A Mathematical Journal
Vol. 6, No 4, (209 - 257). December 2004.

Matrix Liapunov’s Functions Method
and Stability Analysis of Continuous

Systems

A.A. Martynyuk
Stability of Processes Departament of Mechanics

S.P. Timoshenko Institute
Nesterov str.3, 03057, Kiev-57. Ukraine

anmart@stability.kiev.ua

1 Introduction

This paper contains main results of qualitative analysis of motions for large scale
dynamical systems described by ordinary differential equations in terms of matrix-
valued Liapunov’s functions.

The paper is arranged as folows.
Section 2 deals with stability problems for continuous large scale dynamical sys-

tems. The definitions and sufficient conditions for various types of motion stability of
nonautonomous and nonlinear systems are presented. The main theorems of the Sec-
tion are supplied with corollaries which illustrate the generality of the results obtained
and indicate the sources for the assertions.

In Section 3 general theorems of Section 2 are suplied with a constructive algorithm
of constructing the Liapunov functions in terms of matrix-valued auxiliary function.
The conditions for various types of stability of zero solution for a wide class of large-
scale systems are formulated in terms of the property of having a fixed sign of special
matrices.

Section 4 sets out conditions of exponential stability with respect to a part of vari-
ables. These conditions are established in terms of matrix-valued function constructed
by the method proposed in Section 3.

Liapunov functions for linear nonautonomous and autonomous systems in Sec-
tion 5 are constructed by adapting general algorithm from Section 3.



210 A.A. Martynyuk
6, 4(2004)

Section 6 presents a discussion of the algorithm and a numerical example which
demonstrate the efficiency of the proposed method of constructing the Liapunov’s
functions in terms of matrix-valued auxiliary function as compared with the Bellman–
Bailey approach based on the vector Liapunov’s functions.

In final Section 7 some unsolved problems of the method of matrix Liapunov’s
functions are presented.

Thus, this paper provides a development of the direct Liapunov method consisting
in both the establishment of general theorems and proposition of a new method of
constructing of appropriate Liapunov functions for some classes of linear and nonlinear
dynamical systems.

2 The Direct Liapunov’s Method via Matrix-Valued
Functions

In this Section the notions of motion stability corresponding to the motion properties
of nonautonomous systems are presented being necessary in subsequent presentation.
Basic notions of the method of matrix-valued Liapunov functions are discussed and
general theorems and some corollaries are set out.

Throughout this Section, real systems of ordinary differential equations will be
considered. Notations will be used.

2.1 Stability concept in the sense of Liapunov

We consider systems which can appropriately be described by ordinary differential
equations of the form

dyi
dt

= Yi(t, y1, . . . ,yn), i = 1, 2, . . . ,n, (2.1)

or in the equivalent vector form

dy

dt
= Y (t,y), (2.2)

where x ∈ Rn, Y (t,y) = (Y1(t,y), . . . ,Yn(t,y))T, Y T × Rn → Rn. In the present
Section we will assume that the right-hand part of ( 2.2) satisfies the solution existence
and uniqueness conditions of the Cauchy problem

dy

dt
= Y (t,y), y(t0) = y0, (2.3)

for any (t0,y0) ∈ T × Ω, where Ω ⊂ Rn, 0 ∈ Ω and Ω is an open connected subset
of R..

It is clear that the solution of problem ( 2.3) may not exist on R (on R+), even
if the right-hand part Y (t,y) of system ( 2.3) is definite and continuous for all
(t,y) ∈ T × Rn.



6, 4(2004)
Matrix Liapunov’s Functions Method ... 211

For example, the Cauchy problem for the equation

dy

dt
= 1 + y2, y(0) = 0, (2.4)

where y is a scalar, has a unique solution y(t) = tg t, existing on the interval (−π
2
, π

2
),

only, while the right-hand part of equation ( 2.4) is definite on the whole plane (t,y).
Let y(t) = ψ(t; t0,y0) be the solution of system ( 2.2), definite on the interval

[t0,τ) and noncontinuable behind the point τ, i.e˙ y(t) is not definite for t = τ. Then

lim ‖y(t)‖ = +∞ as t → τ − 0. (2.5)

Using solution y(t) and the right-hand part of system ( 2.2) we construct the vector-
function

f(t,x) = Y (t, x + ψ(t)) − Y (t,ψ(t)) (2.6)

and consider the system
dx

dt
= f(t,x). (2.7)

It is easy to verify that the solutions of systems ( 2.2) and ( 2.7) are correlated as

x(t) = y(t) − ψ(t)

on the general interval of existence of solutions y(t) and ψ(t). It is clear that system
( 2.7) has a trivial solution x(t) ≡ 0. This solution corresponds to the solution
y(t) = ψ(t) of system ( 2.2). Obviously, the reduction of system ( 2.2) to system
(2.7) is possible only when the solution y(t) = ψ(t) is known.

Qualitative investigation of solutions of system ( 2.2) relatively solution ψ(t) is
reduced to the investigation of behaviour of solution x(t) to system ( 2.7) which
differs ”little” from the trivial one for t = t0 .

In case when stability of unperturbed motion is discussed with respect to some
continuously differentiable functions Qs(t,ψ1, . . . ,ψn) the perturbed motion equa-
tions are found by the system of equations

dxi
dt

=
∂Qi
∂t

+
n∑

s=1

∂Qi
∂ψs

Ys(t,ψ1, . . . ,ψn) − Ḟ(t),

where xi = Qi(t,ψ1(t), . . . ,ψn(t)) − Fs(t), and Fi(t) = Qi(t, ψ1(t), . . . , ψn(t)) are
some known time functions. The system of equations of ( 2.7) type obtained hereat
satisfies the condition f(t, 0) = 0 for all t ∈ T and therefore these system has a
trivial solution in this case as well.

In motion stability theory system ( 2.7) is called the system of perturbed motion
equations.

Since equations ( 2.7) can generally not be solved analytically in closed from, the
qualitative properties of the equilbrium state are of great practical interest. We begin
with a series of definitions.



212 A.A. