Global Exponential stability for nonlinear delay differential systems Текст научной статьи по специальности «Математика»

DOI: 10.14529/ctcr170214

GLOBAL EXPONENTIAL STABILITY

FOR NONLINEAR DELAY DIFFERENTIAL SYSTEMS

Leonid Berezansky, [email protected] Ben-Gurion University of the Negev, Beer-Sheva, Israel

We give a review on recent results for global stability for nonlinear functional differential equations. Such equations include delay differential equations, integro-differential equations and equations with distributed delay and are applied as mathematical models in Population Dynamics and other sciences. We also consider methods used to study global stability: constructing of Lyapunov functional, applications of special matrices such as M-matrix or special matrix functions such as matrix measure, method of matrix inequalities, which is very popular in papers on Control Theory, fixed point approach and using a notion of nonlinear Volterra operator.

Keywords: the global stability, the Lyapunov functional, the matrix measure, the method of matrix inequalities, the nonlinear Volterra operator.

1. INTRODUCTION

One of the main motivations to study nonlinear delay differential systems is their importance in investigations of artificial neural network models and more generally in Mathematical Biology.

In this review paper we will discuss a global stability problem for linear and nonlinear systems of FDE. Such investigations one can divide by the form of a system: vector o scalar form and also by the method of investigation. The main methods are: constructing of Lyapunov functionals, applications of special matrices such as M-matrix or special matrix functions such as matrix measure, method of matrix inequalities, which is very popular in papers on Control Theory, fixed point approach and using a notion of nonlinear Volterra operator.

In this paper we consider all forms of systems - vector and scalar forms and some of methods - applications of M-matrix and matrix measure, and using an abstract Volterra (causal) operator.

Some worlds about other methods. Lyapunov functionals are usually used for concrete systems arising in applications or for general systems with results formulated in implicit form. In method of matrix inequalities they use very unwieldy matrices, fixed point approach is new and I don't now interesting results, obtained by this method for systems of FDE.

2. VECTOR FORM SYSTEM

2.1. Matrix Measure

Denote by ||x|| a vector norm in M™. Matrix measure(logarithmic matrix norm) can be denoted by the following equality

Application of the matrix measure is one of the main tools in investigations of stability for systems of ODE. For FDE the matrix measure was also applied, see for example [1]. In the recent paper [2] the authors considered the system

pA(t) = lim^0+

— = Ax(t) + F(t, x(t - t1).....x(t - тт), t>to>0,

(2.1)

(2.2)

where A(t) E is a measurable locally essentially bounded matrix function and F(t, u1, ..., um) is a Caratheodory function, i.e., it is a locally essentially bounded matrix function of t, and continuous matrix function at each point (ui, ..., um) for any t, hk are measurable functions, and 0 < t - hk(t) < t, (k = 1, 2, ..., m); with the initial value problem

x(t) = <p(t), t < t0, x(t0) = Xq, (2.3)

where p is a continuous function.

Definition 2.1. The solution of system (2.2)-(2.3) is a locally absolutely continuous function for t > t0, that satisfied equality (2.2) almost everywhere for t > t0 and initial conditions (2.3) for t < t0.

Assume that the unique global solution for system (2.2)-(2.3) exists. Now we want to give the definitions of the global stability.

Definition 2.2. The open set Q c ML" attracts all solutions of (2.2)-(2.3) with the initial conditions in the open set Q0 c M" if for any t0 > 0 and any solution x ofproblem (2.2)-(2.3) with y(t) E Q0, x0 E Q0 there exists t1 > t0 such that x(t) E Q, t > t1.

Definition 2.3. An equilibrium K is globally asymptotically stable, if it is a global attractor for all solutions of (2.2)-(2.3) with the initial conditions in the open set Q0 c M" and it is also locally uniformly stable.

Theorem 2.4. Suppose thatF(t, 0, ..., 0) = 0, Q0 c M" is an open set, such that 0 E Q0, an open set Q c Mn attracts all solutions of system (2.2)-(2.3) with the initial conditions in Q0. Suppose also that there exists a nonnegative sequence Xk (1 < k < m) such that for any uk E Q the following inequalities hold:

TO sup\\F(t , u^,...,

Then the trivial solution is the global attractor for all solutions of (2.2)-(2.3) with the initial conditions in Q0.

