The author takes tbis opportunity to thank S. G. Krein for bis interest and help in this work.
LITERATURE
CITED
A. A. Dezin.
Izv. Akad. Nauk SSSR. Mat.. .;!l. 61-86 (1867).
G. B. Savchenko.
Diff. Uravn..
~. No. 3 (1973).
G. B. Savchenko.
in: A Collection
of Papers by Graduate Students
2.
3.
State
University
[in
Voronezh
STABILITY
ASYMPTOTIC
OF A FAMILY
L
Russian].
OF
OF
SYSTEMS
(1972).
AN
OF
pp.
'
.,.
of the Mathematics
P.OSITION
DIFFERENTIAL
EQUATIONS
Kharitonov
For the trivial
,
FacultyofVoronezh
44-48.
EQUILIBRIUM
LINEAR
2, {. 0..
UDC 517.925.31
solution of a linear-differential-equation
system
x
= Ax.
with constant coefficients to be asymptotically stable, where A is a real n x n matrix, it is necessary aro
sufficient that all zeros of fue characteristic polynomial be in the left half-plane [1, 2].
(1)
If the elements of A are known only approximately, the question of the asymptotic stability of the solution
x :: Orequires the investigation of the family of polynomials
f(z)=zn+alz"-l+..;+an.
I?\
where
Probiem.
Find conditions
Qj~[lXj.llj)(lXj~lIj).j=I.2."...n.
under which all zeros of all polynomials
(3)
(2) and (3) have negative real parts.
This problem was first posed in [3]. Let Gn be the set of all polynomials of the form (2) whose zeros are
in the left hall-planeo let Snbe the family of polynomials (2), (3), and let Sr be the family of the 2n polynomials
in Sn in which each coefficient 4i is equal to eith~r °i of {li.
.THEOREM 1. In order that Sn C Gn, it is necessary aoo sufficient
! The necessity
is obvious because sr
To prove sufficiency
that Sr "C Gn.
C Sn.
we use a lemma.
Q2A+J
Consider the family
~ [a2k+1
1124+11,k;"
of polynomials
0,1,
(2), (3) such that
n-1
2
t (z) =" h (Z") -1-zg (Z2).
The polynomials
h(A) and g(A) are of degree
m if n
1,
(4)
= 2m + 1, and h(A) is of degree m and g(A) is of degree m-
1 if n :: 2m. For f(z) ~ Sn and n = 2m the polynonúal h(A) is fixed, and for n = 2m + 1 the polynomial
fLxed. The fanúly Sr consists of 2( (n+¡) /2] polynomials.
LEMMA. In arder that Sn C Gn, it is sufficient that
Proaf. Suppose that n
= 2m.
The Hermite-Bieler
Spc
g(A) is
Gn.
theorem [4] implies
that,
in arder that f(z) c Gn, it
is nec~
and sufficient that h(A) and g(A) form a positive pair; i.e., the zeros ofthese polynomials must
~ distinct, real, negative, and interlaced as follows:
UI < VI < U. < 11.<
< Um-l < /1m< O;
.A. A. Zhdanov Leningrad State University. Translated from Differentsial'nye Uravneniya, Vol. 14, No.
11,Pp. 2086-2088, November, 1978. Original article submitted November lO, 1975.
0012-2661/78/1411-
1483 $07 .50 ~ 1979 Plenwn Publishing
Corporation
1483
here fue ui are fue zeros of h(A) and fue vi are the zeros of g(A). It follows from a condition of the lemma that
h(A) forms a positive pair with all g(A) corresponding to the polynomials of S? Denote these polynomials by
gl(A), . . . , g m(A)':AIl f(z) = h(z2)+ zgi(z2) (i = 1,2, . . . ,2m) are, by condition, Hurwitz polynomials; hence
all coefficien1s of all polynomials gi (A) are positive. If the absolute value of A is large enough, sign (gi (A» =
sign (f-1)m-l) for i = 1, 2, . . . , 2m.
~
AlI the gi(A) change sign for the first time in the interval (uI' u2)¡ i.e., sign(gi(u2») = sign(r1)m-2¡
(i =
sign(gi(uk)
= sign«-l)m-k)
(i = 1,2,. . ., 2m). Consider an arbitrary polynomial
in Sn; the corresponding polynomial g(A) clearly changes sign in each interval (uk, Uk+.;); i.e., it has zeros in
this intervalo Hence g(A) forms a positive pair with h(A), and thus each polynomial in Sn aro algo in Gn. This
1,2-, . . ., 2m); similarly,
proves the theorem for n = 2m; the proof is similar for n = 2m + 1.
Remark. It is known [5] that, in arder that f(z) ~ Gn when al > O, it is necessary aro sufficient that the
(n
- l)-st
degree polynomial
(-I)"zf(-
cr(z)
z) -(z-2011
f (z)
zn-¡
2 ai
b,zn-
bn -1'
(5)
where
even.
