Exponential Stability of the Solution of Single-component Repairable System with an

Scientific Journal of Control Engineering October 2015, Volume 5, Issue 5, PP.57-62

Exponential Stability of the Solution of Singlecomponent Repairable System with an Identical Cold-standby Component Dongxu Liu1†, Shuyao Sun2 1. Department of Mathematics, College of Science, Yanbian University, Yanji 133002, China 2. Department of Tourism Management, College of Chinese Medicine Materials, Jilin Agricultural University, Changchun 130118, China †

Email: 125025633@qq.com

Abstract The Single-component repairable system with an identical cold-standby component is studied in this paper. With the theory of resolvent positive operator and cofinal, we analyze the essential spectrum of the system operator. The result shows that the dynamic solution of the system is exponentially stable.

Keywords: Repairable System; Essential Spectrum; Cofinal; Exponential Stability

1 INTRODUCTION This repairable system consists of a running component and a cold-standby component. If the running component fails, the cold-standby component will take the place of it and the faulted component will be sent to be repaired as soon as possible. If the standby component fails when the running component is being repaired, the system can’t work and the standby component will be sent to be repaired, too.

In [1], the authors discussed the property of the system operator and proved the upper spectral bound of the operator A + B was 0; In [2], the authors discussed the main operator of the system and its upper spectral bound. In this paper, we will discuss the exponential stability of this system.

According to [1], the state diagram of the system is shown in Figure 1:

FIGURE 1

This system model can be expressed by a group of equations:  dp0 (t ) ∞ −λ p0 (t ) + ∫0 p1 ( x, t ) µ ( x)dx  dt =  ∂p ( x, t ) ∂p ( x, t ) + 1 = −(λ + µ ( x)) p1 ( x, t )  1 ∂x  ∂t  ∂p2 ( x, t ) ∂p2 ( x, t ) + = − µ ( x) p2 ( x, t ) + λ p1 ( x, t )  ∂x  ∂t ∞ λ p0 (t ) + ∫0 p2 ( x, t ) µ ( x)dx, p2 (0, t ) = 0  p1 (0, t ) =  p0 (0) 1, p= p= = 1 ( x, 0) 2 ( x, 0) 0  

The symbols in the equations are defined as follows: State 0: The running component and the cold standby-component are in the normal mode state.

- 57 http://www.sj-ce.org

(1.1)

State 1: The running component fails and the cold-standby component gets into working state.

State 2: The cold-standby component fails and is waiting for being repaired while the running component is still being repaired.

λ : The constant failure rate of the component.

µ ( x) : The repair rate when the failed system is in state j with elapsed repair time x for j = 1, 2 .

P0 (t ) : The probability that the system is in state 0 at time t .

Pj ( x, t ) : The probability density (with respect to repair time) that the failed system is in state j with elapsed repair time x for j = 1, 2 .

Considering the actual background, we suppose that: 0 ≤ µ ( x) < +∞, 0 ≤ x < +∞, ∫0∞ µ j (ξ )d ξ = ∞, lim

x →∞

1 x ∫ µ j (ξ )d ξ < +∞. x 0

Then we define that 1 x ∫ µ (ξ )d ξ x →∞ x 0

µ * = lim We choose the state space as

 T X = P = ( P0 , P1 ( x), P2 ( x) ) P0 ∈ R; Pj ( x) ∈ L1[0, ∞], j = 1, 2.

{

}

We define its norm as  P = P0 + P1 ( x)

L1 ( R+ )

+ P2 ( x)

L1 ( R+ )

It is obvious that ( X , ⋅ ) is a Banach space.

We define operator A , B and their domains:

( (

) (

))

A= diag −λ , − d + λ + µ ( x) , − d + µ ( x) , dx dx   dPj ( x) 1 +  ∞ ∈ L ( R ), P1 (0) = λ p0 + ∫0 P2 ( x) µ ( x)dx, P2 (0) = D( A) = 0, j = 1, 2. ;  P ∈ X Pj ( x) is absolutely continuous, dx  

 0 ∞  µ ( x)dx 0  ∫0   B = 0 0 0 λ 0 0  

and D( B) = X .

