CC BY
8
DOI: 10.14529/ctcr180215
SIMPLE STABILITY TESTS FOR SECOND ORDER DELAY DIFFERENTIAL EQUATIONS
Leonid Berezansky, [email protected] Ben-Gurion University of the Negev, Beer-Sheva, Israel
For linear and nonlinear delay differential equations of the second order with damping terms exponential stability and global asymptotic stability conditions are obtained. The results are based on the new sufficient stability conditions for systems of linear equations. The results are illustrated with numerical examples, and a list of relevant problems for future research is presented.
We proposed a substitution which exploits the parameters of the original model. By using that approach, a broad class of the second order non-autonomous linear equations with delays was examined and explicit easily-verifiable sufficient stability conditions were obtained. There is a natural extension of this approach to stability analysis of high-order models. For the nonlinear second order non-autonomous equations with delays we applied the linearization technique and the results obtained for linear models. Our stability tests are applicable to some milling models and to a non-autonomous Kaldor-Kalecki business cycle model. Several numerical examples illustrate the application of the stability tests. We suggest that a similar technique can be developed for higher order linear delay equations, with or without non-delay terms.
AMS Subject Classification: 34K20, 34K25.
Keywords: second order delay differential equations, exponential stability, reducation to systems.
1. Introduction
In the present paper, a specially designed substitution transforms linear second order equations into a system, to which we apply some known exponential stability results.
This and the linearization techniques are used to devise new global stability tests for nonlinear non-autonomous models. These results are explicit, easily verifiable and can be applied to a general class of second order non-autonomous equations. Some of the theorems complement results [1, 2], as well as the tests obtained in recent papers [3-5]. Consider the following system
*i(t) = -I^iS^a)*; (h?j(t)),i = 1.....m, (1)
where t > 0, m is a natural number, rtj, i,j = 1, ...,m are natural numbers, coefficients a^-: [0,m) — M and delays h^-: [0, m) — M are measurable functions. Let Ai,i = 1,...,m be functions defined as
At(t) - Cx^^IL^ I4(s)ids + iLwZZi i^
where
OiV-ZlhaUt). (2)
Lemma 1. Suppose that at(t) > a0 > 0,maxi=1. mlimsupt-mAi(t) < 1. Then, the system (1) is uniformly exponentially stable.
Lemma 2. Let afi(t) > 0,at(t) > a0 > 0, i = 1,..., m, t > t0,
maxi=i.....m limt-m sup ^^ |a£-(t)I < 1 (3)
and
maX(=1.....m limt-m sup At(t) < 1 + -. (4)
Then, the system (1) is uniformly exponentially stable.
In Lemma 1 does not assumed that afi(t) > 0 but in Lemma 2 the constant in right-hand side of the inequality (4) is better. So these lemmas are independent and we use both of them.
Consider first the following equation:
x(t) + a(t)x(t) + b(t)x(t) + XZLi ck(t)x(hk(t)) = 0, (5)
where
0 < a < a(t) <A,0<b< b(t) < B,\ck(t)\ <Ck,t — hk(t) < rk.
Theorem 1. Suppose lim inft^x ck(t) + a(t) — f) > 0 and the following condition holds
2b(s)
lim sup—---siZk-i |cfc(t)| С ^fla(s) --■
+
+2Z£=i4(t) + a(s)-^)ds +
21^=1 cfc+(s) + a(s) — f)ds + |a(t) — f — + ZZU \cfc(t)\] < 1. (6)
Then equation (5) is exponentially stable. Proof. By substitution to equation (5)
x(t) = —ix + (i—6)y,m = {i—e)№+Tx—i(z—e)y
we have
i\( (t) tb(t) 1 a
y(t) = y~ a ^x(t)+1^Y^=1ck(t)x(hk(t))—Y^=1ck(t)y{hk(t)) — (a(t) — f)y(t). (7)
2 2
By Lemma 1 the following condition implies exponential stability of system (7).
+
-e VI v ' 2 a
2 L 2
a(t)—|— 2£(£)|+Zm=i\Cfc(t)\]<i. (8)
But for small e > 0 inequality (6) implies (8). Hence system (7) and then equation (5) are exponentially stable.
Denote c+ = max{c, 0}, c~ = max{—c, 0}.
Corollary 1. Suppose limsupt_>m (£/T=ick(t) + a(t) — > 0 and there exist t0 > 0,8 > 0 such that at least one of the following conditions hols:
1. a2 > 4B,
Ifc=i\ck(t)\ Jhtfc(t) (2a(s) — a — ^ + 21^=1 ^(s)) ds + Zf=1 cj^(t) <2-^ — S,t>t0. (9)
2. A2 < 4b,
Zfc=i\ck(t)\ /htfc(t) + 21^=14(s)) ds + 1^=1 (t) + ^ < 2a(t) — a — S,t>t0. (10) Then equation (5) is exponentially stable.
Proof. Suppose conditions 1) hold. Inequality a2 > 4B implies that a(s) — ^ — > 0. Hence (6) holds if for some t0 > 0,5 > 0
I£U\Cfc(t)\ Jhtfc(t) (a(s) — I — ^ + 21^=1 + a(s) — f) ds + a(t) — f — ^ + + IT=i\ck(t)\<IT=iCk(t) + a(t)—^ — 8,t>t0. (11)
Inequality (11) after simple transformations coincides with (9). The second case is proved similarly.
Now with an additional assumption ck (t) >0 we can improve Theorem 1. Theorem 2. Suppose ck(t) > 0, k = 1,..., m and the following conditions hold:
a(t)— 1— ^ + Y£=ick(t)<1,
2
limt-œ SUP Z£lCfc(t)+«(t)-S ^ Cfc(t) "G) da(S) -~2-2J?1\ +
+ 2 ck(s) + a(s) -fjds+ \a(t) - f - + ck(t)] < 1+ ±
Then equation (5) is exponentially stable.
The proof is based on Lemma 2 and is similar to the proof of Theorem 1. Corollary 2. Suppose
a(t) = a> 0 ,b(t) = b> 0,0 < ck(t) <Ck,t- hk(t) < rk.
If at least one of the following conditions hols:
1. a2 > 4b,--^ + ;£r=iCfe < 1 and
~ '2 а л
Ckrk(a - ^ +2ir=iCfc) < ^ + i(f + 5T=i Cfc).
2. а2 < + < 1
-<т г _
-ifc=lLfc .
5T=1 CfeTfc g + 21^=1 Cfc) < (1+ ¿)a + ^
Then equation (5) is exponentially stable. Consider here the following equation
m + a(t)x(t)) + EZU bk(t)x(hk(t)) = 0. (12)
Theorem 3. Suppose lim inft_œ (a(t) — | — bk(t)(t — hfc(t))j > 0 and the following conditions holds
limu_m sup-a—™—--X
f_œ F a(t)_2_Zm=i bk(t)(t-hk(t))
X
bfc(t)(t - hfc(t)) (2a(s) - a - ds +
+a(t) _ | + 2ZE5lhfc(t)(t-hfc(t)) + ir=i 6fc(t)(t _ fcfc(t))] < !. (13)
Then equation (12) is exponentially stable.
To prove the theorem we need in the following lemma.
Lemma 3. Suppose ak: [a, to) — M+ = [0, <x>), k = 1, ...,m are measurable essentially bounded functions, hfc: [a, to) — M, hk(t) < t,k = 1, ...,m are measurable functions. Then for any continuous function x: [a, to) — M there exists measurable function h: [a, to) — M, h(t) < t such that minfchfc(t) < h(t) < maxfchfc(t),22l=1afc(t)x(hfc(t)) = (ZZL1afc(t))x(h(t)). Now we are ready to prove the theorem.
Proof. Suppose x is fixed solution of equation (12). Transform equation (12) x(t) + a(t)x(t)) + Zf=1bk(t)x(t) — Y£=1bk(t) j*kit)x(s)ds = 0. (14)
By Lemma 3 there exist rk(t), k = 1, ..., m such that hk(t) < rk(t) < t and
Shk(t)±(s)ds = - hk(t))x{rk(t)). Hence x is a solution of the following equation
z(t) + a(t)z(t)) + I?L16fc(t)z(t) + Z?l=1cfc(t)z(rfc(t)) = 0, (15)
where ck(t) = — bk(t)(t — hfc(t)) < 0 and then c^(t) = 0. Inequality (13) implies that (6) holds, where hk one can replace by rk.
By Theorem 1 equation (15) is exponentially stable. Hence the solution x of of equation (12) tends to zero exponentially.
2. Nonlinear Equations
In this section we examine several nonlinear delay differential equations of the second order which have the following general form
m + Ik=ifk(t,x(Pk(t)),x(gk(t))) + Ilk=1sk(t,x(hk(t))) = 0, (16)
with the following initial function
x(t) = <K0, X(t) = ^(t), t < to, to ^ 0, (17)
where fk(t,u1,u2),k = 1, ...,m,sk(t,u), are Caratheodory functions which are measurable in t and continuous in all the other arguments, condition (a2) holds for delay functions pk, gk, hfc; ^ and ^ are Borel measurable bounded functions.
We will assume that the initial value problem has a unique global solution on [t0, to) for all nonlinear equations considered in this section. Theorem 4. Consider the equation
x(t) + f(t, x(t), x(tj) + s(t, x(tj) + IZLi sfc(t, x(t), X(hk(t))) = 0, (18)
where
f(t,v,0) = 0, s(t, 0) = 0,sfc(t,v,0) = 0,0 < a0 < ^^^ < A,
s(t,u)
sk(t,v,u)
0<bo<^<B, ^^ <Ck,u*0,t-hk(t)< T.
If at least one of the following conditions holds:
1) B<a-f,YZ=1Ck<b0 — a-f(A — a0),
2) bo>a-±(A — ^),YZ=1Ck<?f — B,
then zero is a global attractor for all solutions of problem (18), (17).
Proof. First, by the previous theorem there exists a global solution x of problem (18), (17). Suppose x is a fixed solution of problem (18), (17). Rewrite equation (18) in the form
x(t) + a(t)x(t) + b(t)x(t) + lf=1ck(t)x(hk(t)) = 0, where
mtMtrnt» ±(t) * 0 is(t,x(t)) * 0
a(t) = j m ,x(t)*0, b(t) = \ x(t) ,x(t)*0, ( aQ,x(t) = 0, ( b0,x(t) = 0,
(Sk(t,x(t),x(hk(t))) , 0
Ck(t) = j ,X(hk(t)) * 0
( 0,x(hk(t)) = 0.
Hence the function x is a solution of the linear equation
y(t) + a(t)y(t) + b(t)y(t) + ck(t)y(hk(t)) = 0, (19)
which is exponentially stable by Theorem 1. Thus for any solution y of equation (19) we have limt^OT y(t) = 0. Since x is a solution of (19), we have limt^OT x(t) = 0.
The previous proof is readily adapted to the proof of the following theorems. Theorem 5. Consider the equation
x(t) + f{t, x(t), x(tj) + s(t, x(t)) + 2fc=i Sk(t, X(t), x(hk(t))) = 0, (20)
where
f(t,v,0) = 0,s(t,0) = 0,sk(t,v,0) = 0,0<a0
0<bo<s-^<B,
sk(t,v,u)
< Ck,u^ 0,t - hk(t) < т.
Suppose at least one of the following conditions holds:
2b0-a0(A-a0)
l) B<^,z%=iCk<
2a„
2) b0>j(A-^),Zf=1Ck<^.
Then zero is a global attractor for all solutions ofproblem (20), (17). Theorem 6. Consider the equation
x(t) + f(t, x(t), x(t)) + sk(t, x(hk(t)), x(t)) = 0, (21)
where
f(t, v, 0) = 0, sk(t, 0,u) = 0,0 < a0 < ^^ < A,
0<bk< Sk(t*,u) <Bk,u^0,t- hk(t) < t. Suppose at least one of the following conditions holds:
1) ZZU Bk<f,?f(A-a0)< bk - a0 Bkrk,
2) ir=i bk > I (a - f), Z?=1 Bk (1 + a0Tk) < f.
Then zero is a global attractor for all solutions ofproblem (21), (17). Theorem 7. Consider the equation
x(t) + f(t,x(t),x(t)) + s(t,x(t)) = Ziï=1ck(t)(x(t) - x(hk(t))), (22)
where
f(t, v, 0) = 0, s(t, 0) = 0,0 < a0 < < a,
0<bQ<s-^<B, \ck(t)\ <Ck,u*0,t- hk(t) < Tfc.
Suppose at least one of the following conditions holds:
1) R < ут Г т < 2^о~ао(^~ао) ) — 4 ,2-,k=1 к к 2ао ,
2)
Then zero is a global attractor for all solutions of problem (22), (17). Example 1. To illustrate Part 2) of Theorem 7, consider the equation
x(t) + (1.9 + 0.1 sinx(t))x(t) + (1.1 + 0.1 cosx(t))x(t - 0.19 sin2 t) = 0. (23)
We have m = 1, a0 = 1.8, A = 2, b0 = 1, В = 1.2, т = 0.19; therefore, all conditions of the theorem hold, hence zero is a global attractor for all solutions of equation (23). Consider a generalized Kaldor-Kalecki model
x(t) + [a(t) - P(t)p\x(t))]x(t) + s(t, x(t)) = p(x(t)) - p(x(h(t))), (24)
where а, в are locally essentially bounded functions, s is a Caratheodory function, p is a locally absolutely continuous nondecreasing function,
0 < a0 < «(0 <ai,0<p0< p(t) < ft,
\p'(t)\ < C,a0-piC > 0,0 < b0 <s-^< B,t-h(t) < т.
Denote a0 = a0 - ргС.
Theorem 8. Suppose at least one of the following conditions holds:
1) В < С < 2Ьо~ао(а1~ао)
) ~ 4 , т 2а0 ,
2)
Then zero is a global attractor for all solutions ofproblem (24), (17).
Proof. Suppose x is a fixed solution of problem (24),(17). There exists a function f(t) such that
p(x(t)) - p(h(x(t)) = p'(£(0) (x(t) - x(h(t))). Denote a(t) - P(t)p'(x(t)) = a(t),p'(f(t)) = c(t).
Hence x is a solution of the following equation
y(t) + a(t)y(t) + s(t, y(t)) = c(t) (y(t) - y(h(t))). (25)
Since p'(x) > 0 then 0 < a0 - ргС < a(t) < ax. Equation (25) has a form (22) with f(t, x(t), x(t)) = = a(t)x(t),m = 1. All conditions of Theorem 7 hold, hence for any solution of (25) we have limt^œ y(t) = 0. Then also limt^œ x(t) = 0.
3. Remarks and Open Problems
The technique of reduction of a high-order linear differential equation to a system by the substitution x(fc) = yfc+1 is quite common. However, this substitution does not depend on the parameters of the original equation, and therefore does not offer new insight from a qualitative analysis point of view. Instead, we proposed a substitution which exploits the parameters of the original model. By using that approach, a broad class of the second order non-autonomous linear equations with delays was examined and explicit easily-verifiable sufficient stability conditions were obtained. There is a natural extension of this approach to stability analysis of high-order models. For the nonlinear second order non-autonomous equations with delays we applied the linearization technique and the results obtained for linear models. Our stability tests are applicable to some milling models and to a non-autonomous Kaldor-Kalecki business cycle model. Several numerical examples illustrate the application of the stability tests. We suggest that a similar technique can be developed for higher order linear delay equations, with or without non-delay terms.
Solution of the following problems will complement the results of the present paper:
1. In all stability conditions obtained, we used lower and upper bounds of the coefficients and the delays. It is interesting to obtain stability conditions in an integral form.
2. Apply the technique used in the paper to examine delay differential equations of higher order. Also, the substitution used in this chapter was based on the existence of a non-delay term, it would be interesting to adjust the method for equations which have several delayed terms only.
