# Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone

Authors: D. Ya. Khusainov; J. Diblík; Z. Svoboda; Z. Šmarda
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The present investigation deals with global instability of a generaln-dimensional system
of ordinary differential equations with quadratic right-hand sides. The global instability
of the zero solution in a given cone is proved by Chetaev's method, assuming that the
matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have
negative real parts. The sufficient conditions for global instability obtained are formulated
by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result
is used on the positivity of a general third-degree polynomial in two variables to estimate
the sign of the full derivative of an appropriate function in a cone.


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## Body

## 1. Introduction

Recently, there has been a rapidly growing interest in investigating the instability conditions of differential systems. The number of papers dealing with instability problems is rather low compared with the huge quantity of papers in which the stability of the motion of differential systems is investigated. The first results on the instability of zero solution of differential systems were obtained in a general form by Lyapunov [1] and Chetaev [2].Further investigation on the instability of solutions of systems was carried out to weaken the conditions of the Lyapunov and Chetaev theorems for special-form systems. Some results are presented, for example, in [3–10], but instability problems are analysed only locally. For example, in [7], a linear system of ordinary differential equations in the matrix form is considered, and conditions such that the corresponding forms (of the second and the third power) have fixed sign in some cone of the space ℝn are derived. To investigate this property another problem inverse to the known Lyapunov problem for the construction of Lyapunov functions is solved.In the present paper, instability solutions of systems with quadratic right-hand sides is investigated in a cone dealing with a generaln-dimensional system with quadratic right-hand sides. We assume that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts.Unlike the previous investigations, we prove the global instability of the zero solution in a given cone and the conditions for global instability are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. The main tool is the method of Chetaev and application of a suitable Chetaev-type function. A novelty in the proof of the main result (Theorem3.1) is the utilization of a general third-order polynomial inequality of two variables to estimate the sign of the full derivative of an appropriate function along the trajectories of a given system in a cone.In the sequel, the norms used for vectors and matrices are defined as(1.1)‖x‖=(∑i=1nxi2)1/2,
for a vector x=(x1,…,xn)T and (1.2)‖F‖=(λmax⁡(FTF))1/2,
for any m×n matrix ℱ. Here and throughout the paper, λmax(·) (or λmin(·)) is the maximal (or minimal) eigenvalue of the corresponding symmetric and positive-semidefinite matrix ℱTℱ (see, e.g., [11]).In this paper, we consider the instability of the trivial solution of a nonlinear autonomous differential system with quadratic right-hand sides(1.3)ẋi=∑s=1naisxs+∑s,q=1nbsqixsxq,i=1,…,n,
where coefficients ais and bsqi are constants. Without loss of generality, throughout this paper we assume(1.4)bsqi=bqsi.
As emphasized, for example, in [2, 10–12], system (1.3) can be written in a general vector-matrix form(1.5)ẋ=Ax+XTBx,
where A is an n×n constant square matrix, matrix XT is an n×n2 rectangular matrix (1.6)XT={X1T,X2T,…,XnT},
where the entries of the n×n square matrices Xi, i=1,…,n are equal to zero except the ith row with entries xT=(x1,x2,…,xn), that is, (1.7)XiT=(00⋯0⋯⋯⋯⋯00⋯0x1x2⋯xn00⋯0⋯⋯⋯⋯00⋯0),
and B is a rectangular n2×n matrix such that (1.8)BT={B1,B2,…,Bn},
where matrices Bi={bsqi}, i,s,q=1,…,n, that is, matrices (1.9)Bi=(b11ib12i⋯b1nib21ib22i⋯b2ni⋯⋯⋯⋯bn1ibn2i⋯bnni)
are n×n constant and symmetric. Representation (1.5) permits an investigation of differential systems with quadratic right-hand sides by methods of matrix analysis. Such approach was previously used, for example, in [13].If matrixA admits one simple positive eigenvalue, the system (1.5) can be transformed, using a suitable linear transformation of the dependent variables, to the same form (1.5) but with the matrix A having the form(1.10)A=(A0θθTλ),
where A0 is an (n-1)×(n-1) constant matrix, θ=(0,0,…,0)T is the (n-1)-dimensional zero vector and λ>0. With regard to this fact, we do not introduce new notations for the coefficients bsqi, i,s,q=1,2,…,n in (1.5), assuming throughout the paper that A in (1.5) has the form (1.10), preserving the old notations aij for entries of matrix A0. This means that we assume that A={ais}, i,s=1,2,…,n with ans=asn=0 for s=1,2,…,n-1 and ann=λ, and A0={ais}, i,s=1,2,…,n-1.We will give criteria of the instability of a trivial solution of the system (1.5) if the matrix A of linear terms is defined by (1.10).


## 2. Preliminaries

In this part we collect the necessary material-the definition of a cone, auxiliary Chetaev-type results on instability in a cone and, finally, a third degree polynomial inequality, which will be used to estimate the sign of the full derivative of a Chetaev-type function along the trajectories of system (1.5).


