# Refined Upper Solution Bound of the Continuous Coupled Algebraic Riccati Equation

Authors: Li Wang
Journal: Complexity
 (2020)

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## Abstract

The continuous coupled algebraic Riccati equation (CCARE) has wide applications in control theory and linear systems. In this paper, by a constructed positive semidefinite matrix, matrix inequalities, and matrix eigenvalue inequalities, we propose a new two-parameter-type upper solution bound of the CCARE. Next, we present an iterative algorithm for finding the tighter upper solution bound of CCARE, prove its boundedness, and analyse its monotonicity and convergence. Finally, corresponding numerical examples are given to illustrate the superiority and effectiveness of the derived results.


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## Body

## 1. Introduction

Consider the following optimal control of jump linear system described by(1)x˙t=Artxt+Brtut,xt0=x0,where xt∈Rn is the plant state and ut∈Rm is the control vector. Art∈Rn×n, Brt∈Rn×m, and rt is a finite state Markov jump process on S=1,2,…,s, where s≥2. The quadratic performance index of system (1) is(2)Ju=E∫0∞xTtQrtxt+uTtRrtutdtx0,r0.We haveArt=Ai,Brt=Bi,Qrt=Qi∈Rn×n,andRrt=Ri∈Rm×m with Qi≥0 and Ri>0 when rt=i∈S. The optimal state feedback controller to minimize the quadratic performance index (2) is ut=−Ri−1BiTPixt, where Pi is the symmetric positive semidefinite solution of the continuous coupled algebraic Riccati equation (CCARE):(3)AiTPi+PiAi−PiBiRi−1BiTPi+∑j≠idijPj+Qi=0,where dij are real constants with the properties dii<0,dij≥0i≠j, and ∑j∈Sdij=0. When R=I, the CCARE (3) changes to the following common form:(4)AiTPi+PiAi−PiBiBiTPi+∑j≠idijPj+Qi=0.CCARE (4) is usually encountered in robust and optimal control [1–8], filter design [9], time-delay systems controller design [10], stability analysis [11–16], etc. And in these fields, it often suffices to estimate the tighter solution bounds of the algebraic Riccati equation rather than get the exact solution. For example, in [17, 18], the authors have proposed solution bounds of the continuous algebraic Riccati equation, given their applications of the new bounds in redundant control input systems, and obtained several sufficient conditions to decrease the controller gain of the new systems after the control input extension. In recent years, bounds’ estimation of solution for the CCARE has become an attractive topic, and many research approaches have been devoted to this topic. In [19–21], upper matrix bounds for the solution of the CCARE have been presented and iterative algorithms have been proposed to derive tighter upper matrix bounds. And there are many other works for studying the solutions of the CCARE, such as matrix bounds and properties [22–27], matrix eigenvalue bounds [28–30], numerical solution [31–33], and the explicit solution [34, 35].In this paper, we construct a positive semidefinite matrix, propose a new two-parameter-type upper solution bound of the CCARE by matrix inequalities and matrix eigenvalue inequalities. Examples illuminate that the upper bound improves some recent related results. Then, according to the derived solution bound, we give an iterative algorithm, which can guarantee tighter upper solution bound of the CCARE, and prove its boundedness. Subsequently, we analysis its monotonicity and convergence.Through this paper, letRn×nCn×n denote the set of n×n real (complex) matrices. For A=aij∈Rn×n, the notation A>0A≥0 is used to denote that A is a symmetric positive definite (semidefinite) matrix. Inequality A>≥B means matrix A−B is positive (semi) definite. For i=1,2,…,n, let λiA be the nonincreasing order eigenvalues of A. We assume AT,A−1, and A2 are the transpose, the inverse, and the spectral norm of A, respectively. A is called a Z-matrix if all its off-diagonal elements are nonpositive. It is obvious that any Z-matrix A can be written as sI−B with B is nonnegative. A Z-matrix A is called an M-matrix if s>ρB. For two vectors a=a1,a2,…,anT∈Rn and b=b1,b2,…,bnT∈Rn, a≥b is equivalent to ai≥bii=1,2,…,n. Symbol μA represents the matrix measure of A and is defined as μA=λ1A+AT/2.Lemma 1.
(see [36]). For any given positive semidefinite matrix A∈Rn×n and B∈Rn×n,(5)U=BTA+AB≤0,if and only if the matrix BT+B is negative semidefinite.Lemma 2.
(see [37]). Let A∈Rn×n satisfy that AT+A<0, and let P∈Rn×n be symmetric. Then,(6)ATP+PA≤<0,if and only if the matrix P≥>0.Lemma 3.
(see [38]). For any symmetric matrix X, the following inequality holds:(7)λnXI≤X≤λ1XI.Lemma 4.
(see [38]). Suppose X,Y∈Rn×n are symmetric matrices and 1≤i,j≤n; then,(8)λi+j−1X+Y≤λjX+λiY,i+j≤n+1,λi+j−nX+Y≥λjX+λiY,i+j≥n+1.Lemma 5.
(see [39]). If X∈Rn×n is an M-matrix, then X−1 is nonnegative.Lemma 6.
(see [40]). Let x=x1,…,xnT,y=y1,…,ynT∈Rn and x≥y, A=aij∈Rn×n be nonnegative; then, Ax≥Ay.Lemma 7.
(see [41]). Let A,B∈Rn×n both be symmetric with A≥B. Then,(9)λiA≥λiB,for all 1≤i≤n.Lemma 8.
(see [20]). Let Wkk∈N⊂Rn×n be a given sequence of positive semidefinite matrices. Assume that there exists a positive semidefinite W0∈Rn×n such that for all k,(10)Wk+1≤Wk,Wk≥W0,and then Wkk∈N converges to a unique positive semidefinite W¯∈Rn×n.


## 2. Upper Solution Bounds for the CCARE

In this section, we will propose new upper matrix bounds of the solution for the CCARE (4), which improve the recent results.Theorem 1.
LetPi be the positive semidefinite solution of the CCARE (4). If there exist some positive definite matrices Ki, real numbers ni≥1/2, γi<0, and γi<βi≤−γi such that(11)Ai+AiT<BiBiTniKi+KiBiBiTni,and suppose W is an M-matrix; then, for i=1,…,s,(12)Pi≤1−2μA˜iA˜i+γiI−TA˜i+βiITGiA˜i+βiIA˜i+γiI−1−A˜i+γiI−Tβi2−γi2Gi−2μAi−βi−γiGiA˜i+γiI−1≡Pui,where Ai˜=Ai−BiBiTniKi,Gi=∑j≠idijηjI+Qi+KiBiBiT2ni−1Ki,(13)W=1−d12r1⋯−d1sr1−d21r21⋯−d2sr2⋮⋱⋮−ds1rs−ds2rs⋯1,η1η2⋮ηs=defW−1ξ1ξ2⋮ξs,ri=λ11−2μA˜iA˜i+γiI−TA˜i+βiITA˜i+βiIA˜i+γiI−1−βi2−γi2−2μA˜i−βi−γiA˜i+γiI−TA˜i+γiI−1,ξi=λ11−2μA˜iA˜i+γiI−TA˜i+βiITQi+KiBiBiT2ni−1KiA˜i+βiIA˜i+γiI−1−βi2−γi2−2μA˜i−βi−γiA˜i+γiI−TQi+KiBiBiT2ni−1KiA˜i+γiI−1.Proof.
(14)PiBiBiTPi+εPi2−KiBiBiT+εIniPi−PiBiBiT+εIniKi+KiBiBiT+εI2ni−1Ki=Pi−BiBiT+εIni−1KiTBiBiT+εIPi−BiBiT+εIni−1Ki≥0,where ε is any sufficiently small positive constant. When ε⟶0, from (14), we obtain(15)0≤PiBiBiTPi−KiBiBiTniPi−PiBiBiTniKi+KiBiBiT2ni−1Ki≡Si.
And letQ˜i=Qi+∑j≠idijPj+KiBiBiT2ni−1Ki≥0. According to CARE (4), we obtain(16)A˜iTPi+PiA˜i=Ai−BiBiTniKiTPi+PiAi−BiBiTniKi=AiTPi+PiAi−KiBiBiTniPi−PiBiBiTniKi=PiBiBiTPi−Qi−∑j≠idijPj−KiBiBiTniPi−PiBiBiTniKi=Si−Qi−∑j≠idijPj−KiBiBiT2ni−1Ki=Si−Q˜i.
Let(17)Di=1−2μA˜iA˜i+γiI−TA˜i+βiIT∑j≠idijPj+Qi+KiBiBiT2ni−1KiA˜i+βiIA˜i+γiI−1−A˜i+γiI−Tβi2−γi2∑j≠idijPj+Qi+KiBiBiT2ni−1Ki−2μA˜iA˜i+γiI−1+A˜i+γiI−Tβi−γi∑j≠idijPj+Qi+KiBiBiT2ni−1KiA˜i+γiI−1=A˜i+γiI−TA˜i+βiITQ˜i−2μA˜iA˜i+βiIA˜i+γiI−1−A˜i+γiI−Tβi2−γi2Q˜i−2μA˜iA˜i+γiI−1+A˜i+γiI−Tβi−γiQ˜iA˜i+γiI−1.