(6)
b¡
i ood
be in Gn-1.
We now prove the sufficiency in Theorem 1, applying mathematical induction on n. Suppose that Theorem
1 holds for all families of the forms (2) and (3) when the degrees of the polynomials are lower than n. We sQlill
prove the theórem for families of polynomials of degr~e n. We note that, for n = 1, 2, 3, the assertion of the
theorem is obvious. The lemma implies that, in order that a function f(z) of Sn belong to Gn, it is sufficient
that the correspondingpolynomials of Sr be in Gn. We include f(z) ~ Sr in the family Sn of polynomials such
that a2k ~ [Q2k' fJ2k], k= 1,2,...,
[(n-1)!2].
Corresponding to each ¡Jolynomial in Sn, cor.struct cp(z)l?Yusing (5); this yields a family Sn-1 oí polynomials whose coefficients
bi ~ ['Yí' Ói] (i
'\'~+1
=
= 1,2,
ala~+. -
. . ., n -
a~+a
1), where
al~tIr+t
2.
6~~1=
a,
-'- at~.3
a2
. 01/1""1
. VI/I= 6~= ~.
I
.
By virtue of the induction hypothesis that sn-t c: Gn-t. itis sufficient that sr:"t c: Gn-t. But this means that.
in order that Sn c: G~. it is sufficient that Sr c: Gn. We form sr in the same fashion as Sr. This reasoning
holds for each f(z) 6 sr; hence. for f(z) E: Sn to be in Gn. it is sufficient that Sp c: Gn. This completes the
proof of Theorem 1.
THEOREM 2. In order that S? c: Gn. it is necessary and sufficient that the following tour polynomials of
Sr be in Gn:
ft(z)
On-'"-Dn-lA-l
Un-lA
f~"-2"okeveno
.tan-!'"
k odd,
{~n
k
an-sk-1.k
--
-lit
-lo
even,
odd,
= {~II_"hok even,
,," -sito k odd.
Un-P-l
=
1~n_!"_lo
n-lIl-lo
Qn-tR
k even,
k
odd.
= J~n-th. k even,
t..n-2/¡.kodd.
Qn-tR-l= {lIn_t/¡_l. k even.
~n-2/¡-1'
1.4R4
k odd.
f4(z)
Qn-M
=
{~n_I~'
k
even,
odd.
an-I~.k
Qn-M-1
= {~-I~_I'
k eve~
"n-I~-I.
k odd.
Again the necessity is evident; we will prove the sufficiency. Supposethat n = 2m (the proof is similar
for n = 2m + 1); we use the same type of reasoning as employed in the proof of the lemma. From the gi(A) we
select two: g(A) with the coefficients
and g(A) with the coefficients
It is easily seen that the graph of each g(;\) such that a2k+l ~ [a2k+l' {32k+l] is between the graphs of g(;\) and
g(;\) for ;\ < o; hence. if g(;\) and g(;\) forro positive pairs with h(A). then f(z) = h(z2) + zg(z2) is in Gn. Consider
the following polynomials: h(A) with coefficients
a.\
=
{~".
Qik
=
{~Ih.
k even.
r%JIt.k odd.
k
1'1110 k
even,
od~o
..
and h(A) with coefficients
-
-
;.
-
If h(A) and h(A) forro positive pairs with g(A) and g(A), then each h(X) aro g(A) with coefficients satisfying (3)
forro a positive pair, that is, Sr c: Gn. Theorem 2 follows.
Now consider the original
problem.
Let the matrix A c: U if aij ~ [aij'
f3ij] and aij :S f3ij (i
= 1,2,
..
n), and let f(z) ~ W if i(z) is the characteristio polynomial of Borne matrix of U. Put
a¡=min(a¡}.p¡=max:a¡1
w
w
(i=I,2,
. n)
We have the following resulto
THEOREM 3. For all systems (1) with A ~ U to have the asymptotically stable equilibrium positions
x = O, it is sufficient that the four polynomials described in Theorem 2 be in Gn.
Remark.
Theorems
1 and 2 remain in force for the family
llz) - a,oZ"+alZn-l:-..
where ai ~ [ai'
13i1. ai :5 13i. i
1
.;-a",
o, 1,
LITERATURE
o)
of polynomiaIs
CITED
A. M. Lyapunov, The General Problem of Motion Stability 1in Russian], Gostekhizdat, Moscow (1950).
V. l. Zubov, Mathematical Methods of Investigating Automatic-Control Systems [in Russian], Sudpromgiz, Leningrad (1959).
S. Faedo, Ano. Scuola Norm. Super., Pisa Sci. Fis. Math., 1, Nos. 1 and 2 (1953).
F. R. Gantmakher, Theory of Matrices, Chelsea Publ.
N. G. Chetaev, Motion Stability [in Russian], Nauka, Moscow (1965).