### 2.1. Instability of the Zero Solution of Systems of Differential Equations in a Cone

We consider an autonomous system of differential equations(2.1)ẋ=f(x),
where f:ℝn→ℝn satisfies a local Lipschitz condition and f(0)=0, that is, (2.1) admits the trivial solution. We will consider solutions of (2.1) determined by points (x,t)=(x0,0) where x0∈ℝn. The symbol x(x0,t) denotes the solution x=x(t) of (2.1), satisfying initial condition x(0)=x0.Definition 2.1.
The zero solutionx≡0 of (2.1) is called unstable if there exists ɛ>0 such that, for arbitrary δ>0, there exists an x0∈ℝn with ∥x0∥<δ and T≥0 such that ∥x(x0,T)∥≥ɛ.Definition 2.2.
A setK⊂Rn is called a cone if αx∈K for arbitrary x∈K and α>0.Definition 2.3.
A coneK is said to be a global cone of instability for (2.1) if x(x0,t)∈K for arbitrary x0∈K and t≥0 and limt→∞∥x(x0,t)∥=∞.Definition 2.4.
The zero solutionx≡0 of (2.1) is said to be globally unstable in a cone K if K is a global cone of instability for (2.1).Now, we prove results analogous to the classical Chetaev theorem (see, e.g., [2]) on instability in a form suitable for our analysis. As usual, if 𝒮 is a set, then ∂𝒮 denotes its boundary and 𝒮¯ its closure, that is, 𝒮¯:=𝒮∪∂𝒮.Theorem 2.5.
LetV:ℝn→ℝ, V(0,…,0)=0 be a continuously differentiable function. Assume that the set
(2.2)K={x∈Rn:V(x)>0}
is a cone. If the full derivative of V along the trajectories of (2.1) is positive for every x∈K, that is, if
(2.3)V̇(x):=gradTV(x)f(x)>0,x∈K,
then K is a global cone of instability for the system (2.1).Proof.
Letɛ be a positive number. We define a neighborhood of the origin
(2.4)Uɛ:={x∈Rn:‖x‖<ɛ},
and a constant
(2.5)Mɛ:=max⁡x∈U¯ɛ∩K¯V(x).
Moreover, define a set
(2.6)Wδ:={x∈U¯ɛ∩K¯,V(x)≥δ},
where δ is a positive number such that δ<Mɛ. Then, Wδ≠∅.
Letx0∈Wδ∩K, then V(x0)=δ1∈[δ,Mɛ]. We show that there exists a t=tT=tT(ɛ,x0) such that x(x0,tT)∉U¯ɛ and x(x0,tT)∈K.
Suppose to the contrary that this is not true andx(x0,t)∈U¯ɛ for all t≥0. Since V̇(x)>0, the function V is increasing along the solutions of (2.1). Thus x(x0,t) remains in K. Due to the compactness of Wδ, there exists a positive value β such that for x(x0,t)∈Wδ(2.7)ddtV(x(x0,t))=gradTV(x(x0,t))f(x(x0,t))>β.
Integrating this inequality over the interval [0,t], we get
(2.8)V(x(x0,t))-V(x0)=V(x(x0,t))-δ1>βt.
Then there exists a t=tT=tT(ɛ,x0) satisfying
(2.9)tT>(Mɛ-δ1)β,
such that V(x(x0,tT))>Mɛ and, consequently, x(x0,tT)∉U¯ɛ. This is contrary to our supposition. Since ɛ>0 is arbitrary, we have
(2.10)lim⁡t→∞‖x(x0,t)‖=∞,
that is, the zero solution is globally unstable, and K is a global cone of instability.Theorem 2.6.
LetV:ℝn→ℝ be a continuously differentiable function and let S,Z:ℝn→ℝ, Z(0,…,0)=0 be continuous functions such that V=S·Z. Assume that the set
(2.11)K1={x∈Rn:Z(x)>0}
is a cone, and S(x)>0 for any x∈K1. If the full derivative (2.3) of V along the trajectories of (2.1) is positive for every x∈K1, that is, if V̇(x)>0 for every x∈K1, then K1 is a global cone of instability for the system (2.1).Proof.
The proof is a modification of the proof of Theorem2.5. Let ɛ be a positive number. We define a neighborhood Uɛ of the origin by formula (2.4) and a constant
(2.12)Mɛ:=max⁡x∈U¯ɛ∩K1¯V(x).
Moreover, define a set
(2.13)Wδ:={x∈U¯ɛ∩K1¯,V(x)≥δ},
where δ is a positive number such that δ<Mɛ. Then Wδ≠∅.
Letx0∈Wδ∩K1. Then V(x0)=δ1∈[δ,Mɛ]. We show that there exists a t=tT=tT(ɛ,x0) such that x(x0,tT)∉U¯ɛ and x(x0,tT)∈K1.
Suppose to the contrary that this is not true andx(x0,t)∈U¯ɛ for all t≥0. Since V̇(x)>0, the function V is increasing along the solutions of (2.1). Due to the compactness of Wδ, there exists a positive value β such that for x(x0,t)∈Wδ(2.14)ddtV(x(x0,t))=gradTV(x(x0,t))f(x(x0,t))>β.
Integrating this inequality over interval [0,t], we get
(2.15)V(x(x0,t))-V(x0)=V(x(x0,t))-δ1=S(x(x0,t))Z(x(x0,t))-δ1>βt.
Since S(x(x0,t))>0, the inequality
(2.16)Z(x(x0,t))>δ1+βtS(x(x0,t))>0
is an easy consequence of (2.15). Thus x(x0,t) remains in K1. Apart from this, (2.15) also implies the existence of a t=tT=tT(ɛ,x0) satisfying
(2.17)tT>(Mɛ-δ1)β,
such that V(x(x0,tT))>Mɛ. Consequently, x(x0,tT)∉U¯ɛ. This is contrary to our supposition. Since ɛ>0 is arbitrary, we have
(2.18)lim⁡t→∞‖x(x0,t)‖=∞,
that is, the zero solution is globally unstable and K1 is a global cone of instability.Definition 2.7.
A functionV satisfying all the properties indicated in Theorem 2.5 is called a Chetaev function for the system (2.1). A function V satisfying all the properties indicated in Theorem 2.6 is called a Chetaev-type function for the system (2.1).


