paper studies the exponential stability of switched positive nonlinear systems defined by cooperative and homogeneous vector fields. In order to capture the decay rate of such systems, we first consider the subsystems. A sufficient condition for exponential stability of subsystems with time-varying delays is derived. In particular, for the corresponding delay-free systems, we prove that this sufficient condition is also necessary. Then, we present a sufficient condition of exponential stability under minimum dwell time switching for the switched positive nonlinear systems. Some results in the previous literature are extended. Finally, a numerical example is given to demonstrate the effectiveness of the obtained results.


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## Body

## 1. Introduction

Positive systems are those systems whose state variables and output signals are always contained in the first quadrant whenever both the initial conditions and input signals are nonnegative. Many real-word processes are described by positive systems. Examples of this type can be found in areas such as ecology, biology, and chemical engineering [1–3]. Due to their importance and wide applications, there has been an increasing interest in such systems.Many important and interesting properties of positive systems have been reported and analyzed. It is well known that the most fundamental one is the stability property. Up until now the stability analysis of positive linear time-invariant (LTI) systems is well developed [4, 5]. In [4], it was shown that the delayed positive linear system x˙t=Axt+Bxt-τ was globally asymptotically stable (GAS) for all τ≥0 if the corresponding delay-free system x˙t=A+Bxt was GAS. This implies that such system is insensitive to certain class of time delays. Moreover, the exponential stability of positive linear systems with constant time delays was investigated by Zhu et al. [5], in which it was also shown that the decay rate depended on the magnitude of delays. Recently, some progress on the positive nonlinear systems called cooperative homogeneous systems has been made [6–9]. For example, [6] derived a necessary and sufficient condition for exponential stability of such positive systems with the homogeneity of a degree of one.On the other hand, switched systems have attracted much research attention in control theory field [10–13]. Typically, a switched system is a type of hybrid dynamical system consisting of family (either discrete-time or continuous-time) subsystems and a rule that regulates the switching among them. It has been widely applied in many areas, such as chemical processing and traffic control. For such systems, the stability theory is also one of the fundamental problems. The authors in [12] applied the Lyapunov-based theory, such as the multiple Lyapunov function, to study the stability problems.Furthermore, switched positive systems combining the features of positive systems and switched systems are recently studied [14–18]. As switched positive systems are defined on cones rather than on linear spaces, studying such systems is more challenging than that of general switched systems. In recent years, some results concerning the switched positive systems have been reported. For example, the stability problem of switched positive linear systems has been extensively discussed by the approach of linear copositive functions such as [14, 15, 17]. In [19], the author showed a necessary and sufficient condition for exponential stability of a class of switched positive nonlinear systems under average dwell time switching.It is worth noting that the results in [7] only dealt with asymptotical stability of positive systems. As is known to us, switched systems may be unstable even if we ensure that all the subsystems are stable. Therefore, it is meaningful to combine the positive systems with switched systems. On the other hand, the author of [19] only studied the switched positive systems with the homogeneity of a degree of one, in which the common exponential decay rate for all subsystems independent of initial conditions can be easily found for all subsystems. However, when the degree is not constrained to be one, the exponential decay rate depends on initial conditions, which would make the problem be more complicated. The state at each switched instant should be considered. In this case, it is worth studying whether it is possible to design suitable switching signals to keep the system exponentially stable. In this paper, by using the approach in which the Lyapunov-Krasovskii functional method is not included, we design appropriate time-dependent switching rules under which the switched positive nonlinear system is exponentially stable with the degree 0<α≤1.The main contributions are summarized as follows. We derive a sufficient condition for exponentially stable unswitched positive systems of degree of0<α≤1 with bounded time delays. It is shown that the decay rate depends on the initial conditions and the time delays. In particular, for the corresponding delay-free systems, the sufficient condition is also necessary. Then as special switched positive nonlinear systems, we consider the switched positive homogeneous systems with the degree of 0<α≤1. A sufficient condition for the exponential stability of switched positive homogeneous systems under minimum dwell time (MDT) switching is presented.The layout of the paper is as follows. In Section2, we introduce the notation and review some preliminaries. The main results of this paper are stated in Section 3. Section 4 provides a numerical example to show the validity of our results. Finally, concluding remarks are given in Section 5.


## 2. Notation and Preliminaries

### 2.1. Notation

Throughout the paper, vectors are written in bold lowercase letters. LetR, N, and N0 denote the set of real numbers, natural numbers, and the set of natural numbers including zero, respectively. Rn stands for the n-dimensional Euclidean space. Let R+n be the set of all vectors in Rn with nonnegative entries; that is, R+n≔x∈Rn,xj≥0,1≤j≤n. For two vectors, x,y∈Rn, we write x≥y, if xj≥yj, for 1≤j≤n; x>y, if x≥y and x≠y; x≫y, if xj>yj1≤j≤n. xj is used to denote the jth component of x. Similarly, xpj denotes the jth coordinate of vector xp. In [7], given a vector υ≫0, the weighted l∞ norm is defined by(1)x∞υ=max1≤j≤n⁡xjυj.For matrix A∈Rn×n, let aij denote its entry in row i and column j. A real n×n matrix A=aijn×n is Metzler if and only if its off-diagonal entries aij,i≠j are nonnegative.For a real interval[a,b], let Ca,b,Rn be the space of all continuous functions on [a,b] taking values in Rn. The upper-right Dini derivative of a continuous function h:R→R is denoted by D+h·. (2)D+htt=t0=limΔ→0+⁡sup⁡ht0+Δ-ht0Δ.