If system is written in a scalar form, it is not always suitable to transform it to a vector form. In this case sometimes one can use a notion of Volterra operator.

We denote the space C[t0, <») of continuous bounded vector functions, and the space Lx\t0, <») of measurable essentially bounded vector functions, where both spaces are Banach spaces with usual sup-norm.

Definition 2.5. Suppose B1 and B2 are two vector functional spaces on [t0, w). We say that operator T : B1 ^ B2 is a causal or Volterra operator if for any t1 > t0 equality x(t) = y(t), t E [t0, t1] implies (Tx)(t) = (Ty)(t), t E [t0, t1].

We illustrate how a nonautonomous Nicholson-type model

xt(t) = -ai(t)Xl(t) + b1(t)x2(t) + c1(t)x1(h1(t))exp(-x1(h1(t))), (2.4)

x2(t) = -a2(t)x2(t) + b2(t)Xl(t) + c2(t)x2(h2(t))exp(-x2(h2(t))) with the following initial value problem

x(t) = <p(t) = {(p1(t),(p2(t)}T, t < t0, x(t0) = x0, (2.5)

can be written in a compact form, using causal(Volterra) operators.

Suppose ai, bi, ci, i = 1, 2 are essentially bounded on [0, w) functions, hk, k = 1, ..., m are measurable functions, 0 < t — hk(t) < t, <p(t), t < tQ is a continuous vector-function.

Let x = {x1,x2]T,

-at(t) bi(t)

A(t) = ¿(0 —iV'(i)J, (26)

= (x(m) t>t0 (ph(t) = Mh(t)) t<t

hV J (0 t<to [0 t>t0' V ' (Fvx)(t) = {c1(t)[xihi(t) + <p^(t)] exp(—[xihi(t) + ^(t)]),

c2(t)[x2h2(t) + exp(—[x2h2(t) + <p22(t)])}T. (2.8) Then F^: C[t0, to) ^ Lm[t0, <x>) is a bounded nonlinear Volterra operator and system (2.4)-(2.5) has the operator form

^ = A(t)x(t) + (F<px)(t), t > tQ, x(t0) = x0. (2.9)

Let us fix т > 0 and discuss the global stability conditions for the operator equation (2.9), where for any t0 > 0 and continuous vector function ^ : [t0 — т, t0] ^ R™ operator F(p:C[t0, ж) ^ Lm[t0,<x>) is a bounded Volterra operator.

Theorem 2.6. Suppose that (F<p0)(t) = 0 for ф = 0,Q0 с Rn is an open set, such that 0 E Q0, an open set Q с Rn attracts all solutions of (2.9) with the initial conditions in Q0. Suppose also that there exist numbers X > 0, т > 0 such that for any x E C[t0, ю), x(t) E Q and x(t0) E Q0 the following inequalities hold for sufficiently large t:

\\(F<pX)(t)\\ ^ * supt_T<,r<t||xOf)IU <a:= limt^minf(—^A(t)), where X and т do not depend on ф. Then the trivial solution is a global attractor for all solutions of (2.9) with initial conditions in Q0.

As an application of the previous theorem consider Nicholson system with proportional coefficients xt(t) + r^tXa^t) — b1x2(t) — c1x1(h1(t))exp (—x1(h1(t)))) = 0, (2.10)

x2(t) + r2(t)(a2x2(t) — b2xt(t) — c2x2(h2(t))exp(—x2(h2(t)))) = 0, ri(t) are measurable essentially bounded on [0, да) nonnegative functions; and ai > 0, bi > 0, ci > 0, are constants.

Denote Qi = lim supt^ra r^t) and q¿ = lim inft^ra r¿(í).

Theorem 2.7. Suppose that a positive internal equilibrium (x{,x2) of system (2.10) exists and for some e > 0

max{Q1c1e~2 ,Q2c2e~2 ,Q1c1e~x*^ — e, Q2c2e~xz — e, Q^e'^l — хЦ^2с2е~хЦ1 — хЦ} < < min{q1(a1 — bt), q2(a2 — b2)}. Then this equilibrium is globally asymptotically stable.

