Partial stability analysis of some classes of nonlinear systems

Acta Mathematica Scientia 2017,37B(2):329–341 http://actams.wipm.ac.cn

PARTIAL STABILITY ANALYSIS OF SOME CLASSES OF NONLINEAR SYSTEMS∗ Alexander ALEKSANDROV Elena ALEKSANDROVA Alexey ZHABKO Saint Petersburg State University, Saint Petersburg 199034, Russia E-mail : [email protected]; [email protected]; [email protected]

°)

Yangzhou CHEN (

†

Beijing Key Laboratory of Transportation Engineering, College of Metropolitan Transportation, Beijing University of Technology, Beijing 100124, China E-mail : [email protected] Abstract A nonlinear differential equation system with nonlinearities of a sector type is studied. Using the Lyapunov direct method and the comparison method, conditions are derived under which the zero solution of the system is stable with respect to all variables and asymptotically stable with respect to a part of variables. Moreover, the impact of nonstationary perturbations with zero mean values on the stability of the zero solution is investigated. In addition, the corresponding time-delay system is considered for which delay-independent partial asymptotic stability conditions are found. Three examples are presented to demonstrate effectiveness of the obtained results. Key words

Nonlinear systems; partial asymptotic stability; Lyapunov function; sector nonlinearities; time-delay

2010 MR Subject Classification

1

34D20; 34K20; 93D30

Introduction

The partial stability analysis is fundamental and challenging research problem due to its applications in various branches such as mechanics, electrodynamics, population dynamics, etc.; see, for example, [1–6]. It should be noted that the problem is especially important for satellites attitude control [7–9]. This problem naturally arises in the cases where only a part of variables characterizing the dynamics of the considered system is of interest or where only stability with respect to a part of components of the state vector is in fact possible. The basic approaches to the investigation of stability with respect to a part of variables were developed by V. V. Rumyantsev [2, 10]. It is worth mentioning that he has proposed a counterpart of the Lyapunov functions method for solving issues of partial stability. The Rumyantsev approaches have gotten deep and extensive development, and many interesting ∗ Received

July 10, 2016; revised October 6, 2016. The research was supported by the Saint Petersburg State University (9.42.1045.2016), the Russian Foundation for Basic Research (15-58-53017 and 16-01-00587), and the Natural Science Foundation of China (6141101096, 61573030, and 61273006). † Corresponding author

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and important results were derived with applications to numerous practical problems; see [3– 5, 11–14] and the references cited therein. Nevertheless, during the past few years, we have seen an increasing interest to the problem of partial stability [5, 6, 15–18]. In particular, it should be noted that methods of partial stability analysis is an effective tool for studying the consensus problem for multi-agent systems [19–22]. One of the first results on the partial stability is well-known Lyapunov–Malkin Theorem [23]. In this theorem, conditions of asymptotic stability with respect to a part of variables were found in the critical case of several zero eigenvalues of the matrix of a linear approximation system. Later, some generalizations of this theorem providing partial asymptotic stability conditions by linear approximation were obtained by various authors; see [2, 3, 5] and the references therein. Moreover, in [24], the Lyapunov–Malkin Theorem was extended to the case of an essentially nonlinear homogeneous system of the first approximation. The goal of this article is further development and extension of these results. We consider a complex system describing interaction of two subsystems. It is assumed that the first subsystem is a nonlinear Persidskii-type system [25] with nonlinearities satisfying sector conditions, and the zero solution of the subsystem is asymptotically stable, whereas the second subsystem admits stable zero solution. Using the Lyapunov direct method and the comparison method, conditions are derived under which the zero solution of the complex system is stable with respect to all variables and asymptotically stable with respect to a part of variables. Next, we study an impact of nonstationary perturbations with zero mean values on the considered system. We will show that if the first subsystem is essentially nonlinear, then perturbations do not disturb stability even in the case where their orders coincide with those of the right-hand sides of unperturbed equations. Finally, the corresponding time-delay system is investigated for which, with the aid of the Lyapunov direct method and the Razumikhin approach, delay-independent conditions of stability with respect to all variables and asymptotic stability with respect to a part of variables are obtained.

2

Preliminaries In this section, we present some notations and definitions used in this article.