Then, by (16), we have(18)A˜iTDi−Pi+Di−PiA˜i=−Si+Q˜i+A˜iTDi+DiA˜i=−Si−1βi−γiA˜i+γiITDiA˜i+γiI−A˜i+βiITDiA˜i+βiI+βi2−γi2Di−βi−γiQ˜i=−Si−1βi−γiA˜i+βiITQ˜i−2μA˜iA˜i+βiI−βi2−γi2Q˜i−2μA˜i+βi−γiQ˜i−A˜i+γiI−TA˜i+βiI2TQ˜i−2μA˜iA˜i+βiI2A˜i+γiI−1−A˜i+γiI−TA˜i+βiITβi2−γi2Q˜i−2μA˜iA˜i+βiIA˜i+γiI−1+A˜i+γiI−TA˜i+βiITβi−γiQ˜iA˜i+βiIA˜i+γiI−1+βi2−γi2A˜i+γiI−TA˜i+βiITQ˜i−2μA˜iA˜i+βiIA˜i+γiI−1−A˜i+γiI−Tβi2−γi2Q˜i−2μA˜iA˜i+γiI−1+A˜i+γiI−Tβi−γiQ˜iA˜i+γiI−1−βi−γiQ˜i=−Si−1βi−γiA˜i+γiI−TA˜i+βiITA˜i+γiITQ˜i−2μA˜iA˜i+γiI−A˜i+βiI−TA˜i+γiITβi2−γi2Q˜i−2μA˜iA˜i+γiIA˜i+βiI−1−A˜i+βiITQ˜i−2μA˜iA˜i+βiI+βi2−γi2Q˜i−2μA˜i−βi−γiQ˜i+βi2−γi2Q˜i−2μA˜i−A˜i+βiI−Tβi2−γi22Q˜i−2μA˜iA˜i+βiI−1+A˜i+βiI−Tβi2−γi2βi−γiQ˜iA˜i+βiI−1A˜i+βiIA˜i+γiI−1=−Si−1βi−γiA˜i+γiI−TA˜i+βiITA˜i+γiITQ˜i−2μA˜iA˜i+γiI−A˜i+βiITQ˜i−2μA˜iA˜i+βiI+βi2−γi2Q˜i−2μA˜i−βi−γiQ˜i−βi2−γi2A˜i+βiI−TA˜i+γiITQ˜i−2μA˜iA˜i+γiI−A˜i+βiITQ˜i−2μA˜iA˜i+βiI+βi2−γi2Q˜i−2μA˜i−βi−γiQ˜iA˜i+βiI−1A˜i+βiIA˜i+γiI−1=−Si+A˜i+γiI−TA˜i+βiITA˜iTQ˜i−2μA˜i+Q˜i−2μA˜iA˜i+Q˜i−βi2−γi2A˜i+βiI−TA˜iTQ˜i−2μA˜i+Q˜i−2μA˜iA˜i+Q˜iA˜i+βiI−1A˜i+βiIA˜i+γiI−1=−Si+A˜i+γiI−TA˜i+βiITQ¯i−βi2−γi2A˜i+βiI−TQ¯iA˜i+βiI−1A˜i+βiIA˜i+γiI−1,where Q¯i=A˜iTQ˜i/−2μA˜i+Q˜i/−2μA˜iA˜i+Q˜i. According to condition (11), we get A˜i+A˜iT=Ai+AiT−BiBiTniKi−KiBiBiTni<0; then,(19)A˜iT−2μA˜i+I2+A˜i−2μA˜i+I2=A˜i+A˜iT−2μA˜i+I≤2μA˜iI−2μA˜i+I=0.
ConsideringQ˜i≥0, using Lemma 1 to (19) yields(20)Q¯i=A˜iTQ˜i−2μA˜i+Q˜i−2μA˜iA˜i+Q˜i=A˜iT−2μA˜i+I2Q˜i+Q˜iA˜i−2μA˜i+I2≤0.
Substituting (20) into (18) with βi2−γi2≤0 and Si≥0, we get the right-hand side of (18) is negative semidefinite. Moreover, since A˜i+A˜iT<0, according to Lemma 2, we conclude Di−Pi≥0, that is,(21)Pi≤Di=1−2μA˜iA˜i+γiI−TA˜i+βiIT∑j≠idijPj+Qi+KiBiBiT2ni−1KiA˜i+βiIA˜i+γiI−1−A˜i+γiI−Tβi2−γi2∑j≠idijPj+Qi+KiBiBiT2ni−1Ki−2μA˜iA˜i+γiI−1+A˜i+γiI−Tβi−γi∑j≠idijPj+Qi+KiBiBiT2ni−1KiA˜i+γiI−1.
