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Integral manifolds for uncertain impulsive differential–difference equations with variable impulsive perturbations
Chaos, Solitons & Fractals 65 (2014) 90–96
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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Integral manifolds for uncertain impulsive differential–difference equations with variable impulsive perturbations Gani Tr. Stamov a,⇑, Ivanka M. Stamova b a b
Department of Mathematics, Technical University of Sofia, 8800 Sliven, Bulgaria Department of Mathematics, The University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA
a r t i c l e
i n f o
Article history: Received 1 August 2013 Accepted 2 May 2014
a b s t r a c t In the present paper sufficient conditions for the existence of integral manifolds of uncertain impulsive differential–difference equations with variable impulsive perturbations are obtained. The investigations are carried out by means the concepts of uniformly positive definite matrix functions, Hamilton–Jacobi–Riccati inequalities and piecewise continuous Lyapunov’s functions. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The impulsive differential–difference equations are adequate mathematical models of various real processes and phenomena, characterized by the fact that their state changes by jumps and by the dependence of the process on its history at each moment of time. During the last few decades differential–difference equations have been object of numerous investigations related to the applications of these equations to physics, biology, electronics, control theory, etc. See, for example, [1–3] and references therein. For the basic concept, ideas and theorems of impulsive differential and functional differential equations we refer the reader to [4–6]. The method of integral manifolds is a very powerful instrument for investigating of various problems in the theory of differential equations. For example, the exceptional role of manifolds in the reduction of the dimensions of equations as well as for solving some linearization problems is well known [7–9]. Recently, there exists an intensive work regarding the study of integral manifolds for ⇑ Corresponding author. Tel.: +359 889526191. E-mail addresses: [email protected] (G.Tr. Stamov), ivanka.stamova@ utsa.edu (I.M. Stamova). http://dx.doi.org/10.1016/j.chaos.2014.05.002 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.
specific types of functional differential equations. For example, Akhmet [10] established some criteria for the existence of integral manifolds of differential equations with piecewise constant argument of generalized type. In [11] the stability of invariant sets of functional - equations with delay has been investigated. Naito [12] studied integral manifolds for linear functional differential equations on some Banach space. The linearization near the integral manifold for a system of differential equations with piecewise constant argument has been investigated by Papaschinopoulos in [13]. The stability of integral manifolds of a class of functional differential equations has been discussed in [14]. Also, the theory of integral manifolds is one of the most important part of the qualitative theory of impulsive differential and differential–difference equations. The main results related to the study of the existence of integral manifolds for such equations can be found in [15–19]. The investigations in above mentioned papers have been restricted to equations in their main form. However, it is important to point out that in the real word phenomena exist under the conditions of structural uncertainty. Nonlinear dynamical systems with uncertain parameters may cause changes of equilibrium states which will affect the integral manifold behavior.
G.Tr. Stamov, I.M. Stamova / Chaos, Solitons & Fractals 65 (2014) 90–96
In the meantime, many important results of the control theory for uncertain differential and delay differential systems have been established [20–23]. The work on the control law design relying on an integral manifold is relatively sparse, see [24] and the references therein. In many applications due to impulsive differential equations the uncertainties happen frequently due to modeling errors, measurement inaccuracy, mutations in the evolutionary processes and so on. At the present time there has appeared some results recognized as landmark contributions for uncertain impulsive differential and functional differential systems with fixed moments of impulse effect [25–30]. To study such uncertain impulsive systems, the idea of moving invariant sets has been also employed in [31–34]. However, to the best of our knowledge, there has not been any work so far dedicated to uncertain impulsive differential systems with variable impulsive perturbations. In the investigation of such systems there arise a number of difficulties related to the phenomena ‘‘beating’’ of the solutions, bifurcation, loss of the property of autonomy, etc. [35]. The wider application, however, of this type of equations requires the formulation of effective criteria for existence of integral manifolds. In this paper, motivated by the above considerations, we study an impulsive uncertain differential–difference system with variable impulsive perturbations. The rest of the paper is organized as follows: In Section 2 we give some notations, the problem investigated in this paper is formulated, and some definitions are presented. In Section 3 we establish sufficient conditions for the existence of integral manifolds for linear and nonlinear uncertain systems under consideration. The investigations follow the main ideas from [28] and are carried out by utilizing the concepts of uniformly positive definite matrix functions, Hamilton–Jacobi–Riccati inequalities and piecewise continuous Lyapunov’s type functions. Finally, in Section 4 one example is presented to illustrate the theory. Since the existence of an integral manifold is a generalization of the solutions’ existence of a nonlinear system, our results can be applied in the investigation of many practical problems.
2. Preliminaries Let h ¼ const > 0; Rn be the n-dimensional Euclidean space with norm j: j; Rþ ¼ ½0; 1Þ; t0 2 R. Let J # R and X # Rn ; X – ø. Define the following class of functions: PC½J; X ¼ fr : J ! X : rðtÞg. is a piecewise continuous ~t 2 J at which rð~t Þ function with points of discontinuity and rð~tþ Þ exist and rð~t Þ ¼ rð~tÞ . Let t > t0 . Consider the following system of uncertain impulsive differential–difference equations with variable impulsive perturbations
_ xðtÞ ¼ f ðt; xðtÞ; xðt hÞÞ þ gðt; xðtÞ; xðt hÞÞ; t – sk ðxðtÞÞ;
DxðtÞ ¼ Ik ðxðtÞÞ þ Jk ðxðtÞÞ; t ¼ sk ðxðtÞÞ; k ¼ 1; 2; . . . ; ð2:1Þ where f ; g : ðt 0 ; 1Þ X X ! Rn ; Ik ; J k : X ! Rn ; ðt0 ; 1Þ; k ¼ 1; 2; . . . ; DxðtÞ ¼ xðt þ Þ xðtÞ.
sk : X !
91
The functions gðt; x; yÞ; J k ðxÞ represent the structural uncertainty or uncertain perturbations are characterized by g 2 U g ¼ g : gðt; x; yÞ ¼ eg ðt; x; yÞ:dg ðt;x; yÞ; jdg ðt; x; yÞj 6 jmg ðt; x; yÞj ;
and
J k 2 U J ¼ fJk : J k ðxÞ ¼ ek ðxÞ:dk ðxÞ; jdk ðxÞj 6 jmk ðxÞjg; k ¼ 1; 2; . . . ; where eg : ðt 0 ; 1Þ X X ! Rnm , and ek : X ! Rnm are known matrix functions whose entries are smooth functions of the state, and dg ; dk are unknown vector-valued functions whose norm are bounded, respectively, by the norm of vector-valued functions mg ðt; x; yÞ; mk ðxÞ, respectively. Here mg : ðt0 ; 1Þ X X ! Rm ; mk : X ! Rm ; k ¼ 1; 2; . . . are given smooth functions. Let /0 2 C ½½h; 0; X. Denote by xðtÞ ¼ xðt; t0 ; /0 Þ [5,6] the solution of system (2.1), satisfying the initial conditions
xðt; t0 ; /0 Þ
¼ /0 ðt t0 Þ; t 0 h 6 t 6 t0 ;
xðtþ0 ; t 0 ; /0 Þ ¼ /0 ð0Þ;
ð2:2Þ
J þ ðt0 ; /0 Þ is the maximal interval of the type ½t 0 ; bÞ in which the solution xðt; t 0 ; /0 Þ is defined, and hþ ðt 0 ; /0 ðtÞÞ denotes the integral orbit of the solution xðt; t0 ; /0 Þ for t 2 J þ ðt0 ; /0 Þ. The solutions xðtÞ of system (2.1) are piecewise continuous functions with points of discontinuity of the first kind in which they are left continuous; i.e., at the moments t lk when the integral curve of the solution xðtÞ meets the hypersurfaces
rk ¼ fðt; xÞ 2 ½t0 ; 1Þ X : t ¼ sk ðxÞg the following relations are satisfied:
xðtjk Þ ¼ xðt jk Þ; xðtþjk Þ ¼ xðt jk Þ þ Ijk ðxðt jk ÞÞ þ Jjk ðxðt jk ÞÞ: The points t 1 ; t 2 ; . . .(t0 < t1 < t2 ) are the impulsive moments. Let us note that, in general, k – jk . In other words, it is possible that the integral curve of the problem under consideration does not meet the hypersurface rk at the moment t k . It is clear that the solutions of systems with variable impulsive perturbations have points of discontinuity depending on the solutions, i.e. the different solutions have different points of discontinuity. This leads to a number of difficulties in the investigation of such systems. One of the phenomena occurring with systems of type (2.1) is the so called ‘‘beating’’ of the solutions. This is the phenomenon when the mapping point ðt; xðtÞÞ meets one and the same hypersurface rk several or infinitely many times [5,6,35]. For the proof of the main results we shall use the following nominal system
_ xðtÞ ¼ f ðt; xðtÞ; xðt hÞÞ; t – sk ðxðtÞÞ; DxðtÞ ¼ Ik ðxðtÞÞ; t ¼ sk ðxðtÞÞ; k ¼ 1; 2; . . . ;
ð2:3Þ
We shall introduce the following conditions for the system (2.3): H2.1. f 2 C½ðt 0 ; 1Þ X X; Rn . H2.2. The function f is Lipschitz continuous with respect to its second and third arguments uniformly on t 2 ðt 0 ; 1Þ.