### 2.2. Auxiliary Inequality

Our results will be formulated in terms of global cones of instability. These will be derived using an auxiliary inequality valid in a given cone. Let(x,y)∈ℝ2 and let k be a positive number. We define a cone (2.19)K:={(x,y)∈R2:y>k|x|}.Lemma 2.8.
Leta, b, c, d, and k be given constants such that b>0, d>0, k>0, and |c|≤kd. Assume, moreover, either
(2.20)|a|≤kb,
or
(2.21)|a|>kb,(2.22)|c|≠kd,k≥max⁡{|a+kb|c+kd,|a-kb||c-kd|},
then
(2.23)ax3+bx2y+cxy2+dy3>0,
for every (x,y)∈𝒦.Proof.
We partition𝒦 into two disjoint cones
(2.24)K1:={(x,y)∈R2:y>k|x|,x>0},K2:={(x,y)∈R2:y>k|x|,x≤0},
and rewrite (2.23) as
(2.25)x(ax2+cy2)+y(bx2+dy2)>0.
We prove the validity of (2.23) in each of the two cones separately.
The case of thecone 𝒦1. Suppose that (2.20) holds. Estimating the left-hand side of (2.25), we get(2.26)x(ax2+cy2)+y(bx2+dy2)>x(ax2+cy2)+kx(bx2+dy2)=x[x2(a+kb)+y2(c+kd)]>0,
and (2.23) holds.
If inequalities (2.21) and (2.22) are valid, then, estimating the left-hand side of (2.25), we get(2.27)x(ax2+cy2)+y(bx2+dy2)>x(ax2+cy2)+kx(bx2+dy2)=x[x2(a+kb)+y2(c+kd)]≥x[-|a+kb|x2+(c+kd)y2]=(c+kd)x[y2-|a+kb|c+kdx2]=(c+kd)x[y-|a+kb|c+kdx][y+|a+kb|c+kdx]=(c+kd)x2[k-|a+kb|c+kd][k+|a+kb|c+kd]≥0,
and (2.23) holds again.
The case of thecone 𝒦2. Suppose that (2.20) hold, then, estimating the left-hand side of (2.25), we get
(2.28)x(ax2+cy2)+y(bx2+dy2)=-|x|(ax2+cy2)+y(bx2+dy2)>-|x|(ax2+cy2)+k|x|(bx2+dy2)=-|x|[(a-kb)x2+(c-kd)y2]≥0,
and (2.23) holds.
If inequalities (2.21) and (2.22) are valid, then the estimation of (2.25) implies (we use (2.28))(2.29)x(ax2+cy2)+y(bx2+dy2)>-|x|[(a-kb)x2+(c-kd)y2]=|c-kd||x|[y2-a-kb|c-kd|x2]={≥0ifa-kb<0,|c-kd||x|[y-a-kb|c-kd|x][y+a-kb|c-kd|x]≥|c-kd|x2[k+a-kb|c-kd|][k-a-kb|c-kd|]≥0ifa-kb>0.
Hence, (2.23) holds again.