2.2. M-matrix

M-matrix method is widly used for all classes of systems of differential equations including FDE. A matrix В = (Ь^)™=1 is called a (non-singular) M-matrix if bij < 0, i Ф j and one of the following equivalent conditions holds:

• there exists a positive inverse matrix B> 0;

• the principal minors of matrix B are positive. Consider first several known results.

In the paper [4] the authors consider the autonomous system for the system

x() = — TJl=iaijxj{t — ту), i = 1, ...,m, (2.11)

where тц > 0, the following result holds (below, a+ denotes the positive part of a, i.e., a+ = max{a, 0}). Theorem. Let

0 < aura < 1 + 1/e, i = 1, ..., m and let the m x m matrix H with components

r(l-(aUTU-l/e)+\ = .

hij = Al+(aiiTii-l/e)Jaii,L ¡t к í * j, i, j = 1, ., m be a non-singular M-matrix. Then, system (2.11) is asymptotically stable for any selection of delays т„, i ф j, i, j = 1, ., m.

In the paper [5] the authors consider the non-autonomous system

*() = —TJhaij(t)xj(hij(t)), i = 1.....m, (2.12)

where t E [t0, ж), tQ E E, a^ (t), htj (t) are continuous functions, hij(t) < t, and hij(t) are monotone increasing functions such that limt^ra hy(t) = ж, i,j = 1,...,m.

Theorem. Assume that, for t > tQ, there exist non-negative numbers bj, i,j = 1,...,m, i ф j such that

j) | < bya„(t), i, j = 1, ., m, i ф j, an(t) > 0 and

fra au(s)ds = ж, di = limt^ra sup au(s)ds <3/2, i = 1,...m. Let В = (bij)™=1 be an m x m matrix with entries b¿¿ = 1, i = 1, ., m and, for i ф j, i, j = 1, ., m,

Ьц, if dj < 1, " ' n+2di- -L>1.

(il^Wif dt

\3-2diJ lJ' 1

If B is a nonsingular M-matrix, then system (2.12) is asymptotically stable.

We considered more general systems then previous ones and obtained new results which are independent on known ones.

The following results were obtained in the paper [6]. Consider for any t0 > 0 the system of delay differential equations

Xi(t) = —ai(t)xi(hi(t)) + Y]Jl1Fij (t,Xj (gijtt))}, t>t0, i = \.....m, (2.13)

with the initial conditions

Xi(t) = (pi(t), t0 —a <t<t0, Xi(tQ) = x° , (2.14)

where a > 0 is denoted bellow in (a4) under the following assumptions:

(a1) ai are Lebesgue measurable essentially bounded on [0, w) functions, 0 < ai < ai(t) < Ai almost everywhere (a.e.);

(a2) Fij(t, ) are continuous functions, Fy(-, u) are measurable locally essentially bounded functions,

\Fij(t, u)| < LiJ\u\, a.e. t > 0;

(a3) hi, gij are Borel measurable functions, 0 < t - hi(t) < Ti, 0 < t - giJ(t) < aiJ; (a4) pi are continuous functions on [t0 - a, t0], where a = max{Tk, aiJ- | k, i, j = 1, ., m}.

Assume that conditions (a1)-(a4) hold for problem (2.13), (2.14) and its modifications, and the problem has a unique solution.

We will use some traditional notations. A matrix B = (bij)™=1 is nonnegative if bj > 0 and positive if bj > 0, i,j = 1, ., m; ||a|| is an arbitrary fixed norm of a column vector a = (a1, ., am)T in Mm; ||B|| is the corresponding matrix norm of a matrix B, \a\ = (\a1\, ..., \am\)T and |B| = (|&(_,|)™=1.

Problem (2.13), (2.14) has a unique global solution on [t0; w), if, for example, we assume along with (a1)-(a4) that functions Fj(t; u) are locally Lipschitz in u. The following classical definition of an M-matrix will be used.

Definition 2.8. A matrix B = (bij)]nj=1 is called a (non-singular) M-matrix if bj < 0; i f j and one of the following equivalent conditions holds:

- there exists a positive inverse matrix B-1 > 0;

- the principal minors of matrix B are positive.