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H2.3. sk 2 C½X; ðt 0 ; 1Þ; k ¼ 1; 2; . . .. H2.4. t0 s0 ðxÞ < s1 ðxÞ < s2 ðxÞ < . . . ; x 2 X. H2.5. sk ðxÞ ! 1 as k ! 1 uniformly on x 2 X. H2.6. The functions Ik are Lipschitz continuous in X. H2.7. The functions ðI þ Ik Þ : X ! X; k ¼ 1; 2; . . ., where I is the identity in X. H2.8. jk < jkþ1 < < jkþp < . . ., where jk is the number of the hypersurface met by the integral curve ðt; xðtÞÞ of the problem (2.3) at the moment tk ; k; jk ; p ¼ 1; 2; . . .. We also assume that the following condition holds: H2.9. The integral curves of system (2.3) meet each one of the hypersurfaces r1 ; r2 ; . . .. at most once. The condition H2.9 guarantees absence of the phenomenon ‘‘beating’’ in (2.3). This problem has been investigated by several authors. Efficient sufficient conditions which guarantee the absence of ‘‘beating’’ of the solutions of impulsive functional differential systems are given in [35]. Similar problems are discussed in [36]. It is also clear that the condition H2.9 is satisfied in the special case when sk ðxÞ tk ; k ¼ 1; 2; . . . ; x 2 X, i.e., when the impulses take place at fixed moments. Introduce the following notations:
Gk ¼ fðt; xÞ 2 ½t 0 ; 1Þ X : sk1 ðxÞ < t < sk ðxÞg; k ¼ 1; 2; . . . ; G ¼ [1 k¼1 Gk ; PC 1 ½J; X ¼ fr 2 PC½J; X : rðtÞ is continuously differentiable _ þ everywhere except the points tk at which r_ ðt k Þ and rðt k Þ _ exist and ðr_ ðt Þ ¼ r ðt Þ; k ¼ 1; 2; . . .g; k k
K ¼ fa 2 C½Rþ ; Rþ : aðrÞ is strictly increasin an að0Þ ¼ 0g: Definition 2.1. We call an arbitrary manifold M in the extended phase space ½t0 h; 1Þ X of (2.1) integral manifold, if ðt; /0 ðt t 0 ÞÞ 2 M, for t 2 ½t0 h; t 0 implies, hþ ðt0 ; /0 ðtÞÞ M. In what follows we shall use the class V M of piecewise continuous auxiliary functions V : ½t 0 ; 1Þ X ! Rþ which are analogues of Lyapunov’s functions [19]. Definition 2.2. We shall say that the function V : ½t0 ; 1Þ X ! Rþ belongs to the class V M which kernel is the manifold M in the extended phase space of (2.1), if the following conditions hold: 1. The functions V is continuous in G and locally Lipschitz continuous with respect to its second argument x in each of the sets Gk ; k ¼ 1; 2; . . .. 2. Vðt; xÞ ¼ 0 for ðt; xÞ 2 M; t P t0 and Vðt; xÞ > 0 for ðt; xÞ 2 ½t0 ; 1Þ X n M. 3. For each k ¼ 1; 2; . . . ; and ðtk ; x0 Þ 2 rk there exist the finite limits Vðt Vðt;xÞ; Vðt þ Vðt;xÞ k ;x0 Þ ¼ lim k ;x0 Þ ¼ lim ðt;xÞ!ðt ;x Þ k 0 ðt;xÞ2Gk
ðt;xÞ!ðt ;x Þ k 0 ðt;xÞ2Gkþ1
and the equality Vðt k ; x0 Þ ¼ Vðt k ; x0 Þ is valid.
Given a function V 2 V 0 . For t P t 0 ; t – sk ðxðtÞÞ; k ¼ 1; 2; . . ., introduce the function
Dþ Vðt;xðtÞÞ ¼ limþ sup r!0
1
r
½Vðt þ r;xðtÞ þ rf ðt; xðtÞ;xðt hÞÞÞ Vðt;xðtÞÞ; where ðt; xÞ 2 ½t0 ; 1Þ PC½½t0 ; 1Þ; X. We shall use the next lemma. Lemma 2.3 [19]. Assume that: 1. Conditions H2.1–H2.9 are met. 2. For the system (2.3), there exists a function V 2 V M with a kernel the manifold M, such that:
Vðt þ ; xðtÞ þ Ik ðxðtÞÞÞ 6 Vðt; xðtÞÞ; t ¼ sk ðxðtÞÞ; k ¼ 1; 2; . . . ; ð2:4Þ and the inequality
Dþ Vðt; xðtÞÞ 6 0; t – sk ðxðtÞÞ; t – sk ðxÞ; k ¼ 1; 2; . . . ð2:5Þ is valid whenever Vðt þ s; xðt þ sÞÞ 6 Vðt; xðtÞÞ for t P t 0 and h 6 s 6 0.Then M is an integral manifold of (2.3). Definition 2.4. The matrix function C : R ! Rnn is said to be: (a) a positive definite matrix function, if for any t 2 R; CðtÞ is a positive definite matrix; (b) a positive definite matrix function bounded above, if it is a positive definite matrix function and there exists a positive real number R > 0 such that
kmax ðCðtÞÞ 6 R; t 2 R; where kmax ð:Þ is the greatest eigenvalue of ð:Þ; (c) a uniformly positive definite matrix function, if it is a positive definite matrix function and there exists a positive real number r > 0 such that
kmin ðCðtÞÞ P r; t 2 R; where kmin ð:Þ is the smallest eigenvalue of ð:Þ. The proof of the following lemma is obvious. Lemma 2.5. Let CðtÞ 2 Rnn be a positive definite matrix function and YðtÞ 2 Rnn be a symmetric matrix. Then for any x 2 Rn ; t 2 R the following inequality holds
xT YðtÞx 6 kmax ðC1 ðtÞYðtÞÞxT CðtÞx: 2
ð2:6Þ 2
Lemma 2.6 [28]. Let RðtÞ 2 Rn n be a diagonal matrix function such that RðtÞ ¼ diagfe11 ðtÞ; . . . ; e1n ðtÞ; . . . ; en1 ðtÞ; . . . ; enn ðtÞg, and jeij ðtÞj 6 1; i; j ¼ 1; 2; . . . ; n,. Then for any positive scalar function kðtÞ > 0 and for any 2
n; g 2 Rn , the following inequality holds
2nT RðtÞg 6 k1 ðtÞnT n þ kðtÞgT g:
ð2:7Þ
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if Vðt þ s; xðt þ sÞÞ 6 Vðt; xðtÞÞ for h 6 s 6 0. Then M is an integral manifold of the system (2.1).
3. Main results Introduce the following conditions: H3.1. The function eg is Lipschitz continuous with respect to its second and third arguments uniformly on t 2 ðt0 ; 1Þ. H3.2. The functions ek are Lipschitz continuous in X. H3.3. The functions ðI þ ek Þ : X ! X; k ¼ 1; 2; . . ..
Proof. From (3.1), (3.2), Lemmas 2.5 and 2.6 t ¼ sk ðxÞ; k ¼ 1; 2; . . ., we have
for
Vðt þ ; xðtÞ þ Ik ðxðtÞÞ þ J k ðxðtÞÞÞ 6 Vðtþ ; xðtÞ þ Ik ðxðtÞÞÞ þ G1k dk ðxðtÞ þ dk ðxðtÞÞT G2k dk ðxðtÞÞ 6 Vðt þ ; xðtÞ T T þ Ik ðxðtÞÞÞ þ v1 k G1k G1k þ ðvk þ kmax ðG2k ÞÞmk mk
Lemma 3.1. Let the conditions H2.1–H2.9 and H3.1–H3.3 hold. Then 1. t k ! 1. 2. J þ ðt 0 ; u0 Þ ¼ ½t0 ; 1Þ.