## 2.1. Instability of the Zero Solution of Systems of Differential Equations in a Cone

We consider an autonomous system of differential equations(2.1)ẋ=f(x),
where f:ℝn→ℝn satisfies a local Lipschitz condition and f(0)=0, that is, (2.1) admits the trivial solution. We will consider solutions of (2.1) determined by points (x,t)=(x0,0) where x0∈ℝn. The symbol x(x0,t) denotes the solution x=x(t) of (2.1), satisfying initial condition x(0)=x0.Definition 2.1.
The zero solutionx≡0 of (2.1) is called unstable if there exists ɛ>0 such that, for arbitrary δ>0, there exists an x0∈ℝn with ∥x0∥<δ and T≥0 such that ∥x(x0,T)∥≥ɛ.Definition 2.2.
A setK⊂Rn is called a cone if αx∈K for arbitrary x∈K and α>0.Definition 2.3.
A coneK is said to be a global cone of instability for (2.1) if x(x0,t)∈K for arbitrary x0∈K and t≥0 and limt→∞∥x(x0,t)∥=∞.Definition 2.4.
The zero solutionx≡0 of (2.1) is said to be globally unstable in a cone K if K is a global cone of instability for (2.1).Now, we prove results analogous to the classical Chetaev theorem (see, e.g., [2]) on instability in a form suitable for our analysis. As usual, if 𝒮 is a set, then ∂𝒮 denotes its boundary and 𝒮¯ its closure, that is, 𝒮¯:=𝒮∪∂𝒮.Theorem 2.5.
LetV:ℝn→ℝ, V(0,…,0)=0 be a continuously differentiable function. Assume that the set
(2.2)K={x∈Rn:V(x)>0}
is a cone. If the full derivative of V along the trajectories of (2.1) is positive for every x∈K, that is, if
(2.3)V̇(x):=gradTV(x)f(x)>0,x∈K,
then K is a global cone of instability for the system (2.1).Proof.
Letɛ be a positive number. We define a neighborhood of the origin
(2.4)Uɛ:={x∈Rn:‖x‖<ɛ},
and a constant
(2.5)Mɛ:=max⁡x∈U¯ɛ∩K¯V(x).
Moreover, define a set
(2.6)Wδ:={x∈U¯ɛ∩K¯,V(x)≥δ},
where δ is a positive number such that δ<Mɛ. Then, Wδ≠∅.
Letx0∈Wδ∩K, then V(x0)=δ1∈[δ,Mɛ]. We show that there exists a t=tT=tT(ɛ,x0) such that x(x0,tT)∉U¯ɛ and x(x0,tT)∈K.
Suppose to the contrary that this is not true andx(x0,t)∈U¯ɛ for all t≥0. Since V̇(x)>0, the function V is increasing along the solutions of (2.1). Thus x(x0,t) remains in K. Due to the compactness of Wδ, there exists a positive value β such that for x(x0,t)∈Wδ(2.7)ddtV(x(x0,t))=gradTV(x(x0,t))f(x(x0,t))>β.
Integrating this inequality over the interval [0,t], we get
(2.8)V(x(x0,t))-V(x0)=V(x(x0,t))-δ1>βt.
Then there exists a t=tT=tT(ɛ,x0) satisfying
(2.9)tT>(Mɛ-δ1)β,
such that V(x(x0,tT))>Mɛ and, consequently, x(x0,tT)∉U¯ɛ. This is contrary to our supposition. Since ɛ>0 is arbitrary, we have
(2.10)lim⁡t→∞‖x(x0,t)‖=∞,
that is, the zero solution is globally unstable, and K is a global cone of instability.Theorem 2.6.
LetV:ℝn→ℝ be a continuously differentiable function and let S,Z:ℝn→ℝ, Z(0,…,0)=0 be continuous functions such that V=S·Z. Assume that the set
(2.11)K1={x∈Rn:Z(x)>0}
is a cone, and S(x)>0 for any x∈K1. If the full derivative (2.3) of V along the trajectories of (2.1) is positive for every x∈K1, that is, if V̇(x)>0 for every x∈K1, then K1 is a global cone of instability for the system (2.1).Proof.
The proof is a modification of the proof of Theorem2.5. Let ɛ be a positive number. We define a neighborhood Uɛ of the origin by formula (2.4) and a constant
(2.12)Mɛ:=max⁡x∈U¯ɛ∩K1¯V(x).
Moreover, define a set
(2.13)Wδ:={x∈U¯ɛ∩K1¯,V(x)≥δ},
where δ is a positive number such that δ<Mɛ. Then Wδ≠∅.
Letx0∈Wδ∩K1. Then V(x0)=δ1∈[δ,Mɛ]. We show that there exists a t=tT=tT(ɛ,x0) such that x(x0,tT)∉U¯ɛ and x(x0,tT)∈K1.
Suppose to the contrary that this is not true andx(x0,t)∈U¯ɛ for all t≥0. Since V̇(x)>0, the function V is increasing along the solutions of (2.1). Due to the compactness of Wδ, there exists a positive value β such that for x(x0,t)∈Wδ(2.14)ddtV(x(x0,t))=gradTV(x(x0,t))f(x(x0,t))>β.
Integrating this inequality over interval [0,t], we get
(2.15)V(x(x0,t))-V(x0)=V(x(x0,t))-δ1=S(x(x0,t))Z(x(x0,t))-δ1>βt.
Since S(x(x0,t))>0, the inequality
(2.16)Z(x(x0,t))>δ1+βtS(x(x0,t))>0
is an easy consequence of (2.15). Thus x(x0,t) remains in K1. Apart from this, (2.15) also implies the existence of a t=tT=tT(ɛ,x0) satisfying
(2.17)tT>(Mɛ-δ1)β,
such that V(x(x0,tT))>Mɛ. Consequently, x(x0,tT)∉U¯ɛ. This is contrary to our supposition. Since ɛ>0 is arbitrary, we have
(2.18)lim⁡t→∞‖x(x0,t)‖=∞,
that is, the zero solution is globally unstable and K1 is a global cone of instability.Definition 2.7.
A functionV satisfying all the properties indicated in Theorem 2.5 is called a Chetaev function for the system (2.1). A function V satisfying all the properties indicated in Theorem 2.6 is called a Chetaev-type function for the system (2.1).


## 2.2. Auxiliary Inequality

Our results will be formulated in terms of global cones of instability. These will be derived using an auxiliary inequality valid in a given cone. Let(x,y)∈ℝ2 and let k be a positive number. We define a cone (2.19)K:={(x,y)∈R2:y>k|x|}.Lemma 2.8.
Leta, b, c, d, and k be given constants such that b>0, d>0, k>0, and |c|≤kd. Assume, moreover, either
(2.20)|a|≤kb,
or
(2.21)|a|>kb,(2.22)|c|≠kd,k≥max⁡{|a+kb|c+kd,|a-kb||c-kd|},
then
(2.23)ax3+bx2y+cxy2+dy3>0,
for every (x,y)∈𝒦.Proof.
We partition𝒦 into two disjoint cones
(2.24)K1:={(x,y)∈R2:y>k|x|,x>0},K2:={(x,y)∈R2:y>k|x|,x≤0},
and rewrite (2.23) as
(2.25)x(ax2+cy2)+y(bx2+dy2)>0.
We prove the validity of (2.23) in each of the two cones separately.
The case of thecone 𝒦1. Suppose that (2.20) holds. Estimating the left-hand side of (2.25), we get(2.26)x(ax2+cy2)+y(bx2+dy2)>x(ax2+cy2)+kx(bx2+dy2)=x[x2(a+kb)+y2(c+kd)]>0,
and (2.23) holds.
If inequalities (2.21) and (2.22) are valid, then, estimating the left-hand side of (2.25), we get(2.27)x(ax2+cy2)+y(bx2+dy2)>x(ax2+cy2)+kx(bx2+dy2)=x[x2(a+kb)+y2(c+kd)]≥x[-|a+kb|x2+(c+kd)y2]=(c+kd)x[y2-|a+kb|c+kdx2]=(c+kd)x[y-|a+kb|c+kdx][y+|a+kb|c+kdx]=(c+kd)x2[k-|a+kb|c+kd][k+|a+kb|c+kd]≥0,
and (2.23) holds again.
The case of thecone 𝒦2. Suppose that (2.20) hold, then, estimating the left-hand side of (2.25), we get
(2.28)x(ax2+cy2)+y(bx2+dy2)=-|x|(ax2+cy2)+y(bx2+dy2)>-|x|(ax2+cy2)+k|x|(bx2+dy2)=-|x|[(a-kb)x2+(c-kd)y2]≥0,
and (2.23) holds.
If inequalities (2.21) and (2.22) are valid, then the estimation of (2.25) implies (we use (2.28))(2.29)x(ax2+cy2)+y(bx2+dy2)>-|x|[(a-kb)x2+(c-kd)y2]=|c-kd||x|[y2-a-kb|c-kd|x2]={≥0ifa-kb<0,|c-kd||x|[y-a-kb|c-kd|x][y+a-kb|c-kd|x]≥|c-kd|x2[k+a-kb|c-kd|][k-a-kb|c-kd|]≥0ifa-kb>0.
Hence, (2.23) holds again.