Lemma 2.9. B is an M-matrix if bj < 0; i f j and at least one of the following conditions holds:

1) bu > Ejvi Ibijl; i = 1, ..., m;

2) bjj >Tli*jIbijI;j = 1 m;

3) there exist positive numbers i = 1, ., m such that <f iba > Yjj^i ^jlbijl, i = 1, ., m;

4) there exist positive numbers i = 1, ., m such that ^jbjj > Ei*/ ^ilbijl, j = 1, ., m. Definition 2.10. System (2.13) is globally exponentially stable if there exist constants M > 0 and

X > 0 such that for any solution X(t) ofproblem (2.13), (2.14) the inequality

\\X(t)\\ < Me-X(t-to)(\x(t0)\+supt<to\^(t)\) holds, where M and X do not depend on t0. We define matrix C as follows

C = (c^J=i, cu = i—M+'f*^, c = -^Ulllhi, (2.15)

Theorem 2.11. Suppose C defined by (2.15) is an M-matrix. Then system (2.13) is globally exponentially stable.

As an application of this result we consider non-autonomous BAM (bidirectional associative memory) neural network model

*№ = nit) (-aiXi (fcf1^)) + Я}=1 ai]f](y] + h),

m = n(t) (-biVi (h(2)(t)) + l1l=1bijgj(xj (/^1)(t))) +Л),

(2.16)

i = 1, ..., n, t > 0, with the initial conditions

Xi(t) = (Pi(t), yi(t) = q>i+n(t), t < 0, i = 1, ...,n. (2.17)

We will say that a norm in Mn is monotone if 0 < a < b (a componentwise inequality, i.e. 0 < ai < bi for all i) implies ||a|| < ||b||.

Consider the following algebraic system

uî = YJjhFijiuj), i = 1, ...,m, (2.18)

where |Fy(u) - Fy(v)| < L^u - v|.

Lemma 2.12. Let r(L) be a spectral radius of the matrix L = (Lij)™=1. Ifr(L) < 1 then system (2.18) has a unique solution.

Theorem 2.13. Suppose at least one of the following conditions holds: 1. max ft(A)l < 1, where maximum is taken on all eigenvalues of matrix A.

2. maxt ^ < 1, maxt

f

'j=1 bt

< 1.

3. max, 2f=i"

4. If=1I7=1

< 1, maxjYi=i-

bi

< 1.

+

ЫЩ bi

< 1.

5. max |A(B)| < 1, where maximum is taken on all eigenvalues of matrix B.

\aij\Li

6. maxi Y]1=I——l < 1, maxi

Ji]\b]

< 1.

uj

-v

\b„\Ll

a

1 та \aljWj . .. \ul]\uj . .. 7. maxj < 1, maxj £f=1-L < 1.

Z,i=iZ,j=i

J

ац\Lfj

+

ЫЩ

< 1.

(2.19)

Then system

aixi = Z"=i aiJfj{yJ)+Ii,

biVi = Y]]=1bijgj{xj) +Ji has a unique solution and thus system (2.16) has a unique equilibrium.

Below, assume that system (2.16) has a unique equilibrium (x ; y ). To obtain a global stability condition for this equilibrium, consider the matrix CBAM = (ci7-)fj=i, where

,2(1)

1 aiRi 4 ■ 1 1--= 1, ...,n,

at

b- P? r(l)

1-l-nPRl-ni-n,i = n + 1.....2n,

Pl—n

(2.20)

--—---, i = 1, ...,n,j = n + 1, ...,2n,

atai

Cij =

\bi_nj\Pi_nL3](bi_nPi_nTl(^,n+l)

,i = n+1, ...,2n,j = 1, ...,n,

(2.21)

Pi—nbi—n

v 0, otherwise.

Theorem 2.14. Suppose matrix CBAM is an M-matrix. Then the equilibrium (x*; y*) of system (2.16) is globally exponentially stable.

Corollary 1. Suppose at least one of the following conditions holds:

' J"1 a^j ai '

™ M^i^i) 1 n № <1 ft .....n'

ir=1' ' ^ "-^ < 1 - 7 = 1.....n.

1 1 Pibi aj

3. There exist positive numbers fik; k = 1, ..., 2n such that

^j+n\aij\RiLfj(qiRiT(^) + l) ^ grf^

^j = 1 aiai <Mi(1 a, ),

2,j = i pibi <Vi+n(1 pt ) ,

(i = 1, ...,n).