### 2.2. Preliminaries

Next, we present some definitions and results which are used in this paper.Definition 1.
A vector fieldf:Rn→Rn is called homogeneous of degree α>0, if, for all x∈Rn and λ∈R, fλx=λαfx.Definition 2.
A continuous vector fieldf:Rn→Rn which is continuously differentiable on Rn∖{0} is said to be cooperative if the Jacobian matrix (∂f/∂x)a is Metzler for all a∈R+n∖{0}.It follows from [1, Remark 3.1] that the cooperative systems satisfy the following property.Proposition 3.
Letf:Rn→Rn be cooperative. For any two vectors x and y in R+n with x≥y and xj=yj, one has fjx≥fjy.Definition 4.
A vector field g:Rn→Rn is said to be order-preserving on R+n∖{0}, if gx≥gy for any x,y∈R+n∖{0} such that x≥y.


## 2.1. Notation

Throughout the paper, vectors are written in bold lowercase letters. LetR, N, and N0 denote the set of real numbers, natural numbers, and the set of natural numbers including zero, respectively. Rn stands for the n-dimensional Euclidean space. Let R+n be the set of all vectors in Rn with nonnegative entries; that is, R+n≔x∈Rn,xj≥0,1≤j≤n. For two vectors, x,y∈Rn, we write x≥y, if xj≥yj, for 1≤j≤n; x>y, if x≥y and x≠y; x≫y, if xj>yj1≤j≤n. xj is used to denote the jth component of x. Similarly, xpj denotes the jth coordinate of vector xp. In [7], given a vector υ≫0, the weighted l∞ norm is defined by(1)x∞υ=max1≤j≤n⁡xjυj.For matrix A∈Rn×n, let aij denote its entry in row i and column j. A real n×n matrix A=aijn×n is Metzler if and only if its off-diagonal entries aij,i≠j are nonnegative.For a real interval[a,b], let Ca,b,Rn be the space of all continuous functions on [a,b] taking values in Rn. The upper-right Dini derivative of a continuous function h:R→R is denoted by D+h·. (2)D+htt=t0=limΔ→0+⁡sup⁡ht0+Δ-ht0Δ.


## 2.2. Preliminaries

Next, we present some definitions and results which are used in this paper.Definition 1.
A vector fieldf:Rn→Rn is called homogeneous of degree α>0, if, for all x∈Rn and λ∈R, fλx=λαfx.Definition 2.
A continuous vector fieldf:Rn→Rn which is continuously differentiable on Rn∖{0} is said to be cooperative if the Jacobian matrix (∂f/∂x)a is Metzler for all a∈R+n∖{0}.It follows from [1, Remark 3.1] that the cooperative systems satisfy the following property.Proposition 3.
Letf:Rn→Rn be cooperative. For any two vectors x and y in R+n with x≥y and xj=yj, one has fjx≥fjy.Definition 4.
A vector field g:Rn→Rn is said to be order-preserving on R+n∖{0}, if gx≥gy for any x,y∈R+n∖{0} such that x≥y.


## 3. Main Results

### 3.1. Exponential Stability of Unswitched Positive Homogeneous Systems

We first consider positive system with time-varying delays:(3)x˙t=fxt+gxt-τt,t≥0,xt=θt,t∈-dmax,0,where xt∈Rn is the state variable and f,g:Rn→Rn are continuous vector fields on Rn, continuously differentiable on Rn∖{0}. θt∈C-dmax,0,Rn is the initial condition. In addition, we assume that system (3) satisfies the following conditions.Assumption 5.
(i)
f and g are homogeneous of degree of α>0.
(ii)
f is cooperative and g is order-preserving on R+n∖{0}.Assumption 6.
The time-varying delayτt:R+→R+ is continuous and satisfies (4)0≤τt≤dmax.Here τt is not necessarily continuously differentiable and no restriction on its derivative such as τ˙t<1 is imposed.Remark 7.
Condition (i) of Assumption5 implies that f0=g0=0. Furthermore, as f is cooperative and g is order-preserving on R+n, one has(5)xj=0⟹fjx≥0,∀j∈1,2,…,n,∀x∈R+ngx≥0,∀x∈R+n.It has been shown in [7, Proposition 3.2] that (5) implies that system (3) is positive for any initial condition θ·∈R+n. Throughout this paper, we always assume that the initial conditions are nonnegative.Before giving the sufficient condition for the exponential stability of system (3), a lemma that provides a precise estimation of the upper bound of system (3) is proposed first.Lemma 8 (see [7]).
Consider system (3) under Assumptions 5 and 6. If there exists a vector υ≫0 such that fυ+gυ≪0, then the solution xt satisfies (6)xt∞υ≤θ,t∈0,+∞,where θ=sup-dmax≤t≤0⁡max1≤j≤n⁡θjt/υj.Definition 9.
System (3) is said to be exponentially stable if there exist two constants a>0 and b>0 such that (7)xt≤ae-btx0,t≥0,where · is some norm in Rn.The following theorem states a sufficient condition for exponential stability of system (3) with 0<α≤1.Theorem 10.
For system (3) with the degree of 0<α≤1, suppose that Assumptions 5 and 6 hold. If there exists a vector υ≫0 such that (8)fυ+gυ≪0,then system (3) is exponentially stable for any nonnegative initial conditions. In particular, every solution of system (3) satisfies (9)xt∞υ≤eξdmaxθe-ξt,t≥0,where 0<ξ<min1≤j≤n⁡ξj and ξj is the unique positive solution to(10)θξj+θαfjυυj+eξjdmaxαgjυυj=0.Proof.
Note that (10) has two parameters: the maximum delay dmax and ξj. So, naturally for fixed dmax≥0, define a set of functions with respect to ξ: (11)φjξ=θξ+θαfjυυj+eξdmaxαgjυυj,j=1,2,…,n.It now follows from (10) that (12)φjξj=0∀j∈1,2,…,n.Furthermore, it can be easily checked that φjξ is strictly monotonically increasing in ξ>0, which implies that (13)φ1ξ<0,ξ<ξ1,φ2ξ<0,ξ<ξ2,⋮φnξ<0,ξ<ξn.So if ξ∈0,min1≤j≤n⁡ξj, we have(14)φjξ=θξ+θαfjυυj+eξdmaxαgjυυj<0∀j∈1,2,…,n.Now, inspired by the method developed in [6], let(15)ωjt=eξtxjtvj-eξdmaxθ.Then it follows from Lemma 8 that (16)eξtxjtvj≤eξdmaxxjtvj≤eξdmaxθ∀t∈0,dmax,j∈1,2,…,n.It is immediate to see that, for any j∈1,2,…,n, (17)ωjt≤0,0≤t≤dmax.In order to prove exponential stability of system (3), we still need to prove that, for all j and all t≥dmax,ωjt≤0. Obviously, ωj(t)t=dmax⁡≤0 for all j. As ωjt is continuous function, we claim that it is also holds for all t≥dmax and all j. By contradiction, suppose that this is not always true. Then there exist an index s∈1,2,…,n and a time t∗∈dmax,+∞ such that(18)ωjt≤0∀t∈dmax,t∗,j∈1,2,…,n,(19)ωst∗=0,(20)D+ωstt=t∗≥0.From (18) and (19), we have (21)xjt∗≤eξdmaxe-ξt∗θυj,j=1,2,…,n,j≠s,xst∗=eξdmaxe-ξt∗θυs.Then it follows from cooperativity and homogeneity of f that(22)fsxt∗≤eξdmaxe-ξt∗αθαfsυ.Note that t∗-τt∗∈0,t∗, so we obtain (23)xt∗-τt∗≤eξdmaxe-ξt∗-τt∗θυ.Furthermore, as g is order-preserving and homogeneous, it in turn implies that(24)gsxt∗-τt∗≤eξdmaxe-ξt∗-τt∗αθαgsυ.The upper-right Dini derivative of ωst along the trajectories of system (3) at t=t∗ is given by(25)D+ωstt=t∗=ξeξt∗xst∗vs+eξt∗x˙st∗υs=ξeξt∗xst∗vs+eξt∗fsxt∗+gsxt∗-τt∗υs≤ξeξdmaxθ+eξt∗1-α+ξdmaxαθαfsυυs+eξτt∗αgsυυs≤ξeξdmaxθ+eξt∗+αdmax-t∗θαfsυυs+eξdmaxαgsυυs≤ξeξdmaxθ+eξdmaxθαfsυυs+eξdmaxαgsυυs=eξdmaxθξ+θαfsυυs+eξdmaxαgsυυs<0,where we have used (22) and (24) to get the first inequality. Note also that 0≤τt≤dmax and fsυ/υs+eξdmaxαgsυ/υs<0, so the second and third inequalities are true, and the last inequality is from (14). Now we arrive at a contradiction with (20). Hence, ωjt≤0; for all t≥0, j∈1,2,…,n, which means that (26)xt∞υ≤eξdmaxθe-ξt,t∈0,+∞.This completes the proof.Remark 11.
Theorem10 provides an estimate on how the decay rate depends on initial condition with the degree of 0<α≤1, which is fundamental to study the switched positive systems.Next, consider the corresponding delay-free system:(27)x˙t=fxt+gxt.In the following, we give a sufficient and necessary condition for the exponential stability of system (27).Corollary 12.
Consider system (27) with 0<α≤1 under Assumption 5; then the following statements are equivalent:(1)
There exists a vectorυ≫0 such that fυ+gυ≪0.(2)
System (27) is exponentially stable and every solution xt satisfies (28)xt∞υ≤x0∞υe-γt,where 0<γ<min1≤j≤n⁡γj with γj being the unique positive solution to(29)x0∞υγj+x0∞υαfjυυj+gjυυj=0.Proof.
1
⇒
2
: according to Theorem 10, system (3) is exponentially stable for all delays satisfying Assumption 6. Particularly, let τt=0; then dmax=0 and θ=x0∞υ, which implies that (29) holds.
2
⇒
1
: as system (27) is exponentially stable, it is asymptotically stable. Therefore, it follows from [20, Proposition 3.10, Theorem 3.12] that there exists a vector υ≫0 satisfying fυ+gυ≪0.Remark 13.
From (29), one can verify that(30)x0∞υγ+x0∞υαfjυυj+gjυυj<0.We now consider two cases.Case 1 (x0∞υ≤1).
As0<α≤1, we can get x0∞υ≤x0∞υα. Hence, it follows from fjυ/υj+gjυ/υj<0 that(31)x0∞υγ+x0∞υαfjυυj+gjυυj≤x0∞υγ+fjυυj+gjυυj.Case 2 (x0∞υ>1).
From0<α≤1, we have x0∞υα>1. Thus(32)x0∞υγ+x0∞υαfjυυj+gjυυj≤x0∞υγ+fjυυj+gjυυj.Next, let γj,γj′ be the positive solutions of the following equations: (33)γj+fjυυj+gjυυj=0,x0∞υγj′+fjυυj+gjυυj=0.If γ∈0,min1≤j≤n⁡γj,γj′, then we have(34)γ+fjυυj+gjυυj<0,x0∞υγ+fjυυj+gjυυj<0.Hence, it is easy to verify that (34) ensures that (30) holds. Meanwhile, from the proof of Theorem 10, (30) is the key factor to prove the exponential stability of system (27).


### 3.2. Exponential Stability of Switched Positive Homogeneous Systems

In the following, we consider the following switched positive nonlinear system:(35)x˙t=fσtxt+gσtxt,where xt∈Rn is the state vector and σt:0,+∞→M denotes the switching signal. The finite set M=1,2,…,m is an index set and stands for the collection of subsystems. For all p∈M, fp and gp:Rn→Rn are continuously differentiable on Rn∖{0}.Remark 14.
It should be pointed that, for everyp∈M, as fp is defined to be cooperative and homogeneous and gp is order-preserving and homogeneous, system (35) is positive under arbitrary switching laws. This implies that, for any initial condition x0∈R+n, the corresponding state trajectory xt∈R+n for all t≥0.Lemma 15 (see [19]).
Assume thatx∈Rn; then, for any p,q∈M, (36)x∞υ¯≤x∞υp≤x∞υ_≤βx∞υq,where β=max1≤j≤n⁡(υ¯j/υ_j) with υ¯j=maxp∈M⁡υpj, υ_j=minp∈M⁡υpj, υ¯=υ¯1,υ¯2,…,υ¯n, and υ_=υ1_,υ2_,…,υn_.Based on Theorem10 and Lemma 15, we next establish a sufficient condition for exponential stability of switched system (35) and the symbols that are defined in Lemma 15 continue to be used. In addition, let Q={x∈R+n∖{0}∣xi≤ai,ai∈R,i=1,2,…,n}, where ai>0 is any given constant. Further define (37)ε=supx∈Q⁡x∞υ_.Theorem 16.
For system (35), suppose that, for every p∈M, fp,gp satisfy Assumption 5 with 0<α≤1. If there exists a vector υp≫0 such that (38)fpυp+gpυp≪0,∀p∈M,then system (35) is exponentially stable for any given set Q under MDT switching signal τ satisfying(39)τ>ln⁡βλ,where 0<λ<min⁡μ,η with μ=minp∈M⁡min1≤j≤n⁡λpj and η=minp∈M⁡min1≤j≤n⁡λpj∗ and λpj and λpj∗ are the solutions to the following equations:(40)λpj+fpjυpυpj+gpjυpυpj=0,p∈M,(41)ελpj∗+fpjυpυpj+gpjυpυpj=0,p∈M.Proof.
Consider a switching sequence(42)0=t0<t1<t2<⋯<tk<tk+1<⋯.From (39), we have Δtk=tk+1-tk≥τ>ln⁡β/λ,k∈N0.
The proof now proceeds in two steps.
Step 1. First, for any x0∈Q, we show that(43)xtk∞υσtk-1≤xtk-1∞υσtk-1e-λtk-tk-1,k∈Nxtk∞υσtk≤x0∞υσ0,k∈N.Note that, for every p∈M, there exists υp≫0 such that fpυp+gpυp≪0, so, from Corollary 12, for the first interval t0,t1, we have(44)xt1∞υσ0≤x0∞υσ0e-λt1≤1βx0∞υσ0,where we have used Δt1=t1≥τ>ln⁡β/λ to get the second inequality. Furthermore, we obtain(45)xt1∞υσt1≤βxt1∞υσ0≤β1βx0∞υσ0=x0∞υσ0.Next, consider the second interval t1,t2. It follows from Corollary 12 that(46)xt2∞υσt1≤xt1∞υσt1e-λ1t2-t1.From Remark 13, we have 0<λ1<min1≤j≤n⁡λ¯j,λ¯j′, where λ¯j and λ¯j′ are the positive solutions of the following equations:(47)λ¯j+fυσt1jυσt1υσt1j+gυσt1jυσt1υσt1j=0,(48)xt1∞υσt1λ¯j′+fυσt1jυσt1υσt1j+gυσt1jυσt1υσt1j=0.Observe (40) and (47). For p∈M, let (49)Sp=λpj:λpj+fpjυpυpj+gpjυpυpj=0,j=1,2,…,n,S¯=λ¯j:λ¯j+fυσt1jυσt1υσt1j+gυσt1jυσt1υσt1j=0,j=1,2,…,n.It is easy to verify that S¯⊂⋃p∈MSp; thus (50)minp∈M⁡minλpj∈Sp⁡λpj≤minλ¯j∈S¯⁡λ¯j.That is,(51)μ≤min1≤j≤n⁡λ¯j.On the other hand, from (45), we have xt1∞υσt1≤x0∞υσ0. Furthermore, from the definition of ε, we can get xt1∞υσt1≤ε. Hence, comparing (41) and (48), one can check that (52)η≤min1≤j≤n⁡λ¯j′.This together with (51) implies that (53)min⁡μ,η≤min1≤j≤n⁡λ¯j,λ¯j′.So if for any λ1∈0,min1≤j≤n⁡λ¯j,λ¯j′ (46) holds, then(54)xt2∞υσt1≤xt1∞υσt1e-λt2-t1,where λ∈0,min⁡μ,η. Now it follows from Lemma 15 that(55)xt2∞υσt2≤βxt2∞υσt1≤β1βxt1∞υσt1≤x0∞υσ0,where we have used (39) and (54) to get the second inequality. From (44), (45), (54), and (55), we conclude that (43) hold for k=1,2.
By induction, we assume that (43) hold for a given kk≥3. Next we prove they are true for k+1. Consider the interval tk,tk+1. It follows from Corollary 12 that (56)xtk+1∞υσtk≤xtk∞υσtke-λktk+1-tk,where 0<λk<min1≤j≤n⁡λ~j,λ~j′ with λ~j and λ~j′ satisfying (57)λ~j+fυσtkjυσtkυσtkj+gυσtkjυσtkυσtkj=0,xtk∞υσtkλ~j′+fυσtkjυσtkυσtkj+gυσtkjυσtkυσtkj=0.Then we can prove (58)xtk+1∞υσtk≤xtk∞υσtke-λtk+1-tk,xtk+1∞υσtk+1≤x0∞υσ0.The rest of the proof is similar to the one for the case of k=2 and thus is omitted.
According to the previous arguments, we conclude that (43) hold for each k∈N, which implies that for all subsystems we find the common exponential decay rate λ. In addition, We also prove that xtk∞υσtk≤x0∞υσ0 under minimum dwell time switching, where xk is the initial condition of the k+1th subsystem.
Step 2. We prove that system (35) is exponentially stable. For any t∈0,+∞, assume that t∈tk,tk+1 on which σt=p. Then, from Step 1, we have (59)xt∞υp≤xtk∞υpe-λt-tk.Moreover, applying Lemma 15, we can get (60)xt∞υp≤xtk∞υpe-λt-tk≤βxtk∞υσtk-1e-λt-tk≤βxtk-1∞υσtk-1e-λt-tk-1⋮≤βt/τx0∞υσ0e-λt,where we have used the MDT to obtain the last inequality. On the other hand, note that xt∞υ¯≤xt∞υp. Therefore (61)xt∞υ¯≤βt/τx0∞υσ0e-λt=et/τln⁡βx0∞υσ0e-λt=e-λ-ln⁡β/τtx0∞υσ0≤βx0∞υ¯e-λ-ln⁡β/τt.In summary, system (35) is exponentially stable for any given set Q under MDT switching.
This is the end of the proof.