## 3. Global Cone of Instability

In this part we derive a result on the instability of system (1.5) in a cone. In order to properly formulate the results, we have to define some auxiliary vectors and matrices (some definitions copy the previous ones used in Introduction, but with a dimension of n-1 rather than n). We denote (3.1)x(n-1)=(x1,x2,…,xn-1)T,bi=(b1ni,b2ni,…,bn-1,ni)T,i=1,2,…,n,b̃=(bnn1,bnn2,…,bnnn-1)T.
Apart from this, we define symmetric (n-1)×(n-1) matrices (3.2)Bi0={bsqi},i=1,2,…,n,s,q=1,2,…,n-1,
that is, (3.3)Bi0=(b11ib12i⋯b1,n-1ib21ib22i⋯b2,n-1i⋯⋯⋯⋯bn-1,1ibn-1,2i⋯bn-1,n-1i),B̃=(b1n1b2n1⋯bn-1,n1b1n2b2n2⋯bn-1,n2⋯⋯⋯⋯b1nn-1b2nn-1⋯bn-1,nn-1).
Finally, we define an (n-1)×(n-1)2 matrix (3.4)B¯T={B¯1T,B¯2T,…,B¯n-1T},
where (n-1)×(n-1) matrices B¯iT, i=1,2,…,n-1 are defined as (3.5)B¯iT=(bi11bi21⋯bi,n-11bi12bi22⋯bi,n-12⋯⋯⋯⋯bi1n-1bi2n-1⋯bi,n-1n-1).
We consider a matrix equation(3.6)A0TH+HA0=-C,
where H and C are (n-1)×(n-1) matrices. It is well-known (see, e.g., [14]) that, for a given positive definite symmetric matrix C, (3.6) can be solved for a positive definite symmetric matrix H if and only if the matrix A0 is asymptotically stable.Theorem 3.1 (Main result).
Assume that the matrixA0 is asymptotically stable, bnnn>0 and h is a positive number. Let C be an (n-1)×(n-1) positive definite symmetric matrix and H be a related (n-1)×(n-1) positive definite symmetric matrix solving equation (3.6). Assume that the matrix (-HB̃T-B̃H+h(Bn0)T) is positive definite,
(3.7)‖2hbn-Hb̃‖≤λmin⁡(H)h⋅bnnn,
and, in addition, one of the following conditions is valid:
either(3.8)‖HB¯T‖≤λmin⁡(H)h⋅λmin⁡(-HB̃T-B̃H+h(Bn0)T)
or
(3.9)‖HB¯T‖>λmin⁡(H)h⋅λmin⁡(-HB̃T-B̃H+h(Bn0)T),
a strong inequality holds in (3.7), and
(3.10)λmin⁡(H)h≥max⁡{T1,T2},
where
(3.11)T1=‖HB¯T‖-λmin⁡(H)/h⋅λmin⁡(-HB̃T-B̃H+h(Bn0)T)-‖2hbn-Hb̃‖+λmin⁡(H)h⋅bnnn,T2=‖HB¯T‖+λmin⁡(H)/h⋅λmin⁡(-HB̃T-B̃H+h(Bn0)T)‖2hbn-Hb̃‖+λmin⁡(H)h⋅bnnn.
Then the set
(3.12)K:={(x(n-1)T,xn):hxn>x(n-1)THx(n-1)}
is a global cone of instability for the system (1.5).Proof.
First we make auxiliary computations. For the reader's convenience, we recall that, for two(n-1)×(n-1) matrices 𝒜, 𝒜1, two 1×(n-1) vectors ℓ, ℓ1, two (n-1)×1 vectors 𝒞, 𝒞1 and two 1×1 “matrices" m, m1, the multiplicative rule
(3.13)(AClm)(A1C1l1m1)=(AA1+Cl1AC1+Cm1lA1+ml1lC1+mm1)
holds. This rule can be modified easily for the case of arbitrary rectangular matrices under the condition that all the products are well defined.
We will rewrite system (1.5) in an equivalent form, suitable for further investigation. With this in mind, we define an (n-1)2×(n-1) matrix X(n-1) as (3.14)X(n-1)T=(X1(n-1)T,X2(n-1)T,…,Xn-1(n-1)T),
where all the elements of the (n-1)×(n-1) matrices Xi(n-1)T, i=1,2,…,n-1 are equal to zero except the ith row, which equals x(n-1)T, that is,
(3.15)Xi(n-1)T=(00⋯0⋯⋯⋯⋯00⋯0x1x2⋯xn-100⋯0⋯⋯⋯⋯00⋯0).
Moreover, we define 1×(n-1) vectors Yi, i=1,2,…,n-1 with components equal to zero except the ith element, which equals xn, that is,