Throughout, R stands for the field of real numbers, Rn denotes the n-dimensional Euclidean space, k · k is the Euclidean norm of a vector, and Rn×l is the space of n × l matrices with real entries. Let diag{λ1 , · · · , λn } be the diagonal matrix with the elements λ1 , · · · , λn . We write P T for the transpose of a matrix P . For a symmetric matrix A, the notation A > 0 (A < 0) means that the matrix A is positive (negative) definite. For a given number h > 0, let C([−h, 0], Rn ) be the space of continuous functions ϕ(θ) : [−h, 0] → Rn with the uniform (supremum) norm kϕkh = sup kϕ(θ)k. θ∈[−h,0] n×n

Definition 2.1 ([25]) A matrix C ∈ R is called diagonally stable if there exists a matrix Λ = diag{λ1 , · · · , λn } > 0 such that ΛC + C T Λ < 0.

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Consider the differential equation system x(t) ˙ = F (t, x(t)).

(2.1)

Here, x(t) ∈ Rn is the state vector. Let vectors y ∈ Rk and z ∈ Rm be the partitions of the vector x, respectively. Therefore, x = (y T , z T )T and k + m = n. Assume that the vector function F (t, x) is defined and continuous for t ≥ 0, kyk < H, and kzk < ∆, where H > 0, 0 < ∆ ≤ +∞. Let x(t, t0 , x0 ) = (y T (t, t0 , y0 , z0 ), z T (t, t0 , y0 , z0 ))T denote a solution of (2.1) starting at t = t0 from the point x0 = (y0T , z0T )T . In addition, we assume that system (2.1) admits the zero solution and solutions of (2.1) are z-extendable; see [3]. Definition 2.2 ([3]) The zero solution of system (2.1) is called stable with respect to y (y-stable), if for any ε > 0 and t0 ≥ 0 there exists a number δ > 0 such that k(y0T , z0T )T k < δ implies ky(t, t0 , y0 , z0 )k < ε for any t ≥ t0 . Definition 2.3 ([3]) The zero solution of system (2.1) is called asymptotically stable with respect to y (asymptotically y-stable), if it is y-stable, and for any t0 ≥ 0, there exists a number γ > 0 such that k(y0T , z0T )T k < γ implies ky(t, t0 , y0 , z0 )k → 0 as t → +∞.

3

Problem Statement Let system (2.1) be represented in the form y(t) ˙ = P f (y(t)) + G(t, x(t)),

(3.1)

z(t) ˙ = D(t, z(t)) + R(t, x(t)). T

k

Here, P = {pij }i,j=1 is a constant matrix; f (y) = (f1 (y1 ), · · · , fk (yk )) , where scalar functions fj (yj ) are continuous for |yj | < H and belong to a sector-like constrained set defined as follows: yj fj (yj ) > 0 for yj 6= 0, j = 1, · · · , k; vector functions D(t, z), G(t, x), and R(t, x) are continuous for t ≥ 0, kyk < H, kzk < ∆ and satisfy the conditions D(t, 0) ≡ 0,

kG(t, x)k ≤ β(x) kf (y)k,

and kR(t, x)k ≤ c kykσ ,

(3.2)

where c and σ are positive constants, and β(x) → 0 as kxk → 0. System (3.1) may be treated as a complex system describing the interaction of two isolated subsystems y(t) ˙ = P f (y(t)), (3.3) z(t) ˙ = D(t, z(t)).

(3.4)

Remark 3.1 Subsystem (3.3) is well-known Persidskii type system [25]. Such systems are widely used as mathematical models of neural networks and automatic control systems; see [25–27]. From the properties of function f (y) and conditions (3.2), it follows that systems (3.1), (3.3), and (3.4) admit zero solutions. Let the following assumptions be fulfilled. Assumption 3.1 The matrix P is diagonally stable.

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Assumption 3.2 The zero solution of (3.4) is stable and for this system, there exists a continuously differentiable Lyapunov function V1 (t, z) satisfying the conditions of the Lyapunov Stability Theorem for t ≥ 0, kzk < ∆, seeing [1], and such that k∂V1 (t, z)/∂zk ≤ M for t ≥ 0, kzk < ∆, where M = const > 0. Remark 3.2 Assumption 3.1 implies that the zero solution of (3.3) is asymptotically stable. We will look for stability conditions for the zero solution of system (3.1). Assumption 3.3 Let functions f1 (y1 ), · · · , fk (yk ) be represented in the form ν

