Exponential stability of solutions to one class of nonlinear systems of neutral type with periodic coefficients in linear terms Текст научной статьи по специальности «Математика»

, UiwiMii'iicKiti- CHcreMH, 2015, TOM 5(33), №1-2, 3-11

MSC 2010: 34K20, 34K40

Exponential stability of solutions to one class of nonlinear systems of neutral type with periodic coefficients in linear terms1

G. V. Demidenko, 1.1. Matveeva

Sobolev Institute of Mathematics, Novosibirsk 630090, Russia. Novosibirsk State University, Novosibirsk 630090, Russia. E-mail: [email protected], [email protected]

Abstracts. We study the problem on exponential stability of the zero solution to a class of nonlinear systems of neutral type with periodic coefficients in linear terms. We establish conditions of exponential stability and obtain estimates for solutions.

Keywords: time-delay systems, neutral type, periodic coefficients, exponential stability, Lyapunov-Krasovskii functional.

1. Introduction

Problems on stability of solutions to autonomous delay differential equations are studied well (for instance, see [1, 2, 11, 12, 15, 16, 17, 18, 19, 21] and the bibliography therein). However, there are much less results on the stability theory for nonautonomous delay equations. The main investigations for nonautonomous equations focus on linear delay differential equations with periodic coefficients

d

jtv(t) = A(t)y(t) + B(t)y(t - t), A(t + T) = A(t), B(t + T) = B(t), t> 0. (1.1)

The fundamentals of the stability theory of solutions to these equations were laid in the papers [13, 14, 22, 23] and others. One of the main approaches when studying stability of solutions to (1.1) is the development of the Floquet theory and the use of the monodromy operator. This approach is also applied when studying stability of solutions to linear equations of neutral type with periodic coefficients

d

- (y(t) + Dy(t - t)) = A(t)y(t) + B(t)y(t - t), t> 0. (1.2)

To study asymptotic stability of the zero solution to the time-delay systems (1.1) and (1.2), the authors in [5, 6, 7, 9] proposed to use Lyapunov-Krasovskii functionals of the form

t

V(t,y) = (H(t)(y(t) + Dy(t - t)), (y(t)+ Dy(t - t))) + j (K(t - s)y(s),y(s)) ds. (1.3)

t-r

In particular, the following result was established in [9].

1The work is supported in part by the Russian Foundation for Basic Research (project no. 13-01-00329).

© G. V. DEMIDENKO, 1.1. MATVEEVA

with

Theorem 1. Suppose that there exist matrices H(t) = H;(t) € C 1[0,T\, K(s) = K;(s) € C] such that

d

H (0) = H (T) > 0, K (s) > 0, —K (s) < 0, s € [0,r],

and the matrix

C(t)=( âg )

d

Cn(t) = - JtH(t) - H(t)A(t) - A;(t)H(t) - K(0), d

C\;(t) = --H(t)D - H(t)B(t) - A;(t)H(t)D, d

C;;(t) = -D;-H (t)D - D;H (t)B(t) - B;(t)H (t)D + K (t ) dt

is positive definite for t € [0,T], Then the zero solution to (1.2) is exponentially stable.

Using functionals of the form (1.3), the authors in [5, 6, 9] obtained first analogues of M. G.Krein's estimates [3] characterizing exponential decay of solutions to (1.1) and (1.2) as t ^ rn. In the case of ordinary differential equations with per iodic coefficients (D = B(t) = 0), first estimates of such kind were obtained in [4].

In the present paper we use the Lyapunov-Krasovskii functional (1.3) in order to study stability of the zero solution to the system of nonlinear delay differential equations

d

- (y(t) + Dy(t - t)) = A(t)y(t) + B(t)y(t - t)+ F(t, y(t),y(t - t)), t> 0, (1.5)

where D is a const ant (n x n)-matrix, A(t), B(t) are (n x n)-matrices with continuous T-periodic entries, t > 0 is the time delay, and F(t, u, v) is a continuous vector function satisfying the inequality

\\F (t,u,v)\\ < qi\\u\\ + q;||v||, qi, q; > 0 are const ant. (1.6)

Our aim is to establish conditions of exponential stability and to obtain estimates characterizing exponential decay of solutions to (1.5) at infinity. This paper continues our investigations of nonlinear time-delay systems [8, 10, 20]. Some examples illustrating effectiveness of our approach are given in [10].

2. Main results

Supposing that the conditions of Theorem 1 are satisfied, we formulate our main results

H(t) K(s)

sw=( S;;i!0 ■ (2-i)

d

Sii(t) = --H(t) - H(t)A(t) - A;(t)H(t) - K(0),

Si2(t) = H(t)A(t)D + K(0)D(t) — H(t)B(t), S22(t) = K(T) — D*K(0)D,

q(t) = (qi + y]qi + (qi\\D\\ + q2)^j \\H(t)\\. (2.2)

S(t)

is positive definite if and only if C(t) in (1.4) is positive definite (see Section 3 for details). Denote by I the unit matrix.

qi qi

the matrix (S(t) — q(t)I) is positive definite for t € [0, T]. Then the zero solution to (1.5) is exponentially stable.

Consider the initial value problem for (1.5)

d

-(y(t) + Dy(t — t)) = A(t)y(t) + B(t)y(t — t) + F(t, y(t), y(t — t)), t> 0,

y(t) = <p(t), t € [—t, 0], (2-3)

y(+0) = <p(0),

where ^(t) € C 1[—t, 0] is a given vector function. Below we establish estimates of solutions to the initial value problem (2.3) characterizing the rate of exponential decay as t ^ to.

H(t)

conditions of Theorem 1 then

d

jH(t) + H(t)A(t) + A*(t)H(t) < —K(0).

H(t) > 0 [0, T] T

the matrix H(t) to the whole half-axis {t > 0}, keeping the same notation. Using this matrix H(t) and the matrix K(s), satisfying the conditions of Theorem 1, we consider the functional (1.3) and put

0

Vo(p) = (H(0)(<p(0) + Dy(—T)), (<p(0) + Dy(—T))) + J(K(—s)<p(s),<p(s))ds. (2.4)

— T

We introduce d

P (t) = — jH (t) — H (t)A(t) — A*(t)H (t) — K (0) — q(t)I

— (H (t)A(t)D + K (0)D — H (t)B(t))[K (t ) — D*K (0)D — q(t)I]—1

x (H(t)A(t)D + K(0)D — H(t)B(t))*. (2.5)

Note that P(t) is positive definite if the matrix (S(t) — q(t)I) is positive definite (see Section 3 for details). Denote by pmin(t) > 0 the minimal eigenvalue of P(t) and by hmin(t) > 0 the minimal eigenvalue of H(t). Let k > 0 be the maximal number such that

d

—K(s) + kK(s) < 0, s € [0, t]. (2.6)

We put

j(t)=min {pmin(t), k\\H(t)\\} ,

Ф = max te[-r,Q]

a = max

/

_______ (Ф) ,

te[Q,T] v

ß(t) =

Y (t)

(2.7)

(2.8)

ß + = max ß(t), ß- = min ß(t).

2\\H (t)\\' ' t&[0,T]..... t&[0,T]'

Analogy as in [9], we distinguish three cases when obtaining estimates. It is not hard to show that the spectrum of the matrix D belongs to the unit disk {X € C : |A| < 1} if the conditions of Theorem 1 are fulfilled; i.e., if the matrix C(t) (or S(t)) is positive definite. Consequently, \\Dj\\ ^ 0 as j ^ ro. Let l > 0 be the minimal integer such that \\Dl\\ < 1.

Theorem 3. Let the conditions of Theorem, 2 be satisfied.

I. If \\Dl\\ < e-l3+T then a solution to the initial value problem (2.3) satisfies the estimate

<

i-1

a( 1 -\\Dl\\eiß+r)\\Dj\\ejß+r

j=Q

+ maxj\\D\\eß+r\\Dl\\eiß+r} Ф

4 ß(0 dt

e о , t> 0.

II. If e l3+T < \\Dl\\ < e 13 T then a solution to the initial value problem (2.3) satisfies the estimate

<

a( 1 + I) Ё D\\ejß+r

. V ' j=Q

+ max{l, \\D\\eß+r,\\Dl-1\\e(l-1)ß+r } Ф

4 ß(0 dt

e о , t> 0.