(3.16)Yi=(0,…,0,xn,0,…,0),
and (n-1)×(n-1) zero matrix Θ.
It is easy to see that matricesXT and B in (1.5) can be expressed as (3.17)XT=(X1(n-1)TY1T⋯Xn-1(n-1)TYn-1TΘθθT0⋯θT0x(n-1)Txn),B=(B10b1b1Tbnn1⋯⋯Bn0bnbnTbnnn).
Now we are able to rewrite the system (1.5) under the above assumption regarding the representation of the matrix A in the form (1.10) in an equivalent form
(3.18)(ẋ(n-1)ẋn)=(A0θθTλ)(x(n-1)xn)+(X1(n-1)TY1T⋯Xn-1(n-1)TYn-1TΘθθT0⋯θT0x(n-1)Txn)×(B10b1b1Tbnn1⋯⋯Bn0bnbnTbnnn)(x(n-1)xn).
Finally, since the equalities
(3.19)∑j=1n-1Xj(n-1)TBj0=B¯TX(n-1),∑j=1n-1YjTbjT=B̃xn,∑j=1n-1Xj(n-1)Tbj=B̃x(n-1),∑j=1n-1YjTbnnj=b̃xn
can be verified easily using (3.13), we have
(3.20)(ẋ(n-1)ẋn)=(A0+r11(x(n-1)T,xn)r12(x(n-1)T,xn)r21(x(n-1)T,xn)λ+r22(x(n-1)T,xn))(x(n-1)xn),
where
(3.21)r11(x(n-1)T,xn)=∑j=1n-1[Xj(n-1)TBj0+YjTbjT]=B¯TX(n-1)+B̃xn,r12(x(n-1)T,xn)=∑j=1n-1[Xj(n-1)Tbj+YjTbnnj]=B̃x(n-1)+b̃xn,r21(x(n-1)T,xn)=x(n-1)TBn0+xnbnT,r22(x(n-1)T,xn)=x(n-1)Tbn+xnbnnn.
The remaining part of the proof is based on Theorem 2.6 with a Chetaev-type function V=S·Z and with suitable functions S and Z. Such functions we define as
(3.22)V(x(n-1)T,xn)=(x(n-1)Txn)(-HθθTh)(x(n-1)xn),
that is,
(3.23)V(x(n-1)T,xn)=-x(n-1)THx(n-1)+hxn2,S(x(n-1)T,xn)=x(n-1)THx(n-1)+hxn,Z(x(n-1)T,xn)=-x(n-1)THx(n-1)+hxn.
We will verify the necessary properties. Obviously, V=S·Z, the set
(3.24)K1:={(x(n-1)T,xn)∈Rn:Z(x(n-1),xn)>0}={(x(n-1)T,xn)∈Rn:hxn>x(n-1)THx(n-1)}
is a cone and S(x(n-1)T,xn)>0 for every (x(n-1)T,xn)∈K1.
The full derivative ofV (in the form (3.22)) along the trajectories of the system (1.5) (we use its transformed form (3.20)) equals(3.25)V̇(x(n-1)T,xn)=(ẋ(n-1)Tẋn)(-HθθTh)(x(n-1)xn)+(x(n-1)Txn)(-HθθTh)(ẋ(n-1)ẋn)=(x(n-1)Txn)(A0T+r11T(x(n-1)T,xn)r21T(x(n-1)T,xn)r12T(x(n-1)T,xn)λ+r22(x(n-1)T,xn))(-HθθTh)(x(n-1)xn)+(x(n-1)Txn)(-HθθTh)(A0+r11(x(n-1)T,xn)r12(x(n-1)T,xn)r21(x(n-1)T,xn)λ+r22(x(n-1)T,xn))×(x(n-1)xn).
Using formula (3.13), we get
(3.26)V̇(x(n-1)T,xn)=(x(n-1)Txn)(c11(x(n-1)T,xn)c12(x(n-1)T,xn)c21(x(n-1)T,xn)c22(x(n-1)T,xn))(x(n-1)xn),
where
(3.27)c11(x(n-1)T,xn)=-[A0+r11(x(n-1)T,xn)]TH-H[A0+r11(x(n-1)T,xn)],c12(x(n-1)T,xn)=hr21T(x(n-1)T,xn)-Hr12(x(n-1)T,xn),c21(x(n-1)T,xn)=hr21(x(n-1)T,xn)-r12T(x(n-1)T,xn)H=c12T(x(n-1)T,xn),c22(x(n-1)T,xn)=2h[λ+r22(x(n-1)T,xn)].
We reduce these formulas using (3.21). Then,
(3.28)c11(x(n-1)T,xn)=-(A0TH+HA0)-(B¯TX(n-1)+B̃xn)TH-H(B¯TX(n-1)+B̃xn),c12(x(n-1)T,xn)=h(x(n-1)TBn0+xnbnT)T-H(B̃x(n-1)+b̃xn),c21(x(n-1)T,xn)=h(x(n-1)TBn0+xnbnT)-(B̃x(n-1)+b̃xn)TH,c22(x(n-1)T,xn)=2h(λ+x(n-1)Tbn+xnbnnn).
The derivative (3.26) turns into
(3.29)V̇(x(n-1)T,xn)=x(n-1)Tc11(x(n-1)T,xn)x(n-1)+x(n-1)Tc12(x(n-1)T,xn)xn+xnc21(x(n-1)T,xn)x(n-1)+xnc22(x(n-1)T,xn)xn=x(n-1)T[-(A0TH+HA0)-(B¯TX(n-1)+B̃xn)TH-H(B¯TX(n-1)+B̃xn)]x(n-1)+x(n-1)T[h(x(n-1)TBn0+xnbnT)T-H(B̃x(n-1)+b̃xn)]xn+xn[h(x(n-1)TBn0+xnbnT)-(B̃x(n-1)+b̃xn)TH]x(n-1)+xn[2h(λ+x(n-1)Tbn+xnbnnn)]xn=-x(n-1)T(A0TH+HA0)x(n-1)+2hλxn2-x(n-1)T((B¯TX(n-1))TH+HB¯TX(n-1))x(n-1)-x(n-1)T((B̃xn)TH+HB̃xn)x(n-1)+x(n-1)T(2h(Bn0)T-HB̃-B̃H)x(n-1)xn+2x(n-1)T(hbn-Hb̃)xn2+2h(x(n-1)Tbn+xnbnnn)xn2.