fj (yj ) = αj yj j + ζj (yj ), where αj are positive constants, νj ≥ 1 are rationals with odd numerators and denominators, ν and ζj (yj )/yj j → 0 as yj → 0, j = 1, · · · , k. Remark 3.3 Without loss of generality, we will assume that α1 = · · · = αk = 1 and ν1 ≤ · · · ≤ νk . Hence, the system y˙ i (t) =

k X

ν

pij yj j (t),

i = 1, · · · , k,

j=1

is a system of the first approximation for (3.3). Remark 3.4 The case where ν1 = · · · = νk = 1 and D(t, z) ≡ 0 corresponds to the Lyapunov–Malkin Theorem. If ν1 = · · · = νk > 1, then (3.3) is nonlinear and homogeneous system. For this case, conditions under which the zero solution of (3.1) is stable with respect to all variables and asymptotically stable with respect to y were found in [24]. Therefore, in what follows, we will assume that 1 ≤ ν1 < νk . Our goal is to derive partial asymptotic stability conditions of the zero solution of (3.1) by the nonlinear nonhomogeneous approximation (3.3) and (3.4). Moreover, we will show that, for some classes of nonstationary perturbations, the obtained conditions can be relaxed. In addition, along with (3.1), we will consider the corresponding time-delay system for which delay-independent stability conditions will be found.

4

Stability Analysis

To obtain partial asymptotic stability conditions for (3.1), we will use the Lyapunov direct method and the comparison method. Theorem 4.1 Let Assumptions 3.1–3.3 be fulfilled. If νk < σ + 1,

(4.1)

then the zero solution of (3.1) is stable with respect to all variables and asymptotically y-stable. Proof Consider a positive definite diagonal matrix Λ = diag{λ1 , · · · , λn } satisfying the condition ΛP + P T Λ < 0. Construct a Lyapunov function in the form V2 (y) =

k X i=1

λi

yiνi +1 . νi + 1

(4.2)

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Differentiating functions V1 (t, z) and V2 (y) with respect to system (3.1), we obtain  T ∂V1 (t, z) ∂V1 (t, z) V˙ 1 = + (D(t, z) + R(t, x)) ≤ cM kykσ , ∂t ∂z v u k k k uX X X ν ν j ν i i V˙ 2 ≤ λi pij yi yj + λi pij yi ζj (yj ) + c1 β(x)kf (y)kt yi2νi i,j=1

i,j=1

i=1

for t ≥ 0, kyk < H, and kzk < ∆, where c1 is a positive constant. Hence, one can choose numbers δ > 0 and c2 > 0 such that V˙ 2 ≤ −c2

k X

yi2νi

i=1

for t ≥ 0, kxk < δ. If the value of δ is sufficiently small, then there exist positive constants a1 and a2 such that the estimates 2νk σ ν +1 ν +1 V˙ 2 ≤ −a2 V k (y) V˙ 1 ≤ a1 V k (y), 2

2

hold for t ≥ 0, kxk < δ. Thus, the system σ ν +1

u˙ 1 = a1 u2 k

,

2νk ν +1

u˙ 2 = −a2 u2 k

(4.3)

is a comparison one for (3.1). It is easy to verify that under condition (4.1), the zero solution of (4.3) is stable with respect to all variables and asymptotically stable with respect to u2 . Therefore, seeing [1], the zero solution of (3.1) is stable with respect to all variables and asymptotically y-stable.  Let system (3.1) be of the form y(t) ˙ = P f (y(t)) + G(t, x(t)),

(4.4)

z(t) ˙ = D(t, z(t)) + L(t)f (y(t)). Here, L(t) is a continuous and bounded m × k matrix for t ≥ 0, and the rest notation is the same as that in (3.1). Corollary 4.2 Let Assumptions 3.1–3.3 be fulfilled. If νk < 2ν1 + 1,

(4.5)

then the zero solution of (4.4) is stable with respect to all variables and asymptotically y-stable. Proof

In a similar way as that in Proof of Theorem 4.1, it can be shown that the system ν1

u˙ 1 = a1 u2ν1 +1 ,

2νk ν +1

u˙ 2 = −a2 u2 k

,

(4.6)

is a comparison one for (4.4), where a1 and a2 are positive constants, and under condition (4.5), the zero solution of (4.6) is stable with respect to all variables and asymptotically stable with respect to u2 .  Example 4.3 Consider the scalar equation ... x (t) + (a + bx(t))¨ x(t) + cx˙ µ (t) = 0.