Using Lemma3 to (21), we obtain(22)Pi≤1−2μA˜iA˜i+γiI−TA˜i+βiITA˜i+βiIA˜i+γiI−1−βi2−γi2−2μA˜i−βi−γiA˜i+γiI−TA˜i+γiI−1∑j≠idijλ1Pj+1−2μA˜iA˜i+γiI−TA˜i+βiITQi+KiBiBiT2ni−1KiA˜i+βiIA˜i+γiI−1−βi2−γi2−2μA˜i−βi−γiA˜i+γiI−TQi+KiBiBiT2ni−1KiA˜i+γiI−1.
Applying Lemma4 to (22) yields(23)λ1Pi≤λ11−2μA˜iA˜i+γiI−TA˜i+βiITA˜i+βiIA˜i+γiI−1−βi2−γi2−2μA˜i−βi−γiA˜i+γiI−TA˜i+γiI−1∑j≠idijλ1Pj+λ11−2μA˜iA˜i+γiI−TA˜i+βiITQi+KiBiBiT2ni−1KiA˜i+βiIA˜i+γiI−1−βi2−γi2−2μA˜i−βi−γiA˜i+γiI−TQi+KiBiBiT2ni−1KiA˜i+γiI−1.
Therefore, from (23), we obtain λ1Pi−ri∑j≠idijλ1Pj≤ξi,i=1,2,…,s, i.e.,(24)Wx≤ξ,where x=λ1P1,λ1P2,…,λ1Ps and ξ=ξ1,ξ2,…,ξs. As the coefficient matrix W of (24) is an M-matrix, W−1 is nonnegative in terms of Lemma 5. Consequently, (24) is equivalent to(25)x≤W−1ξ=η,by Lemma 6, where η=η1,η2,…,ηs. Substituting (25) into (22) completes the proof.Corollary 1.
We show that(26)1Pi≤1−2μA˜i∑j≠idijη˜jI+Qi+KiBiBiT2ni−1Ki≡Pui′,2Pui≤Pui′,where γi˜=1/−2μAi˜,ξi˜=1/−2μAi˜λ1Qi+KiBiBiT2ni−1Ki,(27)η˜1η˜2⋮η˜s=defW˜−1ξ˜1ξ˜2⋮ξ˜s,W˜=1−d12γ˜1⋯−d1sγ˜1−d21γ˜21⋯−d2sγ˜2⋮⋱⋮−ds1γ˜s−ds2γ˜s⋯1,is an M-matrix.Proof.
(1)
Since(28)Ai˜TQ˜i−2μA˜i−Pi+Q˜i−2μA˜i−PiAi˜=−Si+Qi˜+Ai˜TQ˜i−2μA˜i+Q˜i−2μA˜iAi˜.Substituting (20) into (28) and combining Si≥0, we get the right-hand side of (28) is negative semidefinite. Since Ai˜+Ai˜T<0, according to Lemma 2, we conclude Qi˜/−2μAi˜−Pi≥0, that is,(29)Pi≤Qi˜−2μAi˜=1−2μAi˜∑j≠idijPj+Qi+KiBiBiT2ni−1Ki≤1−2μAi˜∑j≠idijλ1PjI+Qi+KiBiBiT2ni−1Ki.Using Lemma4 to (29) yields(30)λ1Pi≤1−2μAi˜∑j≠idijλ1Pj+1−2μAi˜λ1Qi+KiBiBiT2n−1Ki,that is,λ1Pi−γi˜∑j≠idijλ1Pj≤ξi˜, i.e.,(31)W˜x≤ξ˜,whereξ˜=ξ1,ξ2,…,ξs. As the coefficient matrix W˜ of (31) is an M-matrix, W˜−1 is nonnegative in terms of Lemma 5. Consequently, (31) is equivalent to(32)x≤W˜−1ξ˜=η˜,by Lemma6, where η˜=η1˜,η2˜,…,ηs˜. Substituting (32) into (29), we get Pi≤Pui′.(2)