Finally, using (3.6), we get
(3.30)V̇(x(n-1)T,xn)=x(n-1)TCx(n-1)+2hλxn2-2x(n-1)THB¯TX(n-1)x(n-1)+2x(n-1)T[-HB̃T-B̃H+h(Bn0)T]x(n-1)xn+2x(n-1)T(2hbn-Hb̃)xn2+2hbnnnxn3.
Let us find the conditions for the positivity of V̇(x(n-1)T,xn) in the cone K1. We use (3.30). If (x(n-1)T,xn)∈K1, then xn≥0 and
(3.31)V̇(x(n-1)T,xn)≥x(n-1)TCx(n-1)+2hλxn2-2‖HB¯T‖⋅‖x(n-1)‖3+2λmin⁡(-HB̃T-B̃H+h(Bn0)T)⋅‖x(n-1)‖2⋅xn-2‖2hbn-Hb̃‖⋅‖x(n-1)‖⋅xn2+2hbnnnxn3.
We set
(3.32)a=-2‖HB¯T‖,b=2λmin⁡(-HB̃T-B̃H+h(Bn0)T),c=-2‖2hbn-Hb̃‖,d=2hbnnn.
If
(3.33)a‖x(n-1)‖3+b‖x(n-1)‖2⋅xn+c‖x(n-1)‖⋅xn2+dxn3>0
in K1, then V̇(x(n-1)T,xn)>0 since C is a positive definite matrix and
(3.34)x(n-1)TCx(n-1)+2hλxn2≥λmin⁡(C)‖x(n-1)‖2+2hλxn2>0.
If (x(n-1)T,xn)∈K1, then
(3.35)xn>x(n-1)THx(n-1)h≥λmin⁡(H)h⋅‖x(n-1)‖,(3.36)K1⊂K*:={(x(n-1)T,xn)∈Rn:xn>λmin⁡(H)h⋅‖x(n-1)‖}.
Now, we use Lemma 2.8 with 𝒦=𝒦*, y=xn, x=∥x(n-1)∥, with coefficients a, b, c, and d defined by formula (3.32) and with k:=λmin(H)/h.
Obviously|c|≤kd because, due to (3.7), inequality (3.37)‖2hbn-Hb̃‖≤λmin⁡(H)h⋅bnnn
holds. Moreover, |a|≤kb if (3.8) holds, that is, if
(3.38)‖HB¯T‖≤λmin⁡(H)h⋅λmin⁡(-HB̃T-B̃H+h(Bn0)T).
Further, |a|>kb if (3.9) holds, that is, if
(3.39)‖HB¯T‖>λmin⁡(H)h⋅λmin⁡(-HB̃T-B̃H+hn(Bn0)T),
and (2.22) holds due to (4.10) and the condition |c|≠kd. Thus the assumptions of Lemma 2.8 are true, the inequality (3.33) holds in the cone 𝒦* and, due to embedding (3.36), in the cone K1 as well.
All the assumptions of Theorem2.6 are fulfilled with regard to system (1.5) and the theorem is proved, because K1=K.Remark 3.2.
We will focus our attention to Lemma2.8 about the positivity of a third-degree polynomial in two variables in the cone 𝒦. We used it to estimate the derivative V̇ expressed by formula (3.30). Obviously, there are other possibilities of estimating its sign. Let us demonstrate one of them. Let us, for example, estimate the right-hand side of (3.31) in the cone K1 using inequality (3.35), then
(3.40)V̇(x(n-1)T,xn)≥x(n-1)TCx(n-1)+2hλxn2-2‖HB̅T‖⋅‖x(n-1)‖3+2λmin⁡(-HB̃T-B̃H+h(Bn0)T)⋅‖x(n-1)‖2⋅xn-2‖2hbn-Hb̃‖⋅‖x(n-1)‖⋅xn2+2hbnnnxn3≥λmin⁡(C)‖x(n-1)‖2+2hλxn2-2‖HB̅T‖⋅‖x(n-1)‖3+2λmin⁡(H)h⋅λmin⁡(-HB̃T-B̃H+h(Bn0)T)⋅‖x(n-1)‖3-2‖2hbn-Hb̃‖⋅‖x(n-1)‖⋅xn2+2λmin⁡(H)h⋅‖x(n-1)‖⋅hbnnn⋅xn2,
and the positivity of V̇(x(n-1)T,xn) will be guaranteed if
(3.41)‖HB̅T‖≤λmin⁡(H)h⋅λmin⁡(-HB̃T-B̃H+h(Bn0)T),‖2hbn-Hb̃‖≤λmin⁡(H)h⋅bnnn.
We see that this approach produces only one set of inequalities for the positivity of V̇(x(n-1)T,xn), namely the case when (3.7) and (3.8) holds. Unfortunately, using such approach, we are not able to detect the second case (3.7) and  (3.9) when V̇(x(n-1)T,xn) is positive. This demonstrates the advantage of detailed estimates using the above third-degree polynomial in two variables.