Then the system (1.1) can be expressed as an Abstract Cauchy Problem in Banach space X :    dP(t )  = ( A + B) P(t ), t ≥ 0   dt T  P(0) = (1, 0, 0)  Here P(t ) = ( P0 (t ), P1 ( x, t ), P2 ( x, t ))T .

(1.2)

2 EXPONENTIAL STABILITY OF THE SYSTEM Theorem 2.1 Let ( A + B)∗ and X ∗ denote the dual operator of A + B and the dual space of X , respectively. Then for - 58 http://www.sj-ce.org

 = ∀Q

( Q0 , Q1 ( x), Q2 ( x) )T ∈ X ∗ , we have     −λ Q0 + Q1 (0)λ    d  = (A + B )∗ Q  Q0 µ ( x) +  − λ − µ ( x)  Q1 ( x) + λ Q2 ( x)     dx    d  Q1 (0) µ ( x) +  − µ ( x)  Q2 ( x)    dx   

dQ j ( x)    And D ( A + B)∗ =Q ∈ X ∗ ∈ L∞ ( R+ ), Q j ( x) ∈ L∞ ( R+ ) is absolutely continuous, j = 1, 2. . dx  

(

)

Proof:

  For any P ∈ D( A + B) and Q ∈ X ∗ , we have   d ∞ ∞ ∞  ( A + B ) P, Q = −λ P0Q0 + Q0 ∫0 P1 ( x) µ ( x)dx + ∫0  −Q1 ( x) P1 ( x) dx − ∫0 [ λ Q1 ( x) + µ ( x)Q1 ( x)] P1 ( x)dx dx   d ∞ ∞ ∞  + ∫0 λ P1 ( x)Q2 ( x)dx − ∫0 Q2 ( x) P2 ( x) dx − ∫0 µ ( x)Q2 ( x) P2 ( x)dx dx   ∞ d  =[ −λ Q0 + Q1 (0)λ ] P0 + ∫0 [Q0 µ ( x) +  − λ − µ ( x)  Q1 ( x) + λ Q2 ( x)]P1 ( x)dx dx   ∞ d  + ∫0 [Q1 (0) µ ( x) +  − µ ( x)  Q2 ( x)]P2 ( x)dx  dx      −λ Q0 + Q1 (0)λ    P0    d  )  P, ( A + B)∗ Q =  P1 ( x)   Q0 µ ( x) +  − λ − µ ( x)  Q1 ( x) + λ Q2 ( x=   P ( x)    dx   2   d  Q1 (0) µ ( x) +  − µ ( x)  Q2 ( x)    dx   

(

Derived from [1], we know that the system operator A + B is a resolvent positive operator and D ( A + B)∗

)

+

is

cofinal in X +∗ . So operator A + B is the infinitesimal generator of a C0 -semigroup and s ( A + B) = ω ( A + B) = 0 .

Here s ( A + B) is the upper spectral bound of operator A + B and ω ( A + B) is the growth bound of the C0 -semigroup T (t ) , where T (t ) denotes the C0 -semigroup which is generated by operator A + B .

We also obtain that the system has the unique dynamic solution   P(t ,) = T (t ) P(0) , for t ∈ (0, +∞)  which satisfies P(t ,) = 1 .