6 Vðt; xðtÞÞ:
On the other hand, for t – sk ðxÞ; k ¼ 1; 2; . . . and from (3.3), we get
Dþ Vðt; xðtÞÞ ¼
@V @V @V @V @V þ ðf þ gÞ ¼ þ fþ eg dg @t @x @t @x @x @V @V k2 @V @V T 1 þ þ 2 mTg mg fþ k eg eTg @t @x @x 2 @x 2kk ) ( 1 @V 1 @V T 1 eg dTg : kk eTg kk dg 2 @x kk @x kk
Proof of Assertion 1. From condition H2.8 we derive the inequalities
¼
j1 < j2 < . . . : Then since jk are positive integers, we conclude that jk ! 1 as k ! 1. Then by condition H2.5 we get to the equalities
lim tk ¼ lim sjk ðxk Þ ¼ 1;
k!1
jk !1
þ
where xk ¼ xðt k ; t0 ; /0 Þ. Proof of Assertion 2. Since by the conditions H2.1, H2.2, H2.7 and H3.1–H3.3 the solution xðt; t 0 ; /0 Þ of the problem (2.1) is defined on each of the intervals ðtk ; tkþ1 ; k ¼ 1; 2; . . ., then from Assertion 1 we conclude that it is continuable for each t P t 0 . Theorem 3.2. Assume that:
(i) there exist G1k : ðt0 ; 1Þ X ! R1m ; G2k : ðt0 ; 1Þ X ! Rmm , where G2k are positive define matrix functions and for z 2 Rm ; k ¼ 1; 2; . . . it follows
Vðt; x þ Ik ðxÞ þ ek ðxÞzÞ 6 Vðt; x þ Ik ðxÞÞ þ G1k ðt; xÞz þ zT G2k ðt; xÞz;
ð3:1Þ
(ii) there exist positive constants vk such that for t 2 ðt0 ; 1Þ; k ¼ 1; 2; . . . ; x 2 PC 1 ½½t 0 ; 1Þ; X, T Vðt þ ; xðtÞ þ Ik ðxðtÞÞÞ þ v1 k G1k G1k
þ ðvk þ kmax ðG2k ÞÞmTk mk 6 Vðt; xðtÞÞ; t 2 rk ; ð3:2Þ
where G1k ¼ G1k ðt; xðtÞÞ; G2k ¼ G2k ðt; xðtÞÞ; mk ¼ mk ðxðtÞÞ; (iii) there exist scalar functions kk 2 C½X; Rþ such that for t 2 ½t0 ; 1Þ it follows @V @V k2 @V @V T 1 T þ fþ k þ m mg 6 0; ðt; xÞ 2 Gk ; eg eTg 2 @x @x 2k2k g @t @x
ð3:3Þ
1 2k2k
fmTg mg dTg dg g 6
@V @V þ f @t @x
k2k @V @V T 1 eg eTg þ 2 mTg mg 6 0; 2 @x @x 2kk
ð3:5Þ
if Vðt þ s; xðt þ sÞÞ 6 Vðt; xðtÞÞ; h 6 s 6 0. Therefore in the light of inequalities (3.4) and (3.5) and condition 1 of the theorem it follows that the conditions of Lemma 2.3 are satisfied for the system (2.1). The proof of Theorem 3.2 is complete. h Now we consider a linear system of uncertain impulsive differential–difference equations
1. The conditions H2.1–H2.9 and H3.1–H3.3 are met. 2. For the system (2.1) there exists a function V 2 V M with a kernel the manifold M, and the following relations are satisfied:
ð3:4Þ
x_ ¼ AðtÞx þ BðtÞxðt hÞ; t – sk ðxðtÞÞ;
DxðtÞ ¼ Ak ðtÞxðtÞ þ Bk ðtÞxðtÞ; t ¼ sk ðxðtÞÞ;
ð3:6Þ
where A; Ak 2 C ½ðt 0 ; 1Þ; Rnn ; k ¼ 1; 2; . . . are known matrix functions, and B; Bk 2 C ½ðt 0 ; 1Þ; Rnn ; k ¼ 1; 2; . . . are interval matrix functions, i.e.
BðtÞ 2 IN ½PðtÞ; Q ðtÞ ¼ BðtÞ 2 Rnn : BðtÞ ¼ ðbij ðtÞÞ; pij ðtÞ 6 bij ðtÞ 6 qij ðtÞ; i; j ¼ 1; 2; . . . ; n : Bk ðtÞ 2 IN½P k ðtÞ; Q k ðtÞ, where PðtÞ ¼ ðpij ðtÞÞ; Q ðtÞ ¼ ðqij ðtÞÞ; Pk ðtÞ, Q k ðtÞ are known matrices, and P;Q ;P k ;Q k 2C ½R;Rnn ; k¼1;2;.... Lemma 3.3 [28]. Let BðtÞ 2 IN ½PðtÞ; Q ðtÞ, where PðtÞ; Q ðtÞ are known matrices. Then BðtÞ can be written
BðtÞ ¼ B0 ðtÞ þ EðtÞRðtÞFðtÞ; where:
B0 ðtÞ ¼
1 1 ðPðtÞ þ Q ðtÞÞ; HðtÞ ¼ ðhij ðtÞÞ ¼ ðQ ðtÞ PðtÞÞ; 2 2
hij ðtÞ P 0; t 2 R; i; j ¼ 1; 2; . . . ; n;
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RðtÞ ¼ diagfe11 ðtÞ; . . . ; e1n ðtÞ; . . . ; en1 ðtÞ; . . . ; enn ðtÞg 2 Rn
2 n2
;
jeij ðtÞj 6 1; i; j ¼ 1; 2; . . . ; n; EðtÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h11 ðtÞe1 ;...; h1n ðtÞe1 ;...; hn1 ðtÞen ;...; hnn ðtÞen 2 Rnn ;
Then V 2 V M and let we consider the set of all solutions x ¼ xðtÞ of (3.6) such that Vðt þ s; xðt þ sÞÞ 6 Vðt; xðtÞÞ whenever h 6 s 6 0; t P t0 . Then for t – sk ðxÞ; x P 0, from the next inequality
Vðt h; xðt hÞÞ 6 Vðt; xðtÞÞ and
qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 FðtÞ ¼ h11 ðtÞe1 ;...; h1n ðtÞen ;...; hn1 ðtÞe1 ;...; hnn ðtÞen 2 Rn n ;
xT ðt hÞCðt hÞxðt hÞÞ 6 xT ðtÞCðtÞxðtÞÞ it follows that
kmin ðCÞx2 ðt hÞ 6 xT ðt hÞCxðt hÞ 6 xT ðtÞCxðtÞ
ei ð0; . . . ; 0; 1; 0; . . . ; 0ÞT 2 Rn ; i ¼ 1; 2; . . . ; n:
6 kmax ðCÞx2 ðtÞ:
By Lemma 3.3 we rewritten the system (3.6) in the form (
The last inequality shows that
¼ AðtÞx þ B0 ðtÞxðt hÞ þ EðtÞRðtÞFðtÞxðt hÞ; t – sk ðxðtÞÞ; DxðtÞ ¼ A~k ðtÞxðtÞ þ E~k ðtÞR~k ðtÞF~k ðtÞxðtÞ; t ¼ sk ðxðtÞÞ; k ¼ 1;2;...;
x_
ð3:7Þ
kmax ðCÞ 2 x ðtÞ: kmin ðCÞ
x2 ðt hÞ 6
ð3:11Þ
On the other hand from (3.8), (3.9) and (3.11) it follows
where
D Vðt;xðtÞÞ 6 x_ T Cx þ xT C_ x þ xT Cx_ þ
A~k ðtÞ ¼ Ak ðtÞ þ B~k0 ðtÞ;
_ ¼ ½AxðtÞ þ Bxðt hÞCxðtÞ þ xT ðtÞCxðtÞ
Bk ðtÞ ¼ B~k0 ðtÞ þ E~k ðtÞR~k ðtÞF~k ðtÞ; k ¼ 1; 2; . . .
þ xT ðtÞC½AxðtÞ þ Bxðt hÞ h i ¼ xT ðtÞ C_ þ AT C þ CA xðtÞ
and let A ¼ AðtÞ; B0 ¼ B0 ðtÞ; E ¼ EðtÞ; R ¼ RðtÞ; F ¼ FðtÞ for ~k ¼ A ~ k ðtÞ; Bk0 ¼ Bk0 ðtÞ; E ~k ¼ E ~k ðtÞ; R ~k ¼ t – sk ðxÞ, and A ~ k ðtÞ; F~k ¼ F~k ðtÞ for t ¼ sk ðxÞ. R
_ þ xT ðtÞBT CxðtÞ þ xT ðtÞCBxðt hÞ þ xT ðt hÞCxðtÞ h i T _ 6 x ðtÞ C þ AT C þ CA xðtÞ þ 2jBjjCjjxðtÞjjxðt hÞj h i 6 xT ðtÞ C_ þ AT C þ CA xðtÞ
Theorem 3.4. Assume that: 1. Conditions H2.3, H2.4, H2.5, H2.8 and H2.9 are met. 2. There exists a continuous real matrix C ¼ CðtÞ; C : ðt0 ;1Þ ! Rnn , which is symmetric and positive such that:
þ 2jBjjCjðx2 ðtÞ þ x2 ðt hÞj 6 xT ðtÞðCÞxðtÞ kmax ðCÞ 2 x ðtÞ þ 2ðjB0 j þ jEjjFjÞjCj x2 ðtÞ þ kmin ðCÞ kmax ðCÞ x2 ðtÞ: 6 kmin ðCÞ þ 2ðjB0 j þ jEjjFjÞkmax ðCÞ 1 þ kmin ðCÞ
(i) CðtÞ is differentiable at t – sk ðxÞ and the following inequality holds:
ð3:12Þ
C_ þ CA0 þ AT C 6 C; where C 2 Rnn is a positive scalar matrix; kmax ðCÞ (ii) ðjB0 j þ jEjjFjÞkmax ðCÞ 1 þ kmin ðCÞ 6 0; kmin ðCÞ
ð3:8Þ
ð3:9Þ
where kmin and kmax are respectively the smallest and the greatest eingenvalues of matrices C and C, respectively. (iii) the following inequalities hold
T
~T ~k ~T ~ ~T IþA IþA bk ¼ kmax C1 IþA Ck þ n1 k Ck E k E k Ck k k k
io ~k Ck E ~k F~T F~k 6 1: þ nk þ kmax E k
ð3:10Þ
where Ck ¼ CðtÞ for t ¼ sk ðxÞ; x P 0. Then the manifold M; M ¼ ½t0 h; 1Þ fx 2 Rn ; x 6 0g is an integral manifold for the system (3.7).