III. Ife-13 T < \\Dl\\ <1 estimate

<

-1 i-1

a\\Dl\\eiß-r (\\Dl\\eiß-r - l) ^ \\Dj\\e

jß-r

j=Q

+ \\Dl\\1-1 max{l, \\D\\,..., \\Di-1 \\} Ф exp (t Ь D \\) ,

I'

t > 0.

Here a, fi(t), f+ f-, and $ are defined in (2.8), f3(t) = min <j f3(t), - — In\\ }> > 0

We prove Theorem 3 in Section 3. Obviously, Theorem 2 immediately follows from this theorem.

3. Proof of the main results

To prove Theorem 3 we need auxiliary results obtained below.

Lemma 1. Let the conditions of Theorem 2 be satisfied. Then, for a solution to the initial value problem (2.3), the following inequality holds

(- f A «1, «>0, (3.1)

where V0(^>) and j(t) are defined by (2.4) and (2.7), respectively, hmin(t) > 0 is the minimal

H(t)

y(t)

H(t) K(s)

C(t)

functional can be written as follows

dv ™-{ c K J 0 4?(—V

+ (H(t)F(t, y(t),y(t — t)), (y(t) + Dy(t — t))) + (H(t)(y(t) + Dy(t — t)),F(t, y(t),y(t — t)))

t

+ / (ddtK(t — s)y(s),y(s)} ds. (3.2)

t— T

Consider the first summand in the right-hand side of (3.2). Since ( y(t) \ = ( I —D \(y(t) + Dy(t — t)\ \y(t — t)j \0 I j\ y(t — t) j

then

<C K J mr 0 —r>))' MM T >) r T0

where the matrix S(t) is defined in (2.1). Obviously, if the matrix C(t) is positive definite S(t)

Now we consider the second and the third summands in the right-hand side of (3.2). In view of (1.6) we have

(H(t)F(t, y(t),y(t — r)), (y(t) + Dy(t — r))) + (H(t)(y(t) + Dy(t — r)),F(t, y(t),y(t — r))) < 2\\H(t)\\(qi\\y(t)\\ + qi\\y(t — r)\\)\\y(t)+ Dy(t — r)\\ < 2qi\\H(t)\\\\y(t)+ Dy(t — r)\\2 + 2(qi\\D\\ + qi)\\H\\\\y(t — r)\\\\y(t) + Dy(t — r)\\

< q(t)(\\y(t)+ Dy(t — r )\\2 + \\y(t — r )\\2),

q(t) Hence,

К J1)) i

y(t)

KV(t - т)Г \y(t - т),

+ (H(t)F(t, y(t),y(t - t)), (y(t) + Dy(t - ())) + (H(t)(y(t) + Dy(t - т)),F(t, y(t),y(t - т)))

-Myit)Жт>))- (3'3>

By the conditions of Theorem 2, the matrix (S(t) — q(t)I) is positive definite. Using the representation

5(t) - q(t)I = ^0 I

I Sl2(t)(S22(t) - q(t)I)-1

v ( Sii(t) - q(t)I - S12(t)(S22(t) - q(t)I)-1S*12(t) 0

XV 0 S22(t) - q(t)I

I0 (S22(t) - q(t)I)-lSl2(t) I

)

we have

(m - «ш >( y(t> +DT- т r) t ')>

y(t - T)

^ [Sn(t) - q(t)I - S\2(t)(S22(t) - q(t)I)-1S*u(t)] (y(t) + Dy(t - t)), (y(t) + Dy(t - t))) .

Since the matrix (S(t) - q(t)I) is positive definite then the matrix

P(t) = Sn(t) - q(t)I - Si2(t)(S22(t) - q(t)I)-1S*u(t)

is positive definite. Taking into account (2.1), the matrix P(t) has the form (2.5). Consequently, from (3.3) we derive

(с К J >)-G

y(t)

-,y(t - T\y(t - T

+ (H(t)F(t, y(t), y(t - t)), (y(t) + Dy(t - t))) + (H(t)(y(t) + Dy(t - t)),F(t, y(t), y(t - t))) < - (P(t)(y(t) + Dy(t - t)), (y(t) + Dy(t - t)))

< -Pmin(t)\\y(t) + Dy(t - T)\\2, (3.4)

where pmin(t) > 0 is the minimal eigenvalue of P(t). Using the matrix H(t), we have

hmin(t) \\y(t) + Dy(t - t)\\2 < (H(t)(y(t) + Dy(t - t)), (y(t) + Dy(t - t)))

<\\H(t)\\\\y(t)+ Dy(t - t)\2,

where hmin(t) > 0 is the minimal eigenvalue of H(t). Using (2.6) and (3.4), from (3.2) we derive

dv(t, y) < -H)| (H(t)(y(t) + Dy(t - T)), (y(t) + Dy(t - T)))

t

- k j (K(t - s)y(s),y(s)) ds.

t-T

Taking into account the definition of the functional (1.3), we obtain

where j(t) is defined by (2.7). From this differential inequality we derive the estimate

t

V (t,y) < Vo(^)exp (-J

\\H (OH

dC

0

Then, using the definition of the functional (1.3), we have

t

The lemma is proved. □

Lemma 2. Let the conditions of Theorem, 2 be satisfied. Then a solution to the initial value problem (2.3) on every segment t € [kr, (k + 1)r), k = 0,1,..., satisfies the following estimate

J t — jT

k f P(i) di

\\y(t)\\< a£ \\Dj\\e 0 + \\Dk+1\\$, (3.5)

j=0

where a /3(t), and $ are defined in (2.8).

Proof. By Lemma 1, a solution to the initial value problem (2.3) satisfies (3.1). In view of the notations (2.8), the estimate has the form

- J P(i) di

\\y(t) + Dy(t — T)\\< ae 0 , t> 0. (3.6)

Obviously, for t € [0, r) we have the inequality

- J p(i) di - J m di

\\y(t)\\< ae 0 + \\Dy(t — r)\\ < ae 0 + \\D\\$,

k=0

Let t € [kr, (k+1)t), k = 1, 2 .... It is not hard to write out the sequence of the inequalities

- J m di

\\y(t)\\< ae 0 + \\Dy(t — r)\\

-J P(i) di 2 2 3

< ae 0 + \\Dy(t — r) + D2y(t — 2r) \\ + \\D2y(t — 2r) + D3y(t — 3r) \\ + ...

+ \\Dk y(t — kr) + Dk+1y(t — (k + 1)t )\\ + \Dk+1y(t — (k + 1)t )\\

- J m di 0

< ae 0 + \\D\\ \\y(t — r)+ Dy(t — 2r)\\ + \\D2\\ \\y(t — 2r) + Dy(t — 3r)\\ + ...

+ \\Dk\\ \\y(t — kr) + Dy(t — (k + 1)t)\\ + \\Dk+1\\ \\y(t — (k + 1)t)\\. By (3.6) we derive the estimate

t t-t t-2T

-/mdi - f rndi - / m)di

< ae 0 + a\\D\\e 0 + a\\D2\\e 0 + ...

t-kT

, - / m di +

+ a\\Dk\\e 0 + \\Dk+1\\$,

which implies (3.5).

The lemma is proved. □

Proof of Theorem, 3. For the linear time-delay system (1.2) with periodic coefficients, the analogues of Theorem 3 (see Theorems 2-4 in [9]) were proved in detail in [9] by the use of the auxiliary assertions (see Lemmas 2-4 in [9]). In the present paper, using Lemmas 1, 2 and repeating the reasoning carried out when proving Theorems 2-4 in [9], we derive the required estimates for solutions to the initial value problem (2.3). □

Using the proof of Theorem 2, we can reformulate the conditions of exponential stability of the zero solution to the nonlinear system (1.5) as follows.

Theorem 4. Suppose that there exist matrices

h(t) = h* (t) e C 1[0,t], h(o) = h(t) > o,

d

K(s)= K*(s) e C 1[0,r], K(s) > o, —K(s) < o, s e [o,t],

ds

such that the matrices (K(t)—D* K(0)D—q(t)I) and P(t) defined by (2.5) are positive definite for t e [0, T], Then the zero solution to (1.5) is exponentially stable.

P(t)

stability can be formulated in terms of the linear matrix inequality and the differential matrix inequality of Riccati type.

Remark 2. Theorem 3 makes it possible to estimate the rate of exponential decay of solutions to (2.3) at infinity; moreover, all the values characterizing the decay rate are obtained in explicit form.

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Получена 03.06.2015