## 3.1. Exponential Stability of Unswitched Positive Homogeneous Systems

We first consider positive system with time-varying delays:(3)x˙t=fxt+gxt-τt,t≥0,xt=θt,t∈-dmax,0,where xt∈Rn is the state variable and f,g:Rn→Rn are continuous vector fields on Rn, continuously differentiable on Rn∖{0}. θt∈C-dmax,0,Rn is the initial condition. In addition, we assume that system (3) satisfies the following conditions.Assumption 5.
(i)
f and g are homogeneous of degree of α>0.
(ii)
f is cooperative and g is order-preserving on R+n∖{0}.Assumption 6.
The time-varying delayτt:R+→R+ is continuous and satisfies (4)0≤τt≤dmax.Here τt is not necessarily continuously differentiable and no restriction on its derivative such as τ˙t<1 is imposed.Remark 7.
Condition (i) of Assumption5 implies that f0=g0=0. Furthermore, as f is cooperative and g is order-preserving on R+n, one has(5)xj=0⟹fjx≥0,∀j∈1,2,…,n,∀x∈R+ngx≥0,∀x∈R+n.It has been shown in [7, Proposition 3.2] that (5) implies that system (3) is positive for any initial condition θ·∈R+n. Throughout this paper, we always assume that the initial conditions are nonnegative.Before giving the sufficient condition for the exponential stability of system (3), a lemma that provides a precise estimation of the upper bound of system (3) is proposed first.Lemma 8 (see [7]).
Consider system (3) under Assumptions 5 and 6. If there exists a vector υ≫0 such that fυ+gυ≪0, then the solution xt satisfies (6)xt∞υ≤θ,t∈0,+∞,where θ=sup-dmax≤t≤0⁡max1≤j≤n⁡θjt/υj.Definition 9.
System (3) is said to be exponentially stable if there exist two constants a>0 and b>0 such that (7)xt≤ae-btx0,t≥0,where · is some norm in Rn.The following theorem states a sufficient condition for exponential stability of system (3) with 0<α≤1.Theorem 10.
For system (3) with the degree of 0<α≤1, suppose that Assumptions 5 and 6 hold. If there exists a vector υ≫0 such that (8)fυ+gυ≪0,then system (3) is exponentially stable for any nonnegative initial conditions. In particular, every solution of system (3) satisfies (9)xt∞υ≤eξdmaxθe-ξt,t≥0,where 0<ξ<min1≤j≤n⁡ξj and ξj is the unique positive solution to(10)θξj+θαfjυυj+eξjdmaxαgjυυj=0.Proof.
Note that (10) has two parameters: the maximum delay dmax and ξj. So, naturally for fixed dmax≥0, define a set of functions with respect to ξ: (11)φjξ=θξ+θαfjυυj+eξdmaxαgjυυj,j=1,2,…,n.It now follows from (10) that (12)φjξj=0∀j∈1,2,…,n.Furthermore, it can be easily checked that φjξ is strictly monotonically increasing in ξ>0, which implies that (13)φ1ξ<0,ξ<ξ1,φ2ξ<0,ξ<ξ2,⋮φnξ<0,ξ<ξn.So if ξ∈0,min1≤j≤n⁡ξj, we have(14)φjξ=θξ+θαfjυυj+eξdmaxαgjυυj<0∀j∈1,2,…,n.Now, inspired by the method developed in [6], let(15)ωjt=eξtxjtvj-eξdmaxθ.Then it follows from Lemma 8 that (16)eξtxjtvj≤eξdmaxxjtvj≤eξdmaxθ∀t∈0,dmax,j∈1,2,…,n.It is immediate to see that, for any j∈1,2,…,n, (17)ωjt≤0,0≤t≤dmax.In order to prove exponential stability of system (3), we still need to prove that, for all j and all t≥dmax,ωjt≤0. Obviously, ωj(t)t=dmax⁡≤0 for all j. As ωjt is continuous function, we claim that it is also holds for all t≥dmax and all j. By contradiction, suppose that this is not always true. Then there exist an index s∈1,2,…,n and a time t∗∈dmax,+∞ such that(18)ωjt≤0∀t∈dmax,t∗,j∈1,2,…,n,(19)ωst∗=0,(20)D+ωstt=t∗≥0.From (18) and (19), we have (21)xjt∗≤eξdmaxe-ξt∗θυj,j=1,2,…,n,j≠s,xst∗=eξdmaxe-ξt∗θυs.Then it follows from cooperativity and homogeneity of f that(22)fsxt∗≤eξdmaxe-ξt∗αθαfsυ.Note that t∗-τt∗∈0,t∗, so we obtain (23)xt∗-τt∗≤eξdmaxe-ξt∗-τt∗θυ.Furthermore, as g is order-preserving and homogeneous, it in turn implies that(24)gsxt∗-τt∗≤eξdmaxe-ξt∗-τt∗αθαgsυ.The upper-right Dini derivative of ωst along the trajectories of system (3) at t=t∗ is given by(25)D+ωstt=t∗=ξeξt∗xst∗vs+eξt∗x˙st∗υs=ξeξt∗xst∗vs+eξt∗fsxt∗+gsxt∗-τt∗υs≤ξeξdmaxθ+eξt∗1-α+ξdmaxαθαfsυυs+eξτt∗αgsυυs≤ξeξdmaxθ+eξt∗+αdmax-t∗θαfsυυs+eξdmaxαgsυυs≤ξeξdmaxθ+eξdmaxθαfsυυs+eξdmaxαgsυυs=eξdmaxθξ+θαfsυυs+eξdmaxαgsυυs<0,where we have used (22) and (24) to get the first inequality. Note also that 0≤τt≤dmax and fsυ/υs+eξdmaxαgsυ/υs<0, so the second and third inequalities are true, and the last inequality is from (14). Now we arrive at a contradiction with (20). Hence, ωjt≤0; for all t≥0, j∈1,2,…,n, which means that (26)xt∞υ≤eξdmaxθe-ξt,t∈0,+∞.This completes the proof.Remark 11.
Theorem10 provides an estimate on how the decay rate depends on initial condition with the degree of 0<α≤1, which is fundamental to study the switched positive systems.Next, consider the corresponding delay-free system:(27)x˙t=fxt+gxt.In the following, we give a sufficient and necessary condition for the exponential stability of system (27).Corollary 12.
Consider system (27) with 0<α≤1 under Assumption 5; then the following statements are equivalent:(1)
There exists a vectorυ≫0 such that fυ+gυ≪0.(2)
System (27) is exponentially stable and every solution xt satisfies (28)xt∞υ≤x0∞υe-γt,where 0<γ<min1≤j≤n⁡γj with γj being the unique positive solution to(29)x0∞υγj+x0∞υαfjυυj+gjυυj=0.Proof.
1
⇒
2
: according to Theorem 10, system (3) is exponentially stable for all delays satisfying Assumption 6. Particularly, let τt=0; then dmax=0 and θ=x0∞υ, which implies that (29) holds.
2
⇒
1
: as system (27) is exponentially stable, it is asymptotically stable. Therefore, it follows from [20, Proposition 3.10, Theorem 3.12] that there exists a vector υ≫0 satisfying fυ+gυ≪0.Remark 13.
From (29), one can verify that(30)x0∞υγ+x0∞υαfjυυj+gjυυj<0.We now consider two cases.Case 1 (x0∞υ≤1).
As0<α≤1, we can get x0∞υ≤x0∞υα. Hence, it follows from fjυ/υj+gjυ/υj<0 that(31)x0∞υγ+x0∞υαfjυυj+gjυυj≤x0∞υγ+fjυυj+gjυυj.Case 2 (x0∞υ>1).
From0<α≤1, we have x0∞υα>1. Thus(32)x0∞υγ+x0∞υαfjυυj+gjυυj≤x0∞υγ+fjυυj+gjυυj.Next, let γj,γj′ be the positive solutions of the following equations: (33)γj+fjυυj+gjυυj=0,x0∞υγj′+fjυυj+gjυυj=0.If γ∈0,min1≤j≤n⁡γj,γj′, then we have(34)γ+fjυυj+gjυυj<0,x0∞υγ+fjυυj+gjυυj<0.Hence, it is easy to verify that (34) ensures that (30) holds. Meanwhile, from the proof of Theorem 10, (30) is the key factor to prove the exponential stability of system (27).