Then for t – sk ðxÞ; t P t 0 from (3.12) it follows that þ
D Vðt; xÞ 6 0; if Vt þ s; xðt þ sÞÞ 6 Vðt; xðtÞÞ whenever h 6 s 6 0 , which shows that the inequality (3.3) of Theorem 3.2 hold. By the similar proof of (3.4) by Lemmas 2.5 and 2.6 we have h iT ~k þ E ~k R ~ k F~k Vðt þ ;xðtÞ þ Ak ðxðtÞÞ þ Bk ðxðtÞÞ ¼ xT ðtÞ I þ A h i ~k þ E ~ k F~k xðtÞ ~k R Ck I þ A
T
T ~ T Ck I þ A ~k þ I þ A ~ k Ck E ~k R ~ k F~k ¼ xT ðtÞ I þ A k
T
T ~k E ~k R ~k R ~ k F~k Ck I þ A ~ k F~k Ck E ~k R ~ k F~k xðtÞ þ E
T
T
~ T Ck I þ A ~ k þ n1 I þ A ~ k Ck E ~k ~k E ~T Ck I þ A 6 xT ðtÞ I þ A k k k i
~k Ck E ~k xT ðtÞF~T F~k xðtÞ þnk F~k F~Tk xðtÞ þ kmax E k
T
1 ~T ~k Þ I þ A ~T ¼ xT ðtÞ I þ A Ck þ n Ck E~k E~T Ck I þ A
Proof. We will show that for the system (3.6) all conditions of Theorem 3.2 hold. Consider the function
Vðt; xÞ ¼
T
x CðtÞx; x P 0; 0; x 6 0:
k
k
þ nk þ kmax
~ k Ck E ~k E
k
k
i F~Tk F~k xðtÞ 6 xT ðtÞCk xðtÞ ¼ Vðt;xðtÞÞ; t P t 0 :
From the last inequality it follows that the condition (3.2) of Theorem 3.2 hold. The proof of Theorem 3.4 is complete. h
G.Tr. Stamov, I.M. Stamova / Chaos, Solitons & Fractals 65 (2014) 90–96
4. An example Consider the following nonlinear system of uncertain impulsive differential–difference equations with variable impulsive perturbations
8 _ ¼ f ðt;xðtÞ; xðt hÞÞ þ gðt; xðtÞ; xðt hÞÞ; t – sk ðxðtÞÞ; xðtÞ > < ! 1 1k 0 > xðtÞ; t ¼ sk ðxðtÞÞ; k ¼ 1; 2; .. . ; D xðtÞ ¼ : 0 1 1k ð4:1Þ where
x¼
x1 x2
;
f ðt; x; yÞ ¼
y¼
y1 y2
;
2y1 11x1 þ 2x2 ðx21 þ x22 Þ y2 9x2 x1 ðx21 þ x22 Þ
! ;
g 2 U g ¼ g : gðt; x; yÞ ¼ eg ðt; x; yÞ:dg ðt;x;yÞ; jdg ðt; x; yÞj 6 jmg ðt; x; yÞj ;
eg ¼ I; mg ¼
! pffiffiffi 2ðx1 þ x2 Þ ; y2
sk ðxÞ ¼ jxj þ k; k ¼ 1; 2;. .. :
We have sk 2 C½R2 ; ð0; 1Þ; k ¼ 1; 2; . . . ; sk ðxÞ ! 1 as k ! 1 uniformly on x 2 R2 , and also
0 < s1 ðxÞ < s2 ðxÞ < . . . ; x 2 R2 : Also, for the given choice of the functions H2.9 is satisfied [35]. Vðt; xÞ ¼ 12 ðx21 þ 2x22 Þ.
Let
Consider
sk condition
the
manifold
M ¼ fðt; 0Þ : t 2 ½h; 1Þg. Then, obviously, V 2 V M and V is differentiable in t 2 Rþ . For ðt; xÞ 2 Gk ; k ¼ 1; 2; . . . we have the following Hamilton–Jacobi inequality
@V @V k2 @V @V T 1 fþ k eg eTg þ þ 2 mTg mg ¼ 2x1 ðtÞx1 ðt hÞ @t @x 2 @x @x 2kk 11x21 ðtÞ þ 2x2 ðtÞx2 ðt hÞ 18x22 ðtÞ þ
x21 ðtÞ þ 2x22 ðtÞ 2
9 þ 2ðx21 ðtÞ þ x22 ðtÞÞ þ x22 ðt hÞ 6 Vðt; xÞ 6 0 2
ð4:2Þ
satisfied whenever Vðt þ s;xðt þ sÞÞ 6 Vðt; xðtÞÞ; h 6 s 6 0. Also, for t ¼ sk ðxÞ; k ¼ 1; 2; . . .
Vðt þ ; xðtÞ þ Ik ðxðtÞÞ þ J k ðxðtÞÞÞ " 2 2 # 1 x1 ðtÞ x2 ðtÞ þ2 ¼ 2 k k ¼
1 2k
2
ðx21 ðtÞ þ 2x22 ðtÞÞ 6 Vðt; xðtÞÞ:
ð4:3Þ
Thus, by Theorem 3.2, we get that M is an integral manifold of the system (4.1). References [1] Diblik J, Koksch N. Existence of global solutions of delayed differential equations via retract approach. Nonlinear Anal 2006;64:1153–70. [2] Faria T. An asymptotic stability result for scalar delayed population models. Proc Am Math Soc 2003;132:1163–9. [3] Hale JK, Verduyn Lunel SM. Introduction to functional differential equations. New York: Springer-Verlag; 1993.