3. Establish necessary stability conditions for the equations considered in this chapter by reduction to a system of delay differential equations.
4. For the sunflower equation and its modifications establish set of conditions to guarantee boundedness of all solutions.
5. Apply the technique used in the paper to examine delay differential equations of higher order.
References
1. Berezansky L., Diblik J., and Smarda Z. Positive Solutions of Second-order Delay Differential Equations with a Damping Term. Computers and Mathematics with Applications, 2010, 60, pp. 13321342. DOI: 10.1016/j.camwa.2010.06.014
2. Berezansky L., Braverman E., Domoshnitsky A. Stability of the Second Order Delay Differential Equations with a Damping Term. Differential Equations and Dynamical Systems, 2008, vol. 16, issue 3, pp. 185-205. DOI: 10.1007/s12591-008-0012-4
3. Cahlon B., Schmidt D. Stability Criteria for Certain Second-order Delay Differential Equations with Mixed Coefficients. Journal of Computational and Applied Mathematics, 2004, vol. 170, issue 1, pp. 79-102. DOI: 10.1016/j.cam.2003.12.043
4. Cahlon B. Unconditional Stability for Certain Delay Differential Equations. Dynamic Systems Applications, 1996, vol. 5, pp. 583-594.
5. Gyori I., Hartung F. Asymptotically Exponential Solutions in Nonlinear Integral and Differential Equations. Journal of Differential Equations, 2010, vol. 249, issue 6, pp. 1322-1352. DOI: 10.1016/j .jde.2010.06.017
Received 10 February 2018
УДК 517.929 DOI: 10.14529/^сг180215
ПРОСТЫЕ ТЕСТЫ УСТОЙЧИВОСТИ ДЛЯ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ ЗАДЕРЖКИ ВТОРОГО ПОРЯДКА
Л. Березанский
Университет им. Бен-Гуриона в Негеве, г. Беэр-Шева, Израиль
Для линейных и нелинейных дифференциальных уравнений запаздывания второго порядка с затухающими членами получены экспоненциальная устойчивость и условия глобальной асимптотической устойчивости. Результаты основаны на новых достаточных условиях устойчивости для систем линейных уравнений. Результаты иллюстрируются численными примерами, а также приводится перечень соответствующих проблем для будущего исследования.
Предложена подстановка, в которой используются параметры исходной модели. Используя этот подход, широкий класс неавтономных линейных уравнений второго порядка с задержками был исследован и получены явные легко проверяемые достаточные условия устойчивости. Приводится естественное продолжение этого подхода к анализу устойчивости моделей высокого порядка. Для нелинейных неавтономных уравнений второго порядка с задержками применен метод линеаризации и получены результаты для линейных моделей. Приведенные тесты стабильности применимы к некоторым моделям фрезерования и к неавтономной модели бизнес-цикла Калдора - Калецкого. Мы предлагаем, чтобы аналогичная методика была разработана для линейных уравнений с условием линейной задержки или без задержки.
Ключевые слова: дифференциальные уравнения запаздывания второго порядка, экспоненциальная устойчивость, редукция систем.
Березанский Леонид, д-р физ.-мат.наук, Департамент математики, Университет им. Бен-Гуриона в Негеве, г. Беэр-Шева, Израиль; [email protected].
Поступила в редакцию 10 февраля 2018 г.
ОБРАЗЕЦ ЦИТИРОВАНИЯ
Berezansky, L. Simple Stability Tests for Second Order Delay Differential Equations / L. Berezansky // Вестник ЮУрГУ. Серия «Компьютерные технологии, управление, радиоэлектроника». - 2018. - Т. 18, № 2. -С. 149-155. DOI: 10.14529/ctcr180215
FOR CITATION
Berezansky L. Simple Stability Tests for Second Order Delay Differential Equations. Bulletin of the South Ural State University. Ser. Computer Technologies, Automatic Control, Radio Electronics, 2018, vol. 18, no. 2, pp. 149-155. DOI: 10.14529/ctcr180215