(4.7)

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Here, x(t) ∈ R; a, b, and c are constant coefficients with a > 0, c > 0; µ > 1 is a rational with the odd numerator ard denominator. We will look for stability conditions for the equilibrium position x = x˙ = x ¨ = 0 of (4.7). Using the substitution x1 (t) = x(t),

x2 (t) = x(t), ˙

x3 (t) = x˙ µ (t) + ρ¨ x(t),

where ρ = const > 0, we transform (4.7) into the system x˙ 1 (t) = x2 (t), x˙ 2 (t) = −

xµ2 (t) x3 (t) + , ρ ρ

(4.8)

x˙ 3 (t) = (a − ρc)xµ2 (t) − ax3 (t) + (x3 (t) − xµ2 (t))



 µ µ−1 x2 (t) − bx1 (t) . ρ

Thus, in this case, y = (x2 , x3 )T , z = x1 , D(t, z) ≡ 0, ν1 = 1, ν2 = µ, σ = 1, and   1 1 −  ρ ρ  P = . a − ρc −a Let ρ > a/c. Then, matrix P is diagonally stable. Applying Theorem 4.1, it is obtained that if 1 < µ < 2, (4.9) then the zero solution of (4.8) is stable with respect to all variables and asymptotically stable with respect to x2 , x3 . Hence, under condition (4.9), the equilibrium position x = x˙ = x ¨ = 0 of (4.7) is stable with respect to all variables and asymptotically stable with respect to x, ˙ x ¨.

5

Partial Asymptotic Stability Conditions for Perturbed Systems Along with (3.1), consider the corresponding perturbed system y(t) ˙ = (P + B(t))f (y(t)) + G(t, x(t)),

(5.1)

z(t) ˙ = D(t, z(t)) + R(t, x(t)). Here, B(t) is a continuous and bounded k × k matrix for t ≥ 0. Thus, in (5.1), the orders of perturbations coincide with those of the right-hand sides of subsystem (3.3). In what follows, we will impose an additional restriction on the matrix B(t). Assumption 5.1 Let 1 T

Z

t+T

B(s)ds → 0 as T → +∞

t

uniformly with respect to t ≥ 0. Hence, mean values of all entries of the matrix B(t) are equal to zero. Remark 5.1 It is well known, seeing [28, 29], that perturbations of such type might destroy asymptotic stability of linear systems, that is, asymptotic stability of a linear timeinvariant system x(t) ˙ = P x(t) does not imply asymptotic stability of the corresponding perturbed system x(t) ˙ = (P + B(t))x(t).

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In this section, we will show that in the case where subsystem (3.3) is essentially nonlinear, that is, its right-hand sides do not contain linear terms with respect to components of the state vector, one can guarantee the preservation of partial asymptotic stability for the zero solution of (5.1). Theorem 5.2 Let Assumptions 3.1–3.3 and 5.1 be fulfilled. If 1 < ν1 ≤ · · · ≤ νk < σ + 1,

(5.2)

then the zero solution of (5.1) is stable with respect to all variables and asymptotically y-stable. Proof In a similar way as in Proof of Theorem 4.1, consider a positive definite diagonal matrix Λ = diag{λ1 , · · · , λn } such that ΛP + P T Λ < 0, and construct the Lyapunov function V2 (y) by formula (4.2). Next, applying the approach proposed in [30, 31], choose a Lyapunov function for the perturbed system in the form Ve2 (t, y) =

Here,

ψij (t, ε) =

k X

k X yiνi +1 ν λi − λi ψij (t, ε)yiνi yj j . ν + 1 i i=1 i,j=1

Z

t

eε(s−t) bij (s)ds,

i, j = 1, · · · , k,

0

ε is a positive parameter, and bij (t) are entries of the matrix B(t). There exists a number δ > 0 such that k k k k X X a ˜3 X 2νi a ˜3 X 2νi yi ≤ Ve2 (t, y) ≤ a ˜2 yiνi +1 + y a ˜1 yiνi +1 − ε i=1 ε i=1 i i=1 i=1 for t ≥ 0, kyk < δ, where a ˜1 , a ˜2 , and a ˜3 are positive constants. Differentiating Ve2 (t, y) with respect to system (5.1), we obtain k k X X ˙ ν Ve 2 = λi pij yiνi yj j + λi (pij + bij (t)) yiνi ζj (yj ) i,j=1

+

i,j=1

k X

λi yiνi Gi (t, x) + ε

i=1

−

k X

i,j=1

k X

ν λi νi ψij (t, ε)yiνi −1 yj j

k X

ν −1 λi νj ψij (t, ε)yiνi yj j

i,j=1

−

k X

!