(i)
First, let us comparePui with Gi/−2μA˜i:(33)Pui−Gi−2μA˜i=1−2μA˜iA˜i+γiI−TA˜i+βiITGiA˜i+βiIA˜i+γi−1−A˜i+γiI−Tβi2−γi2Gi−2μA˜i−βi−γiGiA˜i+γiI−1−Gi−2μA˜=A˜i+γiI−TA˜i+βiITGi−2μA˜A˜i+βiI−βi2−γi2Gi−2μA˜i+βi−γiGi−A˜i+γiITGi−2μA˜A˜i+γiIA˜i+γiI−1=βi−γiA˜i+γiI−TA˜TGi−2μA˜+Gi−2μA˜A˜+GiA˜i+γiI−1.ConsideringGi≥0, using Lemma 1 to (19) yields(34)A˜TGi−2μA˜+Gi−2μA˜A˜+Gi=A˜iT−2μA˜i+I2Gi+GiA˜i−2μA˜i+I2≤0.Substituting (34) into (33) with βi−γi>0, we get the right-hand side of (33) is negative semidefinite. Thus,(35)Pui≤Gi−2μA˜i.(ii)
Next, let us compareGi/−2μA˜i with Pui′. For (35), assume Gi=I; then, (35) changes to(36)1−2μA˜iA˜i+γiI−TA˜i+βiITA˜i+βiIA˜i+γi−1−βi2−γi2−2μA˜i−βi−γiA˜i+γiI−TA˜i+γiI−1≤1−2μA˜iI.
By Lemma7, it is easily to see ri≤γi˜, then W≥W˜. As WandW˜ are M-matrix, we obtain(37)W−1≤W˜−1.
On the contrary, forGi of (35), assume ∑j≠idijηjI=0; then, (35) changes to(38)1−2μA˜iA˜i+γiI−TA˜i+βiITQi+KiBiBiT2ni−1KiA˜i+βiIA˜i+γiI−1−βi2−γi2−2μA˜i−βi−γiA˜i+γiI−TQi+KiBiBiT2ni−1KiA˜i+γiI−1≤1−2μA˜iQi+KiBiBiT2ni−1Ki.
Applying Lemma7 to (38), we conclude ξi≤ξi˜. Thus, we obtain(39)ηi≤ηi˜,with (37). Subsequently, we obtain(40)Gi−2μA˜i≤1−2μA˜i∑j≠idijη˜jI+Qi+KiBiBiT2ni−1Ki=Pui′.
Lastly, combining (35) with (40), we derive Pui≤Pui′.Remark 1.
Due to the different proof methods, it is very hard to compare the upper bounds of Theorem1 with the parallel results theoretically, and we will present examples to illustrate that our upper bounds are tighter than the recent results for some cases.