Theorem 2.2

{

}

Let ρ ( A + B) denote the resolvent set of A + B , then γ ∈ C Re γ > 0 or γ = ia, a ∈ R, a ≠ 0 ⊂ ρ ( A + B)

Proof:    For ∀G = G . Making use of the method in [3], we ( g0 , g1 ( x), g 2 ( x) )T , we consider the equation [γ I − ( A + B)] P =   can obtain that if Re γ > 0 or γ = ia , a ∈ R \ {0} , the equation [γ I − ( A + B)] P = G has the unique solution. It

shows that [γ I − ( A + B)] is a surjective. For [γ I − ( A + B)] is closed and D( A + B) is dense in X , according to inverse operator theorem, we have [γ I − ( A + B) ] which is a bounded linear operator. −1

Theorem 2.3 - 59 http://www.sj-ce.org

0 is an eigenvalue of ( A + B)∗ , whose geometric multiplicity is 1 . And the vector Q∗ = (1,1,1) is the eigenvector corresponding to 0.

T

Proof:

  For ∀Q ∈ D ( A + B)∗ , we consider the operator equation ( A + B)∗ Q = 0:

(

)

 λ Q0 = Q1 (0)λ  dQ1 ( x) = −Q0 µ ( x) − λ Q2 ( x) + ( λ + µ ( x) ) Q1 ( x)   dQdx( x)  2 = µ ( x)Q2 ( x) − Q1 (0) µ ( x)  dx

(2.1)

∗ Solving the equation, we obtain that = Q0 Q= 1 ( x ) Q2 ( x ) . So Q = (1,1,1) is the eigenvector corresponding to 0 , and the geometric multiplicity of 0 is 1 .

T

Theorem 2.4 0 is an eigenvalue of operator A + B and its algebraic multiplicity is 1 .

Proof:

 We consider the operator equation ( A + B ) P = 0 , namely

 ∞ λ P0 = ∫0 P1 ( x) µ ( x)dx  dP1 ( x) = − ( λ + µ ( x) ) P1 ( x)   dPdx( x)  2 = − µ ( x) P2 ( x) + λ P1 ( x)  dx

Solving the equation, we obtain P1 ( x) =

λ f1 f0

P0 , P2 ( x) =

λ f2 f0

P0

where ∞

f 0 = ∫0 µ ( x)e− ∫

x 0

( λ + µ (ξ )) d ξ

(ξ ) ) d ξ , f 2 e− ∫ dx , f1 = e− ∫ ( λ + µ= x 0

x 0

µ (ξ ) d ξ

(1 − e λ ) . − x

We let P0 > 0 , then Pj ( x) > 0 , j = 1, 2 . So we obtain that 0 is a eigenvalue of operator A + B and its eigenvector is    T = P 0, P  1 ( x), P  2 ( x) T is also an eigenvector of 0.  = P P and P > 0 , then P P = ( P0 , P1 ( x), P2 ( x) ) . Let P 0

(

)

 is the steady state solution of the system. The geometric multiplicity of Considering the background of the model, P the eigenvalue 0 is 1 .

Now we prove the algebraic multiplicity of the eigenvalue 0 is 1 . We assume algebraic multiplicity of 0 is 2, then    . According to theorem 2.3, there must be a vector Y such that ( A + B ) Y = = Q∗ (1,1,1)T ∈ X ∗ is the eigenvector of P ∗

∗

0 . eigenvalue 0 of the operator ( A + B ) .

So ( A + B ) Q∗ = Then we have  ∗  ∗ ∗ < ( A + B ) Y , Q >=< Y , ( A + B ) Q >= 0 .   , Q∗ >= 1 . So it’s contradictory. That means the assumption is wrong and the But we know that < ( A + B ) Y , Q∗ >=< P algebraic multiplicity of the eigenvalue 0 is 1 . the

From[2], we know operator A generates a C0 -semigroup which is denoted as S (t ) and ω ( A) = s ( A) = − µ ∗ . Noting that operator B is a finite-rank operator, we know B is a compact operator. From perturbation and compact perturbation theory of semigroup, we can obtain the following result.

- 60 http://www.sj-ce.org

Theorem 2.5

{

}

For any ε > 0 , the number of the spectral points of A + B in the area D = γ ∈ C − µ ∗ + ε ≤ Re γ < 0 is finite. And all the spectral points are isolated. The algebraic multiplicity of every spectral point is finite, too.