## 3.2. Exponential Stability of Switched Positive Homogeneous Systems

In the following, we consider the following switched positive nonlinear system:(35)x˙t=fσtxt+gσtxt,where xt∈Rn is the state vector and σt:0,+∞→M denotes the switching signal. The finite set M=1,2,…,m is an index set and stands for the collection of subsystems. For all p∈M, fp and gp:Rn→Rn are continuously differentiable on Rn∖{0}.Remark 14.
It should be pointed that, for everyp∈M, as fp is defined to be cooperative and homogeneous and gp is order-preserving and homogeneous, system (35) is positive under arbitrary switching laws. This implies that, for any initial condition x0∈R+n, the corresponding state trajectory xt∈R+n for all t≥0.Lemma 15 (see [19]).
Assume thatx∈Rn; then, for any p,q∈M, (36)x∞υ¯≤x∞υp≤x∞υ_≤βx∞υq,where β=max1≤j≤n⁡(υ¯j/υ_j) with υ¯j=maxp∈M⁡υpj, υ_j=minp∈M⁡υpj, υ¯=υ¯1,υ¯2,…,υ¯n, and υ_=υ1_,υ2_,…,υn_.Based on Theorem10 and Lemma 15, we next establish a sufficient condition for exponential stability of switched system (35) and the symbols that are defined in Lemma 15 continue to be used. In addition, let Q={x∈R+n∖{0}∣xi≤ai,ai∈R,i=1,2,…,n}, where ai>0 is any given constant. Further define (37)ε=supx∈Q⁡x∞υ_.Theorem 16.
For system (35), suppose that, for every p∈M, fp,gp satisfy Assumption 5 with 0<α≤1. If there exists a vector υp≫0 such that (38)fpυp+gpυp≪0,∀p∈M,then system (35) is exponentially stable for any given set Q under MDT switching signal τ satisfying(39)τ>ln⁡βλ,where 0<λ<min⁡μ,η with μ=minp∈M⁡min1≤j≤n⁡λpj and η=minp∈M⁡min1≤j≤n⁡λpj∗ and λpj and λpj∗ are the solutions to the following equations:(40)λpj+fpjυpυpj+gpjυpυpj=0,p∈M,(41)ελpj∗+fpjυpυpj+gpjυpυpj=0,p∈M.Proof.
Consider a switching sequence(42)0=t0<t1<t2<⋯<tk<tk+1<⋯.From (39), we have Δtk=tk+1-tk≥τ>ln⁡β/λ,k∈N0.
The proof now proceeds in two steps.
Step 1. First, for any x0∈Q, we show that(43)xtk∞υσtk-1≤xtk-1∞υσtk-1e-λtk-tk-1,k∈Nxtk∞υσtk≤x0∞υσ0,k∈N.Note that, for every p∈M, there exists υp≫0 such that fpυp+gpυp≪0, so, from Corollary 12, for the first interval t0,t1, we have(44)xt1∞υσ0≤x0∞υσ0e-λt1≤1βx0∞υσ0,where we have used Δt1=t1≥τ>ln⁡β/λ to get the second inequality. Furthermore, we obtain(45)xt1∞υσt1≤βxt1∞υσ0≤β1βx0∞υσ0=x0∞υσ0.Next, consider the second interval t1,t2. It follows from Corollary 12 that(46)xt2∞υσt1≤xt1∞υσt1e-λ1t2-t1.From Remark 13, we have 0<λ1<min1≤j≤n⁡λ¯j,λ¯j′, where λ¯j and λ¯j′ are the positive solutions of the following equations:(47)λ¯j+fυσt1jυσt1υσt1j+gυσt1jυσt1υσt1j=0,(48)xt1∞υσt1λ¯j′+fυσt1jυσt1υσt1j+gυσt1jυσt1υσt1j=0.Observe (40) and (47). For p∈M, let (49)Sp=λpj:λpj+fpjυpυpj+gpjυpυpj=0,j=1,2,…,n,S¯=λ¯j:λ¯j+fυσt1jυσt1υσt1j+gυσt1jυσt1υσt1j=0,j=1,2,…,n.It is easy to verify that S¯⊂⋃p∈MSp; thus (50)minp∈M⁡minλpj∈Sp⁡λpj≤minλ¯j∈S¯⁡λ¯j.That is,(51)μ≤min1≤j≤n⁡λ¯j.On the other hand, from (45), we have xt1∞υσt1≤x0∞υσ0. Furthermore, from the definition of ε, we can get xt1∞υσt1≤ε. Hence, comparing (41) and (48), one can check that (52)η≤min1≤j≤n⁡λ¯j′.This together with (51) implies that (53)min⁡μ,η≤min1≤j≤n⁡λ¯j,λ¯j′.So if for any λ1∈0,min1≤j≤n⁡λ¯j,λ¯j′ (46) holds, then(54)xt2∞υσt1≤xt1∞υσt1e-λt2-t1,where λ∈0,min⁡μ,η. Now it follows from Lemma 15 that(55)xt2∞υσt2≤βxt2∞υσt1≤β1βxt1∞υσt1≤x0∞υσ0,where we have used (39) and (54) to get the second inequality. From (44), (45), (54), and (55), we conclude that (43) hold for k=1,2.
By induction, we assume that (43) hold for a given kk≥3. Next we prove they are true for k+1. Consider the interval tk,tk+1. It follows from Corollary 12 that (56)xtk+1∞υσtk≤xtk∞υσtke-λktk+1-tk,where 0<λk<min1≤j≤n⁡λ~j,λ~j′ with λ~j and λ~j′ satisfying (57)λ~j+fυσtkjυσtkυσtkj+gυσtkjυσtkυσtkj=0,xtk∞υσtkλ~j′+fυσtkjυσtkυσtkj+gυσtkjυσtkυσtkj=0.Then we can prove (58)xtk+1∞υσtk≤xtk∞υσtke-λtk+1-tk,xtk+1∞υσtk+1≤x0∞υσ0.The rest of the proof is similar to the one for the case of k=2 and thus is omitted.
According to the previous arguments, we conclude that (43) hold for each k∈N, which implies that for all subsystems we find the common exponential decay rate λ. In addition, We also prove that xtk∞υσtk≤x0∞υσ0 under minimum dwell time switching, where xk is the initial condition of the k+1th subsystem.
Step 2. We prove that system (35) is exponentially stable. For any t∈0,+∞, assume that t∈tk,tk+1 on which σt=p. Then, from Step 1, we have (59)xt∞υp≤xtk∞υpe-λt-tk.Moreover, applying Lemma 15, we can get (60)xt∞υp≤xtk∞υpe-λt-tk≤βxtk∞υσtk-1e-λt-tk≤βxtk-1∞υσtk-1e-λt-tk-1⋮≤βt/τx0∞υσ0e-λt,where we have used the MDT to obtain the last inequality. On the other hand, note that xt∞υ¯≤xt∞υp. Therefore (61)xt∞υ¯≤βt/τx0∞υσ0e-λt=et/τln⁡βx0∞υσ0e-λt=e-λ-ln⁡β/τtx0∞υσ0≤βx0∞υ¯e-λ-ln⁡β/τt.In summary, system (35) is exponentially stable for any given set Q under MDT switching.
This is the end of the proof.