(pil + bil (t)) fl (yl ) + Gi (t, x)

l=1

i,j=1

≤ −˜ a4

ν

λi ψij (t, ε)yiνi yj j

k X

k X

!

(pjl + bjl (t)) fl (yl ) + Gj (t, x)

l=1

yi2νi + a ˜5

i=1

˜ ε) +˜ a7 εψ(t,

k X

|yi |νi |ζj (yj )| + a ˜6 β(x)kf (y)k

i,j=1 k X i=1

yi2νi +

k X

|yi |νi

i=1

k X

a ˜8 kf (y)k |yi |νi −1 |yj |νj ε i,j=1

˜ ε) = for t ≥ 0, kxk < δ. Here, G1 (t, x), · · · , Gk (t, x) are components of the vector G(t, x), ψ(t, max |ψij (t, ε)|, and a ˜r > 0, r = 4, · · · , 8. It should be noted that values of constants i,j=1,··· ,k

a ˜1 , · · · , a ˜8 are independent of ε.

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˜ ε) → 0 as ε → 0 uniformly with respect to t ≥ 0. Using It is known, seeing [32], that εψ(t, ˜ ε) < a this property, fix a value of ε > 0 such that 2˜ a7 εψ(t, ˜4 for any t ≥ 0. As νi > 1 for i = 1, · · · , k, for the choosen ε, one can take δ > 0 such that the estimates k k X a ˜1 X νi +1 e y ≤ V2 (t, y) ≤ 2˜ a2 yiνi +1 , 2 i=1 i i=1 σ

ν +1 V˙ 1 ≤ a1 Ve2 k (t, y),

k 2νk a ˜4 X 2νi ˙ ν +1 Ve 2 ≤ − yi ≤ −a2 Ve2 k (t, y) 4 i=1

hold for t ≥ 0, kxk < δ. Here, a1 and a2 are positive constants. Thus, system (4.3) is a comparison one for the perturbed system (5.1), and under condition (5.2), the zero solution of (4.3) is stable with respect to all variables and asymptotically stable with respect to u2 .  Remark 5.3 In particular, Assumption 5.1 is fulfilled in the case where entries of the matrix B(t) describe periodic or almost periodic oscillations with zero mean values. It is important that, for such perturbations, Theorem 5.2 guarantees partial asymptotic stability without any restrictions imposed on the amplitudes of these oscillations. Example 5.4 Consider the problem of damping the angular motions of a rigid body rotating around its center of inertia. We will use the right-handed Cartesian rectangular coordinate system whose origin is at the center of inertia of the body, and axis of this system coincide with the principal central axes of the body. Motions of the body under the action of a control torque M are described by the dynamical Euler equations Θ ω(t) ˙ + ω(t) × Θω(t) = M ;

(5.3)

see [3]. Here, ω(t) = (ω1 (t), ω2 (t), ω3 (t))T is the vector of angular velocity, Θ = diag{A, B, C} is the inertia tensor of the body, and A, B, C are positive constants. We consider the case, where A > C, B > C, and choose a control torque in the form M = (α1 ω1ν1 , α2 ω2ν2 , 0)T ,

(5.4)

where α1 and α2 are negative constants, and ν1 and ν2 are rationals with the odd numerators and denominators, 1 < ν1 ≤ ν2 . Then, system (5.3) may be rewritten as follows: Aω˙ 1 (t) = (B − C)ω2 (t)ω3 (t) + α1 ω1ν1 (t), B ω˙ 2 (t) = (C − A)ω1 (t)ω3 (t) + α2 ω2ν2 (t),

(5.5)

C ω˙ 3 (t) = (A − B)ω1 (t)ω2 (t). Using the Lyapunov function V (ω) = A(A − C)ω12 + B(B − C)ω22 , then the equilibrium position ω = 0 of system (5.5) is asymptotically stable with respect to ω1 , ω2 . Hence, torque (5.4) provides damping the angular motions of the body with respect to two of the three principal central axes of inertia.