## 3. Iterative Algorithm

According to Section2, we give an iterative algorithm as following. The algorithm is based on upper bounds (12) from Theorem 1. And we will discuss the following conclusions in the same condition as Theorem 1.Algorithm 1.
LetXi0=ηiI, i∈S, and ηi is from Theorem 1. For k=1,2,…,(41)Xik+1=1−2μA˜iA˜i+γiI−TA˜i+βiITUikA˜i+βiIA˜i+γiI−1−βi2−γi2A˜i+γiI−TUik−2μA˜iA˜i+γiI−1+βi−γiA˜i+γiI−TUikA˜i+γiI−1,where Uik=∑j≠idijXjk+Qi+KiBiBiT2ni−1Ki.
Next, we will discuss the boundedness of Algorithm1 and analyse its monotonicity and convergence.Theorem 2.
LetXik be the iteration in Algorithm 1. If the CCARE (4) has positive semidefinite solution Pi, then(42)Pi≤Xik,k=0,1,2,….Proof.
We proof the result by induction.(i)
From (25), we get Pi≤λ1PiI≤ηiI=Xi0. Furthermore,(43)Xi1=1−2μA˜iA˜i+γiI−TA˜i+βiITUi0A˜i+βiIA˜i+γiI−1−βi2−γi2A˜i+γiI−TUi0−2μA˜iA˜i+γiI−1+βi−γiA˜i+γiI−TUi0A˜i+γiI−1,whereUi0=Qi+∑j≠idijXj0+KiBiBiT2ni−1Ki=Qi+∑j≠idijηiI+KiBiBiT2ni−1Ki. According to Theorem 1, we get Pi≤Pui=Xi1.(ii)
SupposePi≤Xik for all k≥1, then Q˜i=Qi+∑j≠idijPj+KiBiBiT2ni−1Ki≤Qi+∑j≠idijXjk+KiBiBiT2ni−1Ki=Uik. In a similar way to the proof of (18), we have(44)A˜iTXik+1−Pi+Xik+1−PiA˜i=−A˜iTPi+PiA˜i+A˜iTXik+1+Xik+1A˜i=−Si+Q˜i+A˜iTXik+1+Xik+1A˜i≤−Si+Uik+A˜iTXik+1+Xik+1A˜i=−Si−1βi−γiA˜i+γiITXik+1A˜i+γiI−A˜i+βiITXik+1A˜i+βiI+βi2−γi2Xik+1−βi−γiUik=−Si+A˜i+γiI−TA˜i+βiITUik¯−βi2−γi2A˜i+βiI−TUik¯A˜i+βiI−1A˜i+βiIA˜i+γiI−1,where Uik¯=A˜iTUik/−2μA˜i+Uik/−2μA˜iA˜i+Uik. Considering Uik≥0, using Lemma 1 to (19) yield(45)Uik¯=A˜TUik−2μA˜+Uik−2μA˜A˜+Uik=A˜iT−2μA˜i+I2Uik+UikA˜i−2μA˜i+I2≤0.
Substituting (45) into (44) with βi2−γi2≤0 and Si≥0, we get the right-hand side of (44) is negative semidefinite. Moreover, since A˜iT+A˜i<0, according to Lemma 2, we get Pi≤Xik+1. This completes the induction.Theorem 3.
LetXik be the iteration in Algorithm 1. If Pui≤ηiI,i∈S, where Pui and ηi are from Theorem 1, then the sequence Xik is monotone decreasing, i.e.,(46)Xik+1≤Xik,k=0,1,2,….Proof.
We proof the result by induction.(i)
Xi1=Pui≤ηiI=Xi0.(ii)
SupposeXik≤Xik−1 for all k≥1, then(47)Xik+1−Xik=1−2μA˜iA˜i+γiI−TA˜i+βiITUik−Uik−1A˜i+βiIA˜i+γiI−1−βi2−γi2A˜i+γiI−TUik−Uik−1−2μA˜iA˜i+γiI−1+βi−γiA˜i+γiI−TUik−Uik−1A˜i+γiI−1=1−2μA˜iA˜i+γiI−TA˜i+βiIT∑j≠idijXjk−Xjk−1A˜i+βiIA˜i+γiI−1−βi2−γi2A˜i+γiI−T∑j≠idijXjk−Xjk−1−2μA˜iA˜i+γiI−1+βi−γiA˜i+γiI−T∑j≠idijXjk−Xjk−1A˜i+γiI−1≤0.
Therefore, the proof is complete.
According to Lemma8, combining Theorems 2 and 3, we get the following conclusion.Theorem 4.
LetXik be the iteration in Algorithm 1. If Pui≤ηiI,i∈S, where Pui and ηi are from Theorem 1, then the sequence Xik is monotone decreasing, and converges to the unique positive semidefinite matrices X¯i.Now, we present examples to illustrate the effectiveness of the main results, and the stopping criterion in Algorithm1 is set as(48)Xik+1−Xik2<10−8.Example 1.
(see [42]). Consider the CCARE (4) with(49)A1=112−2,A2=−322−4,s=1,2,B1=10,B2=23,Q1=4334,Q2=6223,dij=−0.10.10.1−0.1.
Choosen1=1,γ1=−1,β1=1,K1=2.66,0.9;0.9,1.6; n2=1,γ2=−1,β2=1,andK2=0.23,0.1;0.1,0.15, and by Theorem 1, we obtain(50)η=7.31940.9499,W=1.0000−0.0652−0.01111.0000,Pu1=6.50852.16892.16891.5128,Pu2=0.80050.27980.27980.4234.