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[4] Benchohra M, Henderson J, Ntouyas SK. Impulsive differential equations and inclusions. New York: Hindawi Publishing Corporation; 2006. [5] Stamov GT. Almost periodic solutions of impulsive differential equations. Berlin: Springer; 2012. [6] Stamova IM. Stability analysis of impulsive functional differential equations. Berlin: Walter de Gruyter; 2009. [7] Carr J. Applications of center manifold theory. New York: SpringerVerlag; 1981. [8] Lee JA, Verleysen M. Nonlinear dimensionality reduction. New York: Springer; 2007. [9] Palmer KJ. Linearisation near an integral manifold. J Math Anal Appl 1975;51:243–55. [10] Akhmet MU. Integral manifolds of differential equations with piecewise constant argument of generalized type. Nonlinear Anal 2007;66:367–83. [11] Corduneanu C, Ignatyev AO. Stability of invariant sets of functional differential equations with delay. Nonlinear Funct Anal Appl 2005;10:11–24. [12] Naito T. Integral manifolds for linear functional differential equations on some Banach space. Funkcial Ekvac 1970;13: 199–212. [13] Papaschinopoulos G. Linearisation near the integral manifold for a system of differential equations with piecewise constant argument. J Math Anal Appl 1997;215:317–33. [14] Stokes AP. On the stability of integral manifolds of functional differential equations. J Differ Equ 1971;9:405–19. [15] Kostadinov SI, Schneider K, Stamov GT. Integral manifolds of impulsive differential equations defined on torus. Proc Japan Acad Ser A Math Sci 1999;75:47–66. [16] Stamov GT. Affinity integral manifolds of linear singularly perturbed systems of impulsive differential equations. SUT J Math 1996;32:121–31. [17] Stamov GT. Integral manifolds of perturbed linear impulsive differential equations. Appl Anal 1997;64:39–56. [18] Stamov GT. Lyapunov’s functions and existence of integral manifolds for impulsive differential systems with time-varying delay. Methods Appl Anal 2009;16:291–8. [19] Stamov GT, Stamova IM. Second method of Lyapunov and existence of integral manifolds for impulsive differential–difference equations. J Math Anal Appl 2001;258:371–9. [20] Han QL, Gu K, Yu X. An improved estimate of robust stability bound of time-delay systems with norm-bounded uncertainty. IEEE Trans Autom Control 2003;48:1629–34. [21] He Y, Wu M, She JH, Liu GP. Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Syst Control Lett 2004;51:57–65. [22] Leitmann G. One approach to the control of uncertain dynamic systems. Appl Math Comput 1995;70:261–72. [23] Moon YS, Park P, Kwon WH, Lee YS. Delay-dependent robust stabilization of uncertain state-delayed systems. Int J Control 2001;74:1447–55. [24] Khan Q, Bhatti AI, Ferrara A. Dynamic sliding mode control design based on an integral manifold for nonlinear uncertain systems. J Nonlinear Dyn 2014;2014. Article ID 489364. [25] Chen WH, Zheng WX. Robust stability and H1 -control of uncertain impulsive systems with time-delay. Automatica 2009;45:109–17. [26] Ding X, Xu H. Robust stability and stabilization of a class of impulsive switched systems. Dyn Contin Discrete Impuls Syst 2005;2:795–8. [27] Li D, Wang S, Zhang X, Yang D. Impulsive control of uncertain Lotka– Volterra predator–prey system. Chaos Solitons Fract 2009;41:1572–7. [28] Liu B, Liu X, Liao X. Robust stability of uncertain impulsive dynamical systems. J Math Anal Appl 2004;290:519–33. [29] Stamov GT, Alzabut JO. Almost periodic solutions in the PC-space for uncertain impulsive dynamical systems. Nonlinear Anal 2011;74:4653–9. [30] Stamov GT, Stamov A. On almost periodic processes in uncertain impulsive delay models of price fluctuations in commodity markets. Appl Math Comput 2013;219:5376–83. [31] Stamov G. Impulsive integro-differential equations and stability of moving invariant manifolds. Methods Appl Anal 2007;14:69–76. [32] Stamov GT. Second method of Lyapunov and moving invariant manifolds for uncertain integro-differential equations. Nonlinear Stud 2007;14:337–43. [33] Stamov GTr. Stability processes of moving invariant manifolds in uncertain impulsive differential–difference equations. Math Bohem 2009;134:67–76.
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[34] Stamov GT. Uncertain impulsive differential–difference equations and stability of moving invariant manifolds. J Math Sci 2009;161:320–6. [35] Bainov DD, Dishliev AB. The phenomenon beating of the solutions of impulsive functional differential equations. Commun Appl Anal 1997;1:435–41.
[36] Afonso SM, Bonotto EM, Federson M, Gimenes LP. Boundedness of solutions of retarded functional differential equations with variable impulses via generalized ordinary differential equations. Math Nachr 2012;285:545–61.
Contents lists available at ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Integral manifolds for uncertain impulsive differential–difference equations with variable impulsive perturbations Gani Tr. Stamov a,⇑, Ivanka M. Stamova b a b
Department of Mathematics, Technical University of Sofia, 8800 Sliven, Bulgaria Department of Mathematics, The University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA
a r t i c l e
i n f o
Article history: Received 1 August 2013 Accepted 2 May 2014
a b s t r a c t In the present paper sufficient conditions for the existence of integral manifolds of uncertain impulsive differential–difference equations with variable impulsive perturbations are obtained. The investigations are carried out by means the concepts of uniformly positive definite matrix functions, Hamilton–Jacobi–Riccati inequalities and piecewise continuous Lyapunov’s functions. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The impulsive differential–difference equations are adequate mathematical models of various real processes and phenomena, characterized by the fact that their state changes by jumps and by the dependence of the process on its history at each moment of time. During the last few decades differential–difference equations have been object of numerous investigations related to the applications of these equations to physics, biology, electronics, control theory, etc. See, for example, [1–3] and references therein. For the basic concept, ideas and theorems of impulsive differential and functional differential equations we refer the reader to [4–6]. The method of integral manifolds is a very powerful instrument for investigating of various problems in the theory of differential equations. For example, the exceptional role of manifolds in the reduction of the dimensions of equations as well as for solving some linearization problems is well known [7–9]. Recently, there exists an intensive work regarding the study of integral manifolds for ⇑ Corresponding author. Tel.: +359 889526191. E-mail addresses: [email protected] (G.Tr. Stamov), ivanka.stamova@ utsa.edu (I.M. Stamova). http://dx.doi.org/10.1016/j.chaos.2014.05.002 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.
specific types of functional differential equations. For example, Akhmet [10] established some criteria for the existence of integral manifolds of differential equations with piecewise constant argument of generalized type. In [11] the stability of invariant sets of functional - equations with delay has been investigated. Naito [12] studied integral manifolds for linear functional differential equations on some Banach space. The linearization near the integral manifold for a system of differential equations with piecewise constant argument has been investigated by Papaschinopoulos in [13]. The stability of integral manifolds of a class of functional differential equations has been discussed in [14]. Also, the theory of integral manifolds is one of the most important part of the qualitative theory of impulsive differential and differential–difference equations. The main results related to the study of the existence of integral manifolds for such equations can be found in [15–19]. The investigations in above mentioned papers have been restricted to equations in their main form. However, it is important to point out that in the real word phenomena exist under the conditions of structural uncertainty. Nonlinear dynamical systems with uncertain parameters may cause changes of equilibrium states which will affect the integral manifold behavior.
G.Tr. Stamov, I.M. Stamova / Chaos, Solitons & Fractals 65 (2014) 90–96
In the meantime, many important results of the control theory for uncertain differential and delay differential systems have been established [20–23]. The work on the control law design relying on an integral manifold is relatively sparse, see [24] and the references therein. In many applications due to impulsive differential equations the uncertainties happen frequently due to modeling errors, measurement inaccuracy, mutations in the evolutionary processes and so on. At the present time there has appeared some results recognized as landmark contributions for uncertain impulsive differential and functional differential systems with fixed moments of impulse effect [25–30]. To study such uncertain impulsive systems, the idea of moving invariant sets has been also employed in [31–34]. However, to the best of our knowledge, there has not been any work so far dedicated to uncertain impulsive differential systems with variable impulsive perturbations. In the investigation of such systems there arise a number of difficulties related to the phenomena ‘‘beating’’ of the solutions, bifurcation, loss of the property of autonomy, etc. [35]. The wider application, however, of this type of equations requires the formulation of effective criteria for existence of integral manifolds. In this paper, motivated by the above considerations, we study an impulsive uncertain differential–difference system with variable impulsive perturbations. The rest of the paper is organized as follows: In Section 2 we give some notations, the problem investigated in this paper is formulated, and some definitions are presented. In Section 3 we establish sufficient conditions for the existence of integral manifolds for linear and nonlinear uncertain systems under consideration. The investigations follow the main ideas from [28] and are carried out by utilizing the concepts of uniformly positive definite matrix functions, Hamilton–Jacobi–Riccati inequalities and piecewise continuous Lyapunov’s type functions. Finally, in Section 4 one example is presented to illustrate the theory. Since the existence of an integral manifold is a generalization of the solutions’ existence of a nonlinear system, our results can be applied in the investigation of many practical problems.
2. Preliminaries Let h ¼ const > 0; Rn be the n-dimensional Euclidean space with norm j: j; Rþ ¼ ½0; 1Þ; t0 2 R. Let J # R and X # Rn ; X – ø. Define the following class of functions: PC½J; X ¼ fr : J ! X : rðtÞg. is a piecewise continuous ~t 2 J at which rð~t Þ function with points of discontinuity and rð~tþ Þ exist and rð~t Þ ¼ rð~tÞ . Let t > t0 . Consider the following system of uncertain impulsive differential–difference equations with variable impulsive perturbations
_ xðtÞ ¼ f ðt; xðtÞ; xðt hÞÞ þ gðt; xðtÞ; xðt hÞÞ; t – sk ðxðtÞÞ;
DxðtÞ ¼ Ik ðxðtÞÞ þ Jk ðxðtÞÞ; t ¼ sk ðxðtÞÞ; k ¼ 1; 2; . . . ; ð2:1Þ where f ; g : ðt 0 ; 1Þ X X ! Rn ; Ik ; J k : X ! Rn ; ðt0 ; 1Þ; k ¼ 1; 2; . . . ; DxðtÞ ¼ xðt þ Þ xðtÞ.
sk : X !
