Applied Mathematics
Checking a practical asymptotic stability of
an interval matrix
Ayse Bulgak
Research Center of Applied Mathematics, Selcuk University, Konya, Turkey e-mail:abulgak@karatay1.cc.selcuk.edu.tr
Received: January 11, 2001
Summary.
In this paper we investigate an algorithm for checking a practical asymptotic stability of an interval matrix.Key words:
an interval matrix, asymptotic stability, quality of asymptotic stabilityMathematics Subject Classication (1991): 65F30, 65G10, 15A18
1. Introduction
A matrixAis said to be asymptoticallystable if the Lyapunov matrix equation A H +HA+ 2jjAjjI = 0 with the identical matrix I has a unique positive de nite solution H = H H > 0. Moreover, if A
is asymptotically stable then the value {(A) = jjHjj
1 is known as the quality of stability forA1]. For non-asymptotically stable Awe assume that{(A) =1. The parameter{(A) is a continuous function with respect to the entries of A. If {(A) < { then A is known as a practical asymptotically stable ({ -stable) matrix. In 1], 2], and 4]{6], it was shown that the following theorem is true.
Theorem 1.
Let A be an asymptotically stable matrix. Then for an arbitrary matrix B such thatjjBjj=jjAjj 1 15{(A) 1 Here and everywhere in the sequel we use the notation
jjjjto designate the
the sum A+B is also an asymptotically stable matrix and j{(A+B);{(A)j2:15{ 2(A) jjBjj jjAjj 0:85{(A){(A+B) 1:15{(A):
2. Condition Number of an interval matrix
Let
A
=fA
ijg be a N-dimensional square interval matrix and the entriesA
ij = aijaij] ofA are the intervals of the real axis. The realnumber (
A
) = sup A2A fjjAjjjjA ;1 jjgis known as a condition number of the interval matrix
A
. It is clear that for any orthogonal matricesP andQ, and a real nonzero numberz we have
(
A
) =(zA
) (A
) =(PA
Q):Theorem 2.
IfA
is a regular interval matrix then for an arbitrary matrix A2A
we have the following inequalityjjAjj
max
ij=12:::N
fmax(jaijjjaijj)g
(
A
) :Proof. For a real number z the following equality holds (
A
) =(z
A
). Hence, we can assume that there exists a positive memberapq of the interval entries of
A
such thatapq = maxij
=12:::N
fmax(jaijjjaijj)g=(
A
) apq 0:Letbpq be the element ofA,A2
A
, in the position (pq). We consider a matrixB=A+ (apq;bpq)Epq B 2
A
:HereEpq is a real matrix the element ofEpq in position (pq) equals
1 and all other elements ofEpq are equal to 0. It is clear that
jjBjjapq and
1(B) = min jjxjj=1
jjBxjjjjAjj: Hence (B)apq=jjAjj, and jjAjjapq=(B). ut
3. Quality of asymptotic stability of an interval matrix
LetA
=fA
ijg be a N-dimensional square interval matrix and the entriesA
ij = aijaij] ofAare the intervals of the real line. The realnumber
{(
A
) = supA2A
f{(A)g
is known as the quality of stability of the interval matrix
A
. It is clear that for an orthogonal matrixQ and a positive numberz we have{(
A
) ={(zA
) {(A
) ={(QA
Q):Theorem 3.
LetA
be an asymptotically stable interval matrix. Then for A2A
we have jjAjj max ij=12:::N fmax(jaijjjaijj)g { 3=2(A
) :Proof. For an asymptotically stable matrixA the inequality(A) {
3=2(A) holds (see 1] and 3]). Whence and from Theorem 2 we have Theorem 3. ut
4. Practical asymptotically stable and practical
non-asymptotically stable interval matrices
Let
A
=fA
ijg be a N-dimensional square interval matrix and let { be a real number { > 1. If {(A
) < { thenA
is said to be practical asymptotically stable ({ -stable). Otherwise,A
is said to be practical non-asymptotically stable ({ -unstable).Theorem 4.
LetA
be a { -stable interval matrix. Then for A2A
we have jjAjj max ij=12:::N fmax(jaijjjaijj)g ({ ) 3=2 :
5. Algorithm for a matrix with only one interval entry
Let{ be a real number{ >1 andA
= 8 > > > < > > > : a11b11] a12 ::: a1N a21 a22 ::: a2N ... ... ... ... aN1 aN2 ::: aNN 9 > > > = > > >be an interval matrix with only one interval entry in position (1,1). Our aim is to nd the nite sequence of matrices. Analysis of matri-ces from the sequence allows us to guarantee that the initial interval matrix
A
is practical asymptotically stable or not practical asymp-totically stable. Let A= 8 > > > < > > > : a11 a12 ::: a1N a21 a22 ::: a2N ... ... ... ... aN1 aN2::: aNN 9 > > > = > > > : ThenA
=A+0b11 ;a 11] E11withE11 the matrix de ned in the proof of Theorem 2. The algorithm is as follows
1.
We assume thatA0 =A and= max( maxij=12:::N
fjaijjgjb 11
j) ({ )
3=2 :
1.1.
We compute the minimal singular value1(A0) ofA0 and the maximal singular valueN(A0) of A0.1.2.
If N(A0) < or N(A0) > 1(A0)( { )3=2 then the process stops with the answer\the given interval matrix
A
is not prac-tical asymptoprac-tically stable",{(A
)>{ .1.3.
We computeh1 =
N(A0) 15{(A
0) and check the inequality
a11+h1 b
11:
If the inequality holds then the process stops with the answer {(
A
)<1:15{(A 0): Otherwise, we go to Step 2.2.
We assume that A1 =A0+h1 E 11:2.1.
We compute the minimal singular value1(A1) ofA1 and the maximal singular valueN(A1) of A1.2.2.
If N(A1) < or N(A1) > 1(A1)( { )3=2 then the process stops with the answer\the given interval matrix
A
is not prac-tical asymptoprac-tically stable",{(A
)>{ .2.3.
We computeh2 =
N(A1) 15{(A
1) and check the inequality
a11+h1+h2 b
11:
If the inequality holds then the process stops with the answer {(
A
)<1:15maxj=01
{(Aj): Otherwise, we go to Step 3.
After (k;1) steps of the process a matrixAk
;2 and a real number
hk;1 are computed.
k.
We assume thatAk;1 =Ak;2+hk;1 E
11:
k.1.
We compute the minimal singular value1(Ak;1) ofAk;1 and the maximal singular valueN(Ak;1) of Ak;1.k.2.
If N(Ak;1) < or N(Ak;1) > 1(Ak;1)( { )3=2 then the process stops with the answer\the given interval matrix
A
is not practical asymptotically stable",{(A
)>{ .k.3.
We computehk = N(Ak;1) 15{(Ak
;1) and check the inequality
a11+h1+h2+
+hk b 11:
If the inequality holds then the process stops with the answer {(
A
)<1:15 maxj=01:::k;1
{(Aj): Otherwise, we go to Stepk+ 1.
Since hj =(15{ ) it follows that the algorithm will stop after not more thenM11 steps, where
M11= 15{ (b 11 ;a 11) M11 is an integer:
Remark 1.The initial data of the algorithm are an integer N, a real number { ,{ >1, and the interval matrix
A
=A+ 0b11 ;a11] E
11: The algorithm has n stepsnM
11. We can compute the following real numbers =(
A
) = maxj =01:::n;1 {(Aj) andr=r(A
) = min j=12:::n hj r 15{ :The real numberr(
A
) is the smallest radius. As a result of the algo-rithm we have either the inequality{(A
)1:15or the inequality {(A
) > { with the answer \the interval matrixA
is not practicalasymptotically stable".
6. Algorithm for a matrix with two interval entries
Let{ be a real number{ >1 and letA
= 8 > > > < > > > : a11b11] a12 ::: a1N a21b21] a22 ::: a2N ... ... ... ... aN1 aN2 ::: aNN 9 > > > = > > >be an interval matrix with two interval entries in position (1,1) and (2,1). Our aim is to nd the nite sequence of matrices. Analysis of matrices from the sequence allows us to guarantee that the initial interval matrix
A
is practical asymptotically stable or not practical asymptotically stable.Let Abe the N N matrix with the entriesaij. Then
A
=A+ 0b11 ;a 11] E 11+ 0b21 ;a 21] E 21withEj1the matrix de ned in the proof of Theorem 2. The algorithm is as follows
1.
We assume thatA
0 =A+ 0b11 ;a 11] E 11and= max( maxij=12:::N
fjaijjgjb 11 jjb 21 j) ({ ) 3=2
and apply the algorithm de ned in section 5 to the interval matrix
A
0 with only one interval entry. By this way, we can compute the real numbers 1 =(A
0) = max j=01:::n;1 {(Aj) and r 1 =r(A
0) = min j=12:::n hj:By the de nition, we have 1 { and r 1 15{ : If the inequalitya21+r1 b
21holds then we stop the process with the answer{(
A
)<1:151. Otherwise, we go to Step 2.
2.
We assume thatA
1=A
0+r1E21and apply the algorithm de ned in section 5 to the interval matrixA
1 with only one interval entry. By this way, we can compute the real numbers 2 = (A
1) andr2 =r(
A
1). By the de nition, we have2 { and r 2 15{ : If the inequalitya21+r1+r2 b
21 holds then we stop the process with the answer {(
A
) < 1:15maxf12
g. Otherwise, we go to Step 3.
After (p;1) steps of the process the real numbers
12:::p;1 and r1r2:::rp;1 are computed and for j = 12:::p
;1 we have j { and rj 15{ :
p.
We assume thatA
p;1=A
0+(r1+r2+ +rp;1)E21and apply the algorithm de ned in section 5 to the interval matrix
A
p;1with only one interval entry. By this way, we can compute the real numbersp =(
A
p;1) and rp =r(A
p;1). By the de nition, we havep { and rp 15{ : If the inequalitya21+r1+r2+ +rp b
21holds then we stop the process with the answer{(
A
)<1:15maxf12:::p
g. Other-wise, we go to Stepp+ 1.
Since rj =(15{ ) for8j it follows that we will stop the algorithm after not more then M21 steps, where
M21= 15{ (b 21 ;a 21) M21 is an integer:
Remark 2.The initial data of the algorithm are an integer N, a real number { ,{ >1, and the interval matrix
A
=A+ 0b11 ;a 11] E 11+ 0b21 ;a 21] E 21:The algorithm has n stepsnM
21. We can compute the following real numbers =(
A
) = maxj =12:::n j and r=r(A
) = minj =12:::n rj r 15{ :The real number r is the smallest radius. As a result of the algo-rithm we have either the inequality{(
A
)1:15or the inequality {(A
) > { with the answer \the interval matrixA
is not practicalasymptotically stable".
7. Algorithm for a matrix with
m
interval entries
Let{ be a real number{ >1 and letAbe theNN matrix with the real entries aij. We de ne an interval matrix
A
with m intervalentries by the equality
A
=A+ 0b11 ;a 11] E 11 +0b21 ;a 21] E 21+ + 0bij;aij]Eijwith Eij the matrix introduced in section 2. The matrix
A
has minterval entries in position (1,1),(2,1),:::(i;1j), and (ij) m =
i+(j;1)N. We include real numbersb
11,b21,:::bi;1j, andbijin the initial data of the algorithm. In the sequel we also use the algorithm for checking practical asymptotic stability of an interval matrix with (m;1) interval entries. For m = 1 and m = 2 we have considered the algorithms of such kind in sections 5 and 6. The algorithm for
m3 is as follows.
1.
We assume thatA
0=A+ 0b11 ;a 11] E 11 +0b21 ;a 21] E 21+ + 0bi ;1j ;ai ;1j] Ei ;1j and = max( maxkq=12:::N fjaijjg max k=12:::m fjbk ;k=N]Nk=N]+1 jg) ({ ) 3=2and apply the algorithm to the interval matrix
A
0 with m ; 1 interval entries. By this way, we can compute the real numbers1 =(
A
0) andr1=r(A
0). By the de nition, we have1 { and r 1 15{ : If the inequalityaij+r1
bij holds then we stop the process with the answer{(
A
)<1:152.
We assume thatA
1 =A
0+r1Eij and apply the algorithm to the interval matrixA
1 with (m;1) interval entries. By this way, we can compute the real numbers 2 = (
A
1) and r2 = r(A
1). By the de nition, we have2 { and r 2 15{ : If the inequalityaij +r1+r2
bij holds then we stop the process with the answer {(
A
) < 1:15maxf12
g. Otherwise, we go to Step 3.
After (p;1) steps of the process the real numbers
12:::p;1 and r1r2:::rp;1 are computed and for j = 12:::p
;1 we have j { and rj 15{ :
p.
We assume thatA
p;1=A
0+(r1+r2+ +rp;1)Eij and apply the algorithm to the interval matrix
A
p;1with (m;1) interval entries. By this way, we can compute the real numbersp =(
A
p;1) andrp =r(
A
p;1). By the de nition, we havep { and rp
15{
:
If the inequalityaij+r1+r2+
+rp bij holds then we stop the process with the answer{(
A
)<1:15maxf12:::p
g. Other-wise, we go to Stepp+ 1.
Since rj =(15{ ) for8j it follows that we will stop the algorithm after not more then Mij steps, where
Mij = 15{ (bij;aij) Mij is an integer:
Remark 3.The initial data of the algorithm are an integer N, a real number { ,{ >1, and the interval matrix
A
=A+ 0b11 ;a 11] E 11 +0b21 ;a 21] E 21+ + 0bij;aij]EijThe algorithm has nsteps nMij. We can compute the following real numbers = maxj =12:::n j and r= minj =12:::n rj r 15{ :
The real number r is the smallest radius. As a result of the algo-rithm we have either the inequality{(
A
)1:15or the inequality {(A
) > { with the answer \the interval matrixA
is not practicalasymptotically stable".
8. Conclusion
We have suggested the computer algorithm for checking the practical asymptotic stability of an interval matrix.
References
1. Bulgakov, A.Ya. (1980): An eectively calculable parameter for the stability quality of systems of linear dierential equations with constant coecients, Siberian Math. J.21, 339{347.
2. Godunov, S.K. (1990): The problem of guaranteed precision, in: Numerical Methods of Linear Algebra, Amer. Math. Soc. Transl. (2)147, 65{73.
3. Bulgakov, A.Ya. and Godunov, S.K. (1985): Calculation of positive denite solutions of Lyapunov's equation, in: Numerical methods of linear algebra in Russian], Novosibirsk, Nauka, 17{38.
4. Bulgak(ov), A.Ya. (1995):Matrix Computations with Guaranteed Accuracy in Stability Theory, Selcuk University, Research Center of Applied Mathematics, Konya.
5. Godunov, S.K. (1998): Modern Aspects of Linear Algebra, Transl. of Math. Monographs,175,AMS, Providence.
6. Bulgak, H. (1999): Pseudoeigenvalues, Spectral Portrait of a matrix and their connections with dierent criteria of stability, in: Error Control and Adap-tivity in Scientic Computing, Bulgak, H. and Zenger, C. (Eds.), Kluwer Academic Publishers, 95{124.