## 4. Planar Case

Now we consider a particular case of the system (1.5) for n=2. This means that, in accordance with (1.5) and (1.10), we consider a system(4.1)ẋ1(t)=ax1(t)+b111x12(t)+2b121x1(t)x2(t)+b221x22(t),ẋ2(t)=λx2(t)+b112x12(t)+2b122x1(t)x2(t)+b222x22(t),
where a<0 and λ>0. The solution of matrix equation (3.6) for A0=(a), H=(h11), and C=(c) with c>0, that is,(4.2)(ah11)+(h11a)=-(c)
gives (4.3)H=(h11)=(-c2a),
with h11=-c/2a>0. The set K defined by (3.12) where h>0 and x(n-1)=x1 reduces to(4.4)K={(x1,x2):x2>c2|a|h⋅|x1|}.
Now, from Theorem 3.1, we will deduce sufficient conditions indicating K being a global cone of instability for system (4.1). In our particular case, we have (4.5)bi=(b12i),i=1,2,b̃=(b221),Bi0=(b11i),i=1,2,B̃=(b121),B¯T=(b111)=B10.
Now, we compute all necessary expressions used in Theorem 3.1. We have (4.6)-HB̃T-B̃H+h(Bn0)T=-(-c2a)(b121)-(b121)(-c2a)+h(b112)=(hb112-c|a|b121),‖2hbn-Hb̃‖=|2hb122-c2|a|b221|,λmin⁡(H)h=ch2|a|,λmin⁡(H)h=c2|a|h,‖HB¯T‖=|c2|a|b111|=c2|a||b111|,λmin⁡(-HB̃T-B̃H+h(Bn0)T)=hb112-c|a|b121,T1=‖HB¯T‖-λmin⁡(H)/h⋅λmin⁡(-HB̃T-B̃H+h(Bn0)T)-‖2hbn-Hb̃‖+λmin⁡(H)h⋅bnnn=(c/2|a|)|b111|-c/2|a|h⋅(hb112-(c/|a|)b121)-|2hb122-(c/2|a|)b221|+ch/2|a|⋅b222,T2=‖HB¯T‖+λmin⁡(H)/h⋅λmin⁡(-HB̃T-B̃H+h(Bn0)T)‖2hbn-Hb̃‖+λmin⁡(H)h⋅bnnn=(c/2|a|)|b111|+(c/2|a|h)⋅(hb112-(c/|a|)b121)|2hb122-(c/2|a|)b221|+(ch/2|a|)⋅b222.Theorem 4.1 (Planar Case).
Assume thata<0, b222>0, h>0, c>0 and hb112|a|>cb121. Let
(4.7)|2hb122-c2|a|b221|≤ch2|a|⋅b222,
and, in addition, one of the following conditions is valid:
either(4.8)c2|a||b111|≤c2|a|h⋅(hb112-c|a|b121)
or
(4.9)c2|a||b111|>c2|a|h⋅(hb112-c|a|b121),
strong inequality holds in (4.7), and
(4.10)c2|a|h≥max⁡{T1,T2},
where 𝒯1 and 𝒯1 are defined by (4.6). Then the set K defined by (4.4) is a global cone of instability for the system (4.1).It is easy to see that the choiceh=1, c=|a| significantly simplifies all assumptions. Therefore we give such a particular case of Theorem 4.1.Corollary 4.2 (Planar Case).
Assume thata<0, b222>0 and b112>b121. Let
(4.11)|2b122-12b221|≤12⋅b222,
and, in addition, one of the following conditions is valid:
either(4.12)12|b111|≤12⋅(b112-b121)
or
(4.13)12|b111|>12⋅(b112-b121),
strong inequality holds in (4.11), and
(4.14)12≥max⁡{T1,T2},
where
(4.15)T1=(1/2)|b111|-(1/2)⋅(b112-b121)-|2b122-(1/2)b221|+(1/2)⋅b222,T2=(1/2)|b111|+(1/2)⋅(b112-b121)|2b122-(1/2)b221|+(1/2)⋅b222,
Then the set
(4.16)K={(x1,x2):x2>12⋅|x1|}
is a global cone of instability for the system (4.1).Example 4.3.
The setK defined by (4.16) is a global cone of instability for the system
(4.17)ẋ1(t)=ax1(t)+x12(t)+22x1(t)x2(t)+x22(t),ẋ2(t)=λx2(t)+22x12(t)+2x1(t)x2(t)+22x22(t),
where a<0 and λ>0 since inequalities (4.11) and (4.12) in Corollary 4.2 hold.Example 4.4.
The setK defined by (4.16) is a global cone of instability for the system
(4.18)ẋ1(t)=ax1(t)+4x12(t)+22x1(t)x2(t)+x22(t),ẋ2(t)=λx2(t)+22x12(t)+2x1(t)x2(t)+202x22(t),
where a<0 and λ>0 since inequalities (4.11), (4.13), (4.14) in Corollary 4.2 hold.



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