By Corollary1, we obtain(51)W˜=1.0000−0.0652−0.01111.0000,Pu1′=7.29103.51943.51943.2029,Pu2′=0.83630.27770.27770.4849.
By Theorem 3.2 in [21], taking a˜1=3,a˜2=0.1, Ti=a˜iI,ai=3,andn=1, we obtain(52)P˜1=8.94634.97364.97363.5837,P˜2=1.33370.50840.50840.6973.
By Theorem 2.2 in [42], taking K1′=2.600,0.8950,K2′=0.5900,0.1467, γ1′=γ2′=−1,andβ2′=β2′=1, we obtain(53)P¯1=7.46341.86281.86281.7131,P¯2=1.07390.36570.36570.9246.
Obviously, we seePui≺Pui′≺P˜i and Pui≺Pui′≺P¯i, that is, our bounds in Theorem 1 are tighter than [21, 42].
Since(54)X11=Pu1=6.50852.16892.16891.5128<7.3194I=η1I=X10,X21=Pu2=0.80050.27980.27980.4234<0.9499I=η2I=X20,then Algorithm 1 is expected to be strictly monotone decreasing and converges to tighter upper bounds. By computation, the upper solution bounds of CCARE (4) are(55)X16=6.49282.17462.17461.4956,X26=0.79060.30220.30220.3674,after 6 iterations, which are tighter than X10 and X20, respectively.Example 2.
Consider the CCARE (4) with(56)A1=1.81.23.2−3,A2=−2.21.31.4−3.1,A3=1.50.60.25−0.4,B1=3.60.8,B2=3.64.3,B3=1.50.2,Q1=6334,Q2=8557,Q3=5224,dij=−0.20.10.10.1−0.20.10.10.1−0.2,R1=R2=R3=I,s=1,2,3.
Choosingn1=1,γ1=−2,β1=2,K1=0.675,0.135;0.135,0.21, n2=1,γ2=−4,β2=4,K2=0.115,0.05;0.05,0.25, and n3=1,γ3=−2,β3=2,K3=0.93,0.25;0.25,0.65, by Theorem 1, we obtain(57)η=1.54211.16617.2392,W=1.0000−0.0142−0.0142−0.01451.0000−0.0145−0.1098−0.10981.0000,Pu1=1.25240.48010.48010.6508,Pu2=1.00650.20210.20210.7353,Pu3=6.05651.79401.79404.2349.
By Corollary1, we obtain(58)Pu1′=1.94080.66130.66130.8035,Pu2′=1.39520.83880.83881.4212,Pu3′=8.27402.99662.99665.1641.
By Theorem 3.1 in [21], taking a˜1=0.5,a˜2=1.6,a˜3=1.0, Ti=a˜iI,ai=2,andn=1, we obtain(59)P˜1=4.74530.43240.43243.1767,P˜2=2.65290.56090.56092.8528,P˜3=8.53051.24221.24229.4916.
By Theorem 2.2 in [42], taking γ1′=−0.2,β1′=0.2,K1′=3.55,2.9, γ2′=−0.4,β2′=0.4,K2′=0.11,0.75, γ3′=−0.68,β3′=0.68,andK3′=3.33,1.95, we obtain(60)P¯1=2.80580.13160.13162.5964,P¯2=3.37440.09250.09253.5902,P¯3=8.37805.23285.23289.8737.
Obviously, we seePui≺Pui′≺P˜i and Pui≺Pui′≺P¯i, that is, our bounds in Theorem 1 are tighter than [21, 42].
Since(61)X11=Pu1=1.25240.48010.48010.6508<1.5421I=η1I=X10,X21=Pu2=1.00650.20210.20210.7353<1.1661I=η2I=X20,X31=Pu3=6.05651.79401.79404.2349<7.2392I=η3I=X30,then Algorithm 1 is expected to be strictly monotone decreasing and converges to tighter upper bounds. By computation, the upper solution bounds of CCARE (4) are(62)X17=1.23910.49600.49600.5958,X27=0.98090.22950.22950.6990,X37=6.01231.86951.86954.0732,after 7 iterations, which are tighter than X10, X20, and X30, respectively.


## 4. Conclusions

In this paper, a new two-parameter-type upper solution bound of the CCARE has been proposed. Next, an iterative algorithm for finding the tighter upper solution bound of CCARE has been presented, and its boundedness, monotonicity, and convergence have been proved. Finally, corresponding numerical examples are given to illustrate the superiority and effectiveness of the derived results.



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*Source: 8627492-2020-08-04.xml*