Proof: For s ( A) = − µ ∗ , we have sess ( A) ≤ s ( A) = − µ ∗ , where s ( A) and sess ( A) denote the upper spectral bound of A and upper essential spectral bound of A , respectively. B is a finite-rank operator, so B is a compact operator. From [4] and [5], we know that for operator A , the perturbation of a compact operator B won’t change the upper essential spectral bound of it. That means sess ( A= + B) sess ( A) ≤ − µ ∗ . For A + B is closed and D( A + B) = X , the conclusion of this theorem is obtained.

Theorem 2.6 For some proper positive constant M and ω , we have   ≤ Me−ωt P(t ,) − P   is the steady-state solution of the system.

Where P(t ,) is the dynamic solution of the system and P

Proof: From theorem 2.2 and theorem 2.4, we know 0 is the unique nonnegative spectral point. Its algebraic multiplicity is 1 . Let P0 denote the projection operator corresponding to 0. Then from theorem 2.5 and [6], we have T (t ) , the C0 semigroup generated by A + B , which can be expressed as T (t= ) P0 + R(t ) −ωt Where R(t ) is a projection for some proper positive constant M and ω . From  operator and satisfies R(t ) ≤ Me theorem 2.1, we know P(t ,) = T (t ) P(0) . Then we have        + R(t ) P (0) =+  R (t ) P (0) < P(0), Q∗ > P P(t ,) = T (t ) P(0) =P0 + R(t ) P(0) = P   ≤ Me−ωt . The exponential stability of the system is obtained.

Where Q∗ = (1,1,1)T . So we have P(t ,) − P

(

)

ACKNOWLEDGEMENTS The work of Dongxu Liu is partially supported by NNSFC under Grant No. 11461074 and the educational department of Jilin Province of China.

REFERENCES [1]

Yurong Zhao, Shi Liu, Jianyu Zhu,Weiyuan Li, Yufeng Zhang. “Properties of the System Operator of the 1-unit Repairable

[2]

Aidong Jin, Chen Li, Yufeng Zhang. “Properties of the Main Operator of the 1-unit Repairable System with an Identical Cold

[3]

Xin Zhang. “Reliability analysis of the operator of two-different-units parallel repairable system”. Systems Engineering Theory

[4]

Genqi Xu. “The perturbation theory of the essential spectral for strongly continuous semigroups.” Acta Analysis Functionalis

[5]

Genqi Xu. “Estimation of the essential spectral radius of the perturbed (C0) semi-group.” Acta Analysis Functionalis Applicata.

System with an Identical Cold Standby.” Mathematics in Practice and Theory, 2014, 44(2): 262-270 Standby” Mathematics in Practice and Theory, 2014, 44(21): 237-244 and Practice, 2013, 33(10): 2615-2622 Applicata. 1990, 33(6) : 757−763 1993, 36(3): 335−340 [6] Arendt W, Nagel R. One-Parameter Semigroup of Positive Operator: Lceture Notes in Mathematics. Springer Verlag, New York, 1986 [7] Dongxu Liu, Wenyi Si, Zhe Yin. “Exponential Stability Analysis of the Solution of a Repairable Human-Machine System.”

- 61 http://www.sj-ce.org

Scientific Journal of Control Engineering, 2014, 4(3): 86-93 [8] Dongxu Liu, Wenyi Si, Yujiao Yuan. â&#x20AC;&#x153;The exponential stability and reliability analysis of a kind of Single-component Repairable system.â&#x20AC;? Journal of Yanbian University (Natural Science Edition). 2014, 40(1): 15-19

AUTHORS 1

Dongxu Liu, birthed in Jilin City, Jilin Province, July 14, 1984 and earned Master of Science in Yanbian

University in Yanji in 2010, majored in Functional Analysis.

Now he works at Yanbian University and his job title is lecturer. He is interested in complex system and reliability theory.

- 62 http://www.sj-ce.org