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Next, we assume that, on the body, along with the control torque, a disturbing torque acts. Let the perturbed equations be of the form Aω˙ 1 (t) = (B − C)ω2 (t)ω3 (t) + (α1 + β1 sin t)ω1ν1 (t) + β2 cos t ω2ν2 (t), B ω˙ 2 (t) = (C − A)ω1 (t)ω3 (t) + β3 cos t ω1ν1 (t) + (α2 + β4 sin t)ω2ν2 (t),

(5.6)

C ω˙ 3 (t) = (A − B)ω1 (t)ω2 (t). Here, β1 , β2 , β3 , β4 are constant coefficients. In this case, Assumption 5.1 is fulfilled. Using Theorem 5.2, we obtain the fact that if 1 < ν1 ≤ ν2 < 3, then, for arbitrary values of coefficients β1 , β2 , β3 , β4 , the equilibrium position ω = 0 of (5.6) is stable with respect to all variables and asymptotically stable with respect to ω1 , ω2 .

6

Stability Analysis of Time-Delay Systems

It is worth mentioning that realistic models of various real-world systems should incorporate aftereffect phenomena in their dynamics; see, for instance, [3, 26, 27, 33–35] and the bibliography therein. Such models may be described by difference-differential equations. It is well known [33] that time delay might crucially affect on system’s stability. Therefore, in applications, it is important to have conditions on delay values under which stability can be guaranteed. It should be noted that, while stability theory of difference-differential systems is well developed, there are few results on the partial stability of time-delay systems; see [3, 5, 34]. In this section, a class of nonlinear time-delay systems is studied. Using the comparison method and the Razumikhin approach [3, 33], we will derive delay-independent partial asymptotic stability conditions for the zero solutions of the considered systems. Let positive numbers h and H be given, and ΩH denote the set of functions ϕ(θ) ∈ C([−h, 0], Rn ) such that kϕkh < H. Consider the difference-differential system y(t) ˙ = P f (y(t)) + Qf (y(t − τ (t))) + G(t, xt ),

(6.1)

z(t) ˙ = D(t, z(t)) + R(t, xt ). k

Here, Q = {qij }i,j=1 is a constant matrix; the functionals G(t, ϕ) and R(t, ϕ) are continuous for t ≥ 0, ϕ(θ) ∈ ΩH , the delay τ (t) is a continuous nonnegative and bounded function for t ≥ 0 such that sup τ (t) ≤ h, (6.2) t≥0

and the rest notation is the same as that in (3.1).
T Let x(t, t0 , ϕ) = y T (t, t0 , ϕ), z T (t, t0 , ϕ) stand for a solution of (6.1) with the initial
T conditions t0 ≥ 0, ϕ(θ) ∈ ΩH , and xt (t0 , ϕ) = ytT (t0 , ϕ), ztT (t0 , ϕ) denote the restriction of the solution to the segment [t − h, t], that is, xt (t0 , ϕ) : θ → x(t + θ, t0 , ϕ), θ ∈ [−h, 0]. When the initial conditions are not important, or are well defined from the context, we write x(t) and xt , instead of x(t, t0 , ϕ) and xt (t0 , ϕ), respectively. We assume that



σ 



D(t, 0) ≡ 0, kG(t, ϕ)k ≤ β(ϕ) max f ϕ(y) (θ) , kR(t, ϕ)k ≤ c ϕ(y) θ∈[−h,0]

h

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for t ≥ 0, ϕ(θ) ∈ ΩH . Here, c and σ are positive constants, ϕ(θ) =

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T
T T ϕ(y) (θ) , ϕ(z) (θ) ,

ϕ(y) (θ) ∈ C([−h, 0], Rk ), ϕ(z) (θ) ∈ C([−h, 0], Rm ), and β(ϕ) → 0 as kϕkh → 0. Hence, system (6.1) admits the zero solution. Definition 6.1 ([3]) The zero solution of system (6.1) is called stable with respect to y (y-stable), if for any ε > 0 and t0 ≥ 0, there exists a number δ > 0 such that kϕkh < δ implies ky(t, t0 , ϕ)k < ε for all t ≥ t0 . Definition 6.2 ([3]) The zero solution of system (6.1) is called asymptotically stable with respect to y (asymptotically y-stable), if it is y-stable, and for any t0 ≥ 0 there exists a number γ > 0 such that kϕkh < γ implies ky(t, t0 , ϕ)k → 0 as t → +∞. We will look for conditions providing delay-independent partial asymptotic stability of the solution. Theorem 6.3 Let Assumptions 3.2 and 3.3 be fulfilled, and the matrix P + Q be diagonally stable. If inequalities (5.2) are valid, then, for any h > 0 and any continuous nonnegative function τ (t) satisfying condition (6.2), the zero solution of (6.1) is stable with respect to all variables and asymptotically y-stable. Proof Let Λ = diag{λ1 , · · · , λn } be a positive definite diagonal matrix such that Λ(P + Q) + (P + Q)T Λ < 0. Using diagonal entries of the matrix Λ, construct the Lyapunov function V2 (y) by formula (4.2). Differentiating functions V1 (t, z) and V2 (y) with respect to system (6.1), we obtain  T ∂V1 (t, z(t)) ˙ V1 ≤ R(t, xt ) ≤ cM kyt kσh , (6.3) ∂z V˙ 2 =