91
The functions gðt; x; yÞ; J k ðxÞ represent the structural uncertainty or uncertain perturbations are characterized by g 2 U g ¼ g : gðt; x; yÞ ¼ eg ðt; x; yÞ:dg ðt;x; yÞ; jdg ðt; x; yÞj 6 jmg ðt; x; yÞj ;
and
J k 2 U J ¼ fJk : J k ðxÞ ¼ ek ðxÞ:dk ðxÞ; jdk ðxÞj 6 jmk ðxÞjg; k ¼ 1; 2; . . . ; where eg : ðt 0 ; 1Þ X X ! Rnm , and ek : X ! Rnm are known matrix functions whose entries are smooth functions of the state, and dg ; dk are unknown vector-valued functions whose norm are bounded, respectively, by the norm of vector-valued functions mg ðt; x; yÞ; mk ðxÞ, respectively. Here mg : ðt0 ; 1Þ X X ! Rm ; mk : X ! Rm ; k ¼ 1; 2; . . . are given smooth functions. Let /0 2 C ½½h; 0; X. Denote by xðtÞ ¼ xðt; t0 ; /0 Þ [5,6] the solution of system (2.1), satisfying the initial conditions
xðt; t0 ; /0 Þ
¼ /0 ðt t0 Þ; t 0 h 6 t 6 t0 ;
xðtþ0 ; t 0 ; /0 Þ ¼ /0 ð0Þ;
ð2:2Þ
J þ ðt0 ; /0 Þ is the maximal interval of the type ½t 0 ; bÞ in which the solution xðt; t 0 ; /0 Þ is defined, and hþ ðt 0 ; /0 ðtÞÞ denotes the integral orbit of the solution xðt; t0 ; /0 Þ for t 2 J þ ðt0 ; /0 Þ. The solutions xðtÞ of system (2.1) are piecewise continuous functions with points of discontinuity of the first kind in which they are left continuous; i.e., at the moments t lk when the integral curve of the solution xðtÞ meets the hypersurfaces
rk ¼ fðt; xÞ 2 ½t0 ; 1Þ X : t ¼ sk ðxÞg the following relations are satisfied:
xðtjk Þ ¼ xðt jk Þ; xðtþjk Þ ¼ xðt jk Þ þ Ijk ðxðt jk ÞÞ þ Jjk ðxðt jk ÞÞ: The points t 1 ; t 2 ; . . .(t0 < t1 < t2 ) are the impulsive moments. Let us note that, in general, k – jk . In other words, it is possible that the integral curve of the problem under consideration does not meet the hypersurface rk at the moment t k . It is clear that the solutions of systems with variable impulsive perturbations have points of discontinuity depending on the solutions, i.e. the different solutions have different points of discontinuity. This leads to a number of difficulties in the investigation of such systems. One of the phenomena occurring with systems of type (2.1) is the so called ‘‘beating’’ of the solutions. This is the phenomenon when the mapping point ðt; xðtÞÞ meets one and the same hypersurface rk several or infinitely many times [5,6,35]. For the proof of the main results we shall use the following nominal system
_ xðtÞ ¼ f ðt; xðtÞ; xðt hÞÞ; t – sk ðxðtÞÞ; DxðtÞ ¼ Ik ðxðtÞÞ; t ¼ sk ðxðtÞÞ; k ¼ 1; 2; . . . ;
ð2:3Þ
We shall introduce the following conditions for the system (2.3): H2.1. f 2 C½ðt 0 ; 1Þ X X; Rn . H2.2. The function f is Lipschitz continuous with respect to its second and third arguments uniformly on t 2 ðt 0 ; 1Þ.
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H2.3. sk 2 C½X; ðt 0 ; 1Þ; k ¼ 1; 2; . . .. H2.4. t0 s0 ðxÞ < s1 ðxÞ < s2 ðxÞ < . . . ; x 2 X. H2.5. sk ðxÞ ! 1 as k ! 1 uniformly on x 2 X. H2.6. The functions Ik are Lipschitz continuous in X. H2.7. The functions ðI þ Ik Þ : X ! X; k ¼ 1; 2; . . ., where I is the identity in X. H2.8. jk < jkþ1 < < jkþp < . . ., where jk is the number of the hypersurface met by the integral curve ðt; xðtÞÞ of the problem (2.3) at the moment tk ; k; jk ; p ¼ 1; 2; . . .. We also assume that the following condition holds: H2.9. The integral curves of system (2.3) meet each one of the hypersurfaces r1 ; r2 ; . . .. at most once. The condition H2.9 guarantees absence of the phenomenon ‘‘beating’’ in (2.3). This problem has been investigated by several authors. Efficient sufficient conditions which guarantee the absence of ‘‘beating’’ of the solutions of impulsive functional differential systems are given in [35]. Similar problems are discussed in [36]. It is also clear that the condition H2.9 is satisfied in the special case when sk ðxÞ tk ; k ¼ 1; 2; . . . ; x 2 X, i.e., when the impulses take place at fixed moments. Introduce the following notations:
Gk ¼ fðt; xÞ 2 ½t 0 ; 1Þ X : sk1 ðxÞ < t < sk ðxÞg; k ¼ 1; 2; . . . ; G ¼ [1 k¼1 Gk ; PC 1 ½J; X ¼ fr 2 PC½J; X : rðtÞ is continuously differentiable _ þ everywhere except the points tk at which r_ ðt k Þ and rðt k Þ _ exist and ðr_ ðt Þ ¼ r ðt Þ; k ¼ 1; 2; . . .g; k k
K ¼ fa 2 C½Rþ ; Rþ : aðrÞ is strictly increasin an að0Þ ¼ 0g: Definition 2.1. We call an arbitrary manifold M in the extended phase space ½t0 h; 1Þ X of (2.1) integral manifold, if ðt; /0 ðt t 0 ÞÞ 2 M, for t 2 ½t0 h; t 0 implies, hþ ðt0 ; /0 ðtÞÞ M. In what follows we shall use the class V M of piecewise continuous auxiliary functions V : ½t 0 ; 1Þ X ! Rþ which are analogues of Lyapunov’s functions [19]. Definition 2.2. We shall say that the function V : ½t0 ; 1Þ X ! Rþ belongs to the class V M which kernel is the manifold M in the extended phase space of (2.1), if the following conditions hold: 1. The functions V is continuous in G and locally Lipschitz continuous with respect to its second argument x in each of the sets Gk ; k ¼ 1; 2; . . .. 2. Vðt; xÞ ¼ 0 for ðt; xÞ 2 M; t P t0 and Vðt; xÞ > 0 for ðt; xÞ 2 ½t0 ; 1Þ X n M. 3. For each k ¼ 1; 2; . . . ; and ðtk ; x0 Þ 2 rk there exist the finite limits Vðt Vðt;xÞ; Vðt þ Vðt;xÞ k ;x0 Þ ¼ lim k ;x0 Þ ¼ lim ðt;xÞ!ðt ;x Þ k 0 ðt;xÞ2Gk
ðt;xÞ!ðt ;x Þ k 0 ðt;xÞ2Gkþ1
and the equality Vðt k ; x0 Þ ¼ Vðt k ; x0 Þ is valid.
Given a function V 2 V 0 . For t P t 0 ; t – sk ðxðtÞÞ; k ¼ 1; 2; . . ., introduce the function
Dþ Vðt;xðtÞÞ ¼ limþ sup r!0
1
r
½Vðt þ r;xðtÞ þ rf ðt; xðtÞ;xðt hÞÞÞ Vðt;xðtÞÞ; where ðt; xÞ 2 ½t0 ; 1Þ PC½½t0 ; 1Þ; X. We shall use the next lemma. Lemma 2.3 [19]. Assume that: 1. Conditions H2.1–H2.9 are met. 2. For the system (2.3), there exists a function V 2 V M with a kernel the manifold M, such that:
Vðt þ ; xðtÞ þ Ik ðxðtÞÞÞ 6 Vðt; xðtÞÞ; t ¼ sk ðxðtÞÞ; k ¼ 1; 2; . . . ; ð2:4Þ and the inequality
Dþ Vðt; xðtÞÞ 6 0; t – sk ðxðtÞÞ; t – sk ðxÞ; k ¼ 1; 2; . . . ð2:5Þ is valid whenever Vðt þ s; xðt þ sÞÞ 6 Vðt; xðtÞÞ for t P t 0 and h 6 s 6 0.Then M is an integral manifold of (2.3). Definition 2.4. The matrix function C : R ! Rnn is said to be: (a) a positive definite matrix function, if for any t 2 R; CðtÞ is a positive definite matrix; (b) a positive definite matrix function bounded above, if it is a positive definite matrix function and there exists a positive real number R > 0 such that
kmax ðCðtÞÞ 6 R; t 2 R; where kmax ð:Þ is the greatest eigenvalue of ð:Þ; (c) a uniformly positive definite matrix function, if it is a positive definite matrix function and there exists a positive real number r > 0 such that
kmin ðCðtÞÞ P r; t 2 R; where kmin ð:Þ is the smallest eigenvalue of ð:Þ. The proof of the following lemma is obvious. Lemma 2.5. Let CðtÞ 2 Rnn be a positive definite matrix function and YðtÞ 2 Rnn be a symmetric matrix. Then for any x 2 Rn ; t 2 R the following inequality holds
xT YðtÞx 6 kmax ðC1 ðtÞYðtÞÞxT CðtÞx: 2
ð2:6Þ 2
Lemma 2.6 [28]. Let RðtÞ 2 Rn n be a diagonal matrix function such that RðtÞ ¼ diagfe11 ðtÞ; . . . ; e1n ðtÞ; . . . ; en1 ðtÞ; . . . ; enn ðtÞg, and jeij ðtÞj 6 1; i; j ¼ 1; 2; . . . ; n,. Then for any positive scalar function kðtÞ > 0 and for any 2
n; g 2 Rn , the following inequality holds
2nT RðtÞg 6 k1 ðtÞnT n þ kðtÞgT g:
ð2:7Þ
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if Vðt þ s; xðt þ sÞÞ 6 Vðt; xðtÞÞ for h 6 s 6 0. Then M is an integral manifold of the system (2.1).