4. There exist positive numbers fik; k = 1, ..., 2n such that

^l+n\glj\RlLfj(alRlT^) + l) ^ bjPj*f\

Li = 1 am <№ P] ^

2^1)

ajRjTj

Li=i Pibi < Vj+n \ 1

0' = 1,...,n).

Then the equilibrium (x*; y*) of system (2.16) is globally exponentially stable.

3. SCALAR FORM SYSTEM WITHOUT APPLICATIONS OF SPECIAL MATRIX

The results of this part were obtained in the paper [7].

The aim of this part is to obtain easily checked explicit exponential stability conditions for the following non-autonomous linear delay differential system

*i(t) = -Z^iZfcViKO*,- (^(O), i = 1.....rn, (3.1)

where t > 0, m and rij, i, j = 1, ..., m are natural numbers, coefficients a^-: [0, ^ M and delays [0, ^ M are measurable functions. Define auxiliary functions

at(t) == 1^4(0, i = 1.....m, te[0,M).

Theorem 3.1. Assume that, for t > t0,

«¿(i) > a0 > 0, i = 1, ...,m (3.2)

and

i

maxi=1.....m ess supt>to

x

X

Then, system (3.3) is uniformly exponentially stable.

< 1. (3.3)

In our new paper which is now on preparation we improve the result of the previous paper by re-ing th tigations.

placing the constant 1 by the constant 1 + j which is one of the best known constants in stability inves-

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6. Berezansky L., Idels L., Troib L. Global Dynamics of Nicholson-Type Delay Systems with Applications. Nonlinear Anal. Real World Appl, 2011, vol. 12, no. 1, pp. 436-445.

7. Berezansky L., Diblik J., Svoboda Z., Smarda Z. Simple Uniform Exponential Stability Conditions for a System of Linear Delay Differential Equations. Appl. Math. Comput., 2015, vol. 250, pp. 605-614. DOI: 10.1016/j.amc.2014.10.117

Received 25 February 2017

УДК 51-77 DOI: 10.14529/^сг170214

ГЛОБАЛЬНАЯ ЭКСПОНЕНЦИАЛЬНАЯ УСТОЙЧИВОСТЬ ДЛЯ ДИФФЕРЕНЦИАЛЬНЫХ СИСТЕМ С НЕЛИНЕЙНЫМИ ЗАДЕРЖКАМИ

Л. Березанский

Университет имени Бен-Гуриона, Беэр-Шева, Израиль

Даётся обзор последних результатов по глобальной стабильности для нелинейного уравнения функционального дифференциала. Такие уравнения включают дифференциальные задержки, интегро-дифференциальные уравнения и уравнения с распределенным запаздыванием и применяются в качестве математических моделей в области динамики народонаселения и других наук. Также рассмотрены методы, используемые для изучения глобальной стабильности: построение функционалов Ляпунова, применение специальных матриц, таких как М-матрица или специальных матричных функций, таких как матричная мера, метод матричных неравенств, которые очень популярны в работах по теории контроля, метод неподвижной точки и использование понятия нелинейного оператора Вольтерра.

Ключевые слова: глобальная стабильность, функционал Ляпунова, матричная мера, метод матричных неравенств, нелинейный оператор Вольтерра.

Березанский Леонид, отделение математики, Университет имени Бен-Гуриона, Беэр-Шева, Израиль; [email protected].

Поступила в редакцию 25 февраля 2017 г.

ОБРАЗЕЦ ЦИТИРОВАНИЯ

Berezansky, L. Global Exponential Stability for Nonlinear Delay Differential Systems / L. Berezansky // Вестник ЮУрГУ. Серия «Компьютерные технологии, управление, радиоэлектроника». - 2017. - Т. 17, № 2. -С. 149-155. DOI: 10.14529/ctcr170214

FOR CITATION

Berezansky L. Global Exponential Stability for Nonlinear Delay Differential Systems. Bulletin of the South Ural State University. Ser. Computer Technologies, Automatic Control, Radio Electronics, 2017, vol. 17, no. 2, pp. 149-155. DOI: 10.14529/ctcr170214