k X

ν

λi (pij + qij ) yiνi (t)yj j (t)

i,j=1

+

+

k X

i,j=1

k
X ν ν λi qij yiνi (t) yj j (t − τ (t)) − yj j (t) + λi pij yiνi (t)ζj (yj (t))

k X

k X

i,j=1

λi qij yiνi (t)ζj (yj (t − τ (t))) +

i,j=1

≤ −c1

k X

i=1

yi2νi (t) + c2

i=1

+c3

λi yiνi (t)Gi (t, xt )

k X

k X

i,j=1

ν ν |yi (t)|νi yj j (t − τ (t)) − yj j (t)

|yi (t)|νi |ζj (yj (t))| + c4

i,j=1

k X

|yi (t)|νi |ζj (yj (t − τ (t)))|

i,j=1

+c5 β(xt ) max kf (y(t + θ))k θ∈[−h,0]

k X

|yi (t)|νi

i=1

for t ≥ 0, kxt kh < H. Here, G1 (t, xt ), · · · , Gk (t, xt ) are components of the vector G(t, xt ), and cr > 0, r = 1, · · · , 5. Choose a positive integer N and a number δ ∈ (0, H). Let, for a solution x(t) = y T (t),
T z T (t) of system (6.1), the inequality kx(t)k < δ hold for t ∈ [t0 − h, T ), where t0 ≥ 0 and T > t0 + N h.

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Using the approach proposed in [31, 36], it can be shown that if the value of N is sufficiently large, whereas the value of δ is sufficiently small, and there exists a time instant t˜ ∈ [t0 + (N − 1)h, T ) such that V2 (y(ξ)) ≤ 2V2 (y(t˜)) for ξ ∈ [t˜ − N h, ˜t], then c1 V˙ 2 (y(t˜)) ≤ − 2

k X

2νk νk +1

yi2νi (t˜) ≤ −˜ c V2

(y(t˜)),

c˜ = const > 0.

(6.4)

i=1

In [37], a technique for the estimation of the convergence rate of solutions of nonlinear time-delay systems was developed. Applying the technique, then from (6.4), it follows that one can choose positive numbers d1 , d2 and δ˜ such that kxt0 kh < δ˜ implies  − ν 1−1 (ν1 +1)(νk −1) ν1 +1 k ky(t)k ≤ d1 δ˜ νk +1 1 + d2 δ˜ νk +1 (t − t0 ) (6.5) for t ∈ [t0 , T ). With the aid of estimates (6.3) and (6.5), we have Z t V1 (t, z(t)) ≤ V1 (t0 , z(t0 )) + cM kys kσh ds t0

≤ M δ˜ + cM

Z

t0 +h

kys kσh ds + cM

t0

≤ M δ˜ + hcM dσ1 δ˜ +cM dσ1 δ˜ ≤ M δ˜ +

σ(ν1 +1) νk +1

hcM dσ1 δ˜

t

Z

t0 +h

kys kσh ds

σ(ν1 +1) νk +1

Z

+∞

t0 +h

σ(ν1 +1) νk +1

for t ∈ [t0 , T ), where cˆ = cM dσ1

Z

 − ν σ−1 (ν1 +1)(νk −1) k νk +1 ˜ 1 + d2 δ (s − t0 − h) ds + cˆδ˜

(σ−νk +1)(ν1 +1) νk +1

+∞ −ν

(1 + d2 u)

σ k −1

(6.6)

du.