3. Main results Introduce the following conditions: H3.1. The function eg is Lipschitz continuous with respect to its second and third arguments uniformly on t 2 ðt0 ; 1Þ. H3.2. The functions ek are Lipschitz continuous in X. H3.3. The functions ðI þ ek Þ : X ! X; k ¼ 1; 2; . . ..
Proof. From (3.1), (3.2), Lemmas 2.5 and 2.6 t ¼ sk ðxÞ; k ¼ 1; 2; . . ., we have
for
Vðt þ ; xðtÞ þ Ik ðxðtÞÞ þ J k ðxðtÞÞÞ 6 Vðtþ ; xðtÞ þ Ik ðxðtÞÞÞ þ G1k dk ðxðtÞ þ dk ðxðtÞÞT G2k dk ðxðtÞÞ 6 Vðt þ ; xðtÞ T T þ Ik ðxðtÞÞÞ þ v1 k G1k G1k þ ðvk þ kmax ðG2k ÞÞmk mk
Lemma 3.1. Let the conditions H2.1–H2.9 and H3.1–H3.3 hold. Then 1. t k ! 1. 2. J þ ðt 0 ; u0 Þ ¼ ½t0 ; 1Þ.
6 Vðt; xðtÞÞ:
On the other hand, for t – sk ðxÞ; k ¼ 1; 2; . . . and from (3.3), we get
Dþ Vðt; xðtÞÞ ¼
@V @V @V @V @V þ ðf þ gÞ ¼ þ fþ eg dg @t @x @t @x @x @V @V k2 @V @V T 1 þ þ 2 mTg mg fþ k eg eTg @t @x @x 2 @x 2kk ) ( 1 @V 1 @V T 1 eg dTg : kk eTg kk dg 2 @x kk @x kk
Proof of Assertion 1. From condition H2.8 we derive the inequalities
¼
j1 < j2 < . . . : Then since jk are positive integers, we conclude that jk ! 1 as k ! 1. Then by condition H2.5 we get to the equalities
lim tk ¼ lim sjk ðxk Þ ¼ 1;
k!1
jk !1
þ
where xk ¼ xðt k ; t0 ; /0 Þ. Proof of Assertion 2. Since by the conditions H2.1, H2.2, H2.7 and H3.1–H3.3 the solution xðt; t 0 ; /0 Þ of the problem (2.1) is defined on each of the intervals ðtk ; tkþ1 ; k ¼ 1; 2; . . ., then from Assertion 1 we conclude that it is continuable for each t P t 0 . Theorem 3.2. Assume that:
(i) there exist G1k : ðt0 ; 1Þ X ! R1m ; G2k : ðt0 ; 1Þ X ! Rmm , where G2k are positive define matrix functions and for z 2 Rm ; k ¼ 1; 2; . . . it follows
Vðt; x þ Ik ðxÞ þ ek ðxÞzÞ 6 Vðt; x þ Ik ðxÞÞ þ G1k ðt; xÞz þ zT G2k ðt; xÞz;
ð3:1Þ
(ii) there exist positive constants vk such that for t 2 ðt0 ; 1Þ; k ¼ 1; 2; . . . ; x 2 PC 1 ½½t 0 ; 1Þ; X, T Vðt þ ; xðtÞ þ Ik ðxðtÞÞÞ þ v1 k G1k G1k
þ ðvk þ kmax ðG2k ÞÞmTk mk 6 Vðt; xðtÞÞ; t 2 rk ; ð3:2Þ
where G1k ¼ G1k ðt; xðtÞÞ; G2k ¼ G2k ðt; xðtÞÞ; mk ¼ mk ðxðtÞÞ; (iii) there exist scalar functions kk 2 C½X; Rþ such that for t 2 ½t0 ; 1Þ it follows @V @V k2 @V @V T 1 T þ fþ k þ m mg 6 0; ðt; xÞ 2 Gk ; eg eTg 2 @x @x 2k2k g @t @x
ð3:3Þ
1 2k2k
fmTg mg dTg dg g 6
@V @V þ f @t @x
k2k @V @V T 1 eg eTg þ 2 mTg mg 6 0; 2 @x @x 2kk
ð3:5Þ
if Vðt þ s; xðt þ sÞÞ 6 Vðt; xðtÞÞ; h 6 s 6 0. Therefore in the light of inequalities (3.4) and (3.5) and condition 1 of the theorem it follows that the conditions of Lemma 2.3 are satisfied for the system (2.1). The proof of Theorem 3.2 is complete. h Now we consider a linear system of uncertain impulsive differential–difference equations
1. The conditions H2.1–H2.9 and H3.1–H3.3 are met. 2. For the system (2.1) there exists a function V 2 V M with a kernel the manifold M, and the following relations are satisfied:
ð3:4Þ
x_ ¼ AðtÞx þ BðtÞxðt hÞ; t – sk ðxðtÞÞ;
DxðtÞ ¼ Ak ðtÞxðtÞ þ Bk ðtÞxðtÞ; t ¼ sk ðxðtÞÞ;
ð3:6Þ
where A; Ak 2 C ½ðt 0 ; 1Þ; Rnn ; k ¼ 1; 2; . . . are known matrix functions, and B; Bk 2 C ½ðt 0 ; 1Þ; Rnn ; k ¼ 1; 2; . . . are interval matrix functions, i.e.
BðtÞ 2 IN ½PðtÞ; Q ðtÞ ¼ BðtÞ 2 Rnn : BðtÞ ¼ ðbij ðtÞÞ; pij ðtÞ 6 bij ðtÞ 6 qij ðtÞ; i; j ¼ 1; 2; . . . ; n : Bk ðtÞ 2 IN½P k ðtÞ; Q k ðtÞ, where PðtÞ ¼ ðpij ðtÞÞ; Q ðtÞ ¼ ðqij ðtÞÞ; Pk ðtÞ, Q k ðtÞ are known matrices, and P;Q ;P k ;Q k 2C ½R;Rnn ; k¼1;2;.... Lemma 3.3 [28]. Let BðtÞ 2 IN ½PðtÞ; Q ðtÞ, where PðtÞ; Q ðtÞ are known matrices. Then BðtÞ can be written
BðtÞ ¼ B0 ðtÞ þ EðtÞRðtÞFðtÞ; where:
B0 ðtÞ ¼
1 1 ðPðtÞ þ Q ðtÞÞ; HðtÞ ¼ ðhij ðtÞÞ ¼ ðQ ðtÞ PðtÞÞ; 2 2
hij ðtÞ P 0; t 2 R; i; j ¼ 1; 2; . . . ; n;
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RðtÞ ¼ diagfe11 ðtÞ; . . . ; e1n ðtÞ; . . . ; en1 ðtÞ; . . . ; enn ðtÞg 2 Rn
2 n2
;
jeij ðtÞj 6 1; i; j ¼ 1; 2; . . . ; n; EðtÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h11 ðtÞe1 ;...; h1n ðtÞe1 ;...; hn1 ðtÞen ;...; hnn ðtÞen 2 Rnn ;
Then V 2 V M and let we consider the set of all solutions x ¼ xðtÞ of (3.6) such that Vðt þ s; xðt þ sÞÞ 6 Vðt; xðtÞÞ whenever h 6 s 6 0; t P t0 . Then for t – sk ðxÞ; x P 0, from the next inequality
Vðt h; xðt hÞÞ 6 Vðt; xðtÞÞ and
qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 FðtÞ ¼ h11 ðtÞe1 ;...; h1n ðtÞen ;...; hn1 ðtÞe1 ;...; hnn ðtÞen 2 Rn n ;
xT ðt hÞCðt hÞxðt hÞÞ 6 xT ðtÞCðtÞxðtÞÞ it follows that
kmin ðCÞx2 ðt hÞ 6 xT ðt hÞCxðt hÞ 6 xT ðtÞCxðtÞ
ei ð0; . . . ; 0; 1; 0; . . . ; 0ÞT 2 Rn ; i ¼ 1; 2; . . . ; n:
6 kmax ðCÞx2 ðtÞ:
By Lemma 3.3 we rewritten the system (3.6) in the form (
The last inequality shows that
¼ AðtÞx þ B0 ðtÞxðt hÞ þ EðtÞRðtÞFðtÞxðt hÞ; t – sk ðxðtÞÞ; DxðtÞ ¼ A~k ðtÞxðtÞ þ E~k ðtÞR~k ðtÞF~k ðtÞxðtÞ; t ¼ sk ðxðtÞÞ; k ¼ 1;2;...;
x_
ð3:7Þ
kmax ðCÞ 2 x ðtÞ: kmin ðCÞ
x2 ðt hÞ 6
ð3:11Þ
On the other hand from (3.8), (3.9) and (3.11) it follows
where
D Vðt;xðtÞÞ 6 x_ T Cx þ xT C_ x þ xT Cx_ þ
A~k ðtÞ ¼ Ak ðtÞ þ B~k0 ðtÞ;
_ ¼ ½AxðtÞ þ Bxðt hÞCxðtÞ þ xT ðtÞCxðtÞ
Bk ðtÞ ¼ B~k0 ðtÞ þ E~k ðtÞR~k ðtÞF~k ðtÞ; k ¼ 1; 2; . . .