## 4. Numerical Simulation

Consider the switched nonlinear positive system consisting of two subsystems given by(62)Γ1:x˙t=f1xt+g1xt,Γ2:x˙t=f2xt+g2xt,where(63)f1xt=x1t+2x2t1/2-4x1t+x2t1/22x1t+x2t1/2-x1t+4x2t1/2,g1xt=16x1t1/218x2t1/2,f2xt=x1t+x2t1/2-3x1t+x2t1/2x1t+x2t1/2-x1t+3x2t1/2,g2xt=17x1t+x2t1/21102x1t+x2t1/2.It is easy to verify that f1, f2, g1, and g2 satisfy the conditions of Theorem 16 with homogeneous degree of α=1/2 and(64)f110,8+g110,8=-1.3021,-0.8357T≪0,f28,10+g28,10=-0.9822,-1.4119T≪0.Let Q=x∈R+n∣xi≤8,i=1,2; then ε=1. It follows from Theorem 16 that the corresponding switched nonlinear system is exponentially stable under minimum dwell time τ>2.2314. Figure 1 shows the actual rate and the upper bound which agree with the implication of Theorem 16.Figure 1
The simulation results of the actual decay ratext∞υ¯ and the guaranteed upper bound we calculated with the initial condition x0=7,8T and the MDT τ=3.


## 5. Conclusion

In this paper, we derive a sufficient condition for the exponential stability of switched positive homogeneous systems with degree of0<α≤1. Some results in literature [19] are extended. In addition, a sufficient condition is presented for the exponential stability of the unswitched positive systems with time-varying delays. A numerical example is given to demonstrate the main result. While the stability problem of such system with delays is the subject of ongoing work, we hope to report the result of which in the future.



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