0

Thus, if the value of δ˜ is sufficiently small, then inequalities (6.5) and (6.6) are valid for all t ≥ t0 .  Remark 6.4 In Theorem 6.3, it is assumed that the components of the vector f (y) are essentially nonlinear (νi > 1, i = 1, · · · , k). However, it should be noted that the approach proposed in this section permits us to obtain delay-independent partial asymptotic stability conditions, where some of the exponents ν1 , · · · , νk are equal to one, provided that linear terms are delay-free. Example 6.5 Consider the nonlinear control system with delayed feedback x˙ 1 (t) = α1 x1 (t) + α2 f (η(t − τ (t))) + α3 x1 (t)x2 (t), x˙ 2 (t) = α4 x1 (t) + α5 f (η(t − τ (t))),

(6.7)

η(t) ˙ = α6 x1 (t) + α7 f (η(t − τ (t))). Here, x1 (t), x2 (t), and η(t) are scalar variables; α1 , · · · , α7 are constant coefficients; the scalar nonlinearity f (η) is continuous for |η| < H (H = const > 0) and satisfies the condition ηf (η) > 0 for η 6= 0; the delay τ (t) is a continuous nonnegative and bounded function for t ≥ 0. In addition, we will assume that the function f (η) can be represented as follows: f (η) = η µ + ζ(η), where µ > 1 is a rational with the odd numerator and denominator, and ζ(η)/η µ → 0 as η → 0.

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Vol.37 Ser.B

Thus, in this case, y(t) = (x1 (t), η(t))T , z(t) = x2 (t), ν1 = 1, ν2 = µ,     α1 0 0 α2 , . P = Q= α6 0 0 α7 In a similar way as in Proof of Theorem 6.3, it can be shown that if 1 < µ < 3, and the matrix   α1 α2  P +Q= α6 α7 is diagonally stable, then, for any continuous nonnegative and bounded delay τ (t) for t ≥ 0, the zero solution of system (6.7) is stable with respect to all variables and asymptotically stable with respect to x1 and η.

7

Conclusion

In this article, well-known Lyapunov–Malkin Theorem on the asymptotic stability with respect to a part of variables is extended to the case of a nonlinear system of the first approximation. Moreover, classes of nonstationary systems are found for which the derived stability conditions can be relaxed. For the corresponding time-delay systems, conditions are obtained to provide partial asymptotic stability for an arbitrary continuous nonnegative and bounded delay. An important direction of the future research is an application of the proposed approaches to study the consensus problem for nonlinear multi-agent systems. References [1] Rouche N, Habets P, Laloy M. Stability Theory by Liapunov’s Direct Method. New York etc.: Springer, 1977 [2] Rumyantsev V V, Oziraner A S. Stability and Stabilization of Motion with Respect to Part of the Variables. Moscow: Nauka, 1987 (in Russian) [3] Vorotnikov V I. Partial Stability and Control. Boston: Birkhauser, 1998 [4] Chellaboina V S, Haddad W M. A unification between partial stability and stability theory for time-varying systems. IEEE Control Syst Mag, 2002, 22(6): 66–75 [5] Vorotnikov V I. Partial stability and control: The state-of-the-art and development prospects. Automation and Remote Control, 2005, 66(4): 511–561 [6] Hu W, Wang J, Li X. An approach of partial control design for system control and synchronization. Chaos, Solitons and Fractals, 2009, 39(3): 1410–1417 [7] Likhachev V N, Sazonov V V, Ul’yashin A I. Single-axis solar orientation of a satellite of the earth. Cosmic Research, 2003, 41(2): 159–170 [8] Tikhonov A A. Resonance phenomena in oscillations of a gravity-oriented rigid body. Part 4: Multifrequency resonances. Vestnik Sankt-Peterburgskogo Universiteta. Ser 1. Matematika Mekhanika Astronomiya, 2000, (1): 131–137 [9] Tikhonov A A. Secular evolution of rotary motion of a charged satellite in a decaying orbit. Cosmic Research, 2005, 43(2): 107–121 [10] Rumyantsev V V. On asymptotic stability and instability of motion with respect to a part of the variables. J of Appl Math Mech, 1971, 35(1): 10–30 [11] Peiffer K, Rouche N. Liapunov second method applied to partial stability. J Mecanique, 1969, 6: 20–29 [12] Corduneanu C. Some problems concerning partial stability//Proc Symp Math V 6, Meccanica Non-Lineare e Stabilitri, 23–26 February, 1970, 243–265. New York: Academic Press, 1971

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