þ xT ðtÞC½AxðtÞ þ Bxðt hÞ h i ¼ xT ðtÞ C_ þ AT C þ CA xðtÞ
and let A ¼ AðtÞ; B0 ¼ B0 ðtÞ; E ¼ EðtÞ; R ¼ RðtÞ; F ¼ FðtÞ for ~k ¼ A ~ k ðtÞ; Bk0 ¼ Bk0 ðtÞ; E ~k ¼ E ~k ðtÞ; R ~k ¼ t – sk ðxÞ, and A ~ k ðtÞ; F~k ¼ F~k ðtÞ for t ¼ sk ðxÞ. R
_ þ xT ðtÞBT CxðtÞ þ xT ðtÞCBxðt hÞ þ xT ðt hÞCxðtÞ h i T _ 6 x ðtÞ C þ AT C þ CA xðtÞ þ 2jBjjCjjxðtÞjjxðt hÞj h i 6 xT ðtÞ C_ þ AT C þ CA xðtÞ
Theorem 3.4. Assume that: 1. Conditions H2.3, H2.4, H2.5, H2.8 and H2.9 are met. 2. There exists a continuous real matrix C ¼ CðtÞ; C : ðt0 ;1Þ ! Rnn , which is symmetric and positive such that:
þ 2jBjjCjðx2 ðtÞ þ x2 ðt hÞj 6 xT ðtÞðCÞxðtÞ kmax ðCÞ 2 x ðtÞ þ 2ðjB0 j þ jEjjFjÞjCj x2 ðtÞ þ kmin ðCÞ kmax ðCÞ x2 ðtÞ: 6 kmin ðCÞ þ 2ðjB0 j þ jEjjFjÞkmax ðCÞ 1 þ kmin ðCÞ
(i) CðtÞ is differentiable at t – sk ðxÞ and the following inequality holds:
ð3:12Þ
C_ þ CA0 þ AT C 6 C; where C 2 Rnn is a positive scalar matrix; kmax ðCÞ (ii) ðjB0 j þ jEjjFjÞkmax ðCÞ 1 þ kmin ðCÞ 6 0; kmin ðCÞ
ð3:8Þ
ð3:9Þ
where kmin and kmax are respectively the smallest and the greatest eingenvalues of matrices C and C, respectively. (iii) the following inequalities hold
T
~T ~k ~T ~ ~T IþA IþA bk ¼ kmax C1 IþA Ck þ n1 k Ck E k E k Ck k k k
io ~k Ck E ~k F~T F~k 6 1: þ nk þ kmax E k
ð3:10Þ
where Ck ¼ CðtÞ for t ¼ sk ðxÞ; x P 0. Then the manifold M; M ¼ ½t0 h; 1Þ fx 2 Rn ; x 6 0g is an integral manifold for the system (3.7).
Then for t – sk ðxÞ; t P t 0 from (3.12) it follows that þ
D Vðt; xÞ 6 0; if Vt þ s; xðt þ sÞÞ 6 Vðt; xðtÞÞ whenever h 6 s 6 0 , which shows that the inequality (3.3) of Theorem 3.2 hold. By the similar proof of (3.4) by Lemmas 2.5 and 2.6 we have h iT ~k þ E ~k R ~ k F~k Vðt þ ;xðtÞ þ Ak ðxðtÞÞ þ Bk ðxðtÞÞ ¼ xT ðtÞ I þ A h i ~k þ E ~ k F~k xðtÞ ~k R Ck I þ A
T
T ~ T Ck I þ A ~k þ I þ A ~ k Ck E ~k R ~ k F~k ¼ xT ðtÞ I þ A k
T
T ~k E ~k R ~k R ~ k F~k Ck I þ A ~ k F~k Ck E ~k R ~ k F~k xðtÞ þ E
T
T
~ T Ck I þ A ~ k þ n1 I þ A ~ k Ck E ~k ~k E ~T Ck I þ A 6 xT ðtÞ I þ A k k k i
~k Ck E ~k xT ðtÞF~T F~k xðtÞ þnk F~k F~Tk xðtÞ þ kmax E k
T
1 ~T ~k Þ I þ A ~T ¼ xT ðtÞ I þ A Ck þ n Ck E~k E~T Ck I þ A
Proof. We will show that for the system (3.6) all conditions of Theorem 3.2 hold. Consider the function
Vðt; xÞ ¼
T
x CðtÞx; x P 0; 0; x 6 0:
k
k
þ nk þ kmax
~ k Ck E ~k E
k
k
i F~Tk F~k xðtÞ 6 xT ðtÞCk xðtÞ ¼ Vðt;xðtÞÞ; t P t 0 :
From the last inequality it follows that the condition (3.2) of Theorem 3.2 hold. The proof of Theorem 3.4 is complete. h
G.Tr. Stamov, I.M. Stamova / Chaos, Solitons & Fractals 65 (2014) 90–96
4. An example Consider the following nonlinear system of uncertain impulsive differential–difference equations with variable impulsive perturbations
8 _ ¼ f ðt;xðtÞ; xðt hÞÞ þ gðt; xðtÞ; xðt hÞÞ; t – sk ðxðtÞÞ; xðtÞ > < ! 1 1k 0 > xðtÞ; t ¼ sk ðxðtÞÞ; k ¼ 1; 2; .. . ; D xðtÞ ¼ : 0 1 1k ð4:1Þ where
x¼
x1 x2
;
f ðt; x; yÞ ¼
y¼
y1 y2
;
2y1 11x1 þ 2x2 ðx21 þ x22 Þ y2 9x2 x1 ðx21 þ x22 Þ
! ;
g 2 U g ¼ g : gðt; x; yÞ ¼ eg ðt; x; yÞ:dg ðt;x;yÞ; jdg ðt; x; yÞj 6 jmg ðt; x; yÞj ;
eg ¼ I; mg ¼
! pffiffiffi 2ðx1 þ x2 Þ ; y2
sk ðxÞ ¼ jxj þ k; k ¼ 1; 2;. .. :
We have sk 2 C½R2 ; ð0; 1Þ; k ¼ 1; 2; . . . ; sk ðxÞ ! 1 as k ! 1 uniformly on x 2 R2 , and also
0 < s1 ðxÞ < s2 ðxÞ < . . . ; x 2 R2 : Also, for the given choice of the functions H2.9 is satisfied [35]. Vðt; xÞ ¼ 12 ðx21 þ 2x22 Þ.
Let
Consider
sk condition
the
manifold
M ¼ fðt; 0Þ : t 2 ½h; 1Þg. Then, obviously, V 2 V M and V is differentiable in t 2 Rþ . For ðt; xÞ 2 Gk ; k ¼ 1; 2; . . . we have the following Hamilton–Jacobi inequality
@V @V k2 @V @V T 1 fþ k eg eTg þ þ 2 mTg mg ¼ 2x1 ðtÞx1 ðt hÞ @t @x 2 @x @x 2kk 11x21 ðtÞ þ 2x2 ðtÞx2 ðt hÞ 18x22 ðtÞ þ
x21 ðtÞ þ 2x22 ðtÞ 2
9 þ 2ðx21 ðtÞ þ x22 ðtÞÞ þ x22 ðt hÞ 6 Vðt; xÞ 6 0 2
ð4:2Þ
satisfied whenever Vðt þ s;xðt þ sÞÞ 6 Vðt; xðtÞÞ; h 6 s 6 0. Also, for t ¼ sk ðxÞ; k ¼ 1; 2; . . .
Vðt þ ; xðtÞ þ Ik ðxðtÞÞ þ J k ðxðtÞÞÞ " 2 2 # 1 x1 ðtÞ x2 ðtÞ þ2 ¼ 2 k k ¼
1 2k
2
ðx21 ðtÞ þ 2x22 ðtÞÞ 6 Vðt; xðtÞÞ:
ð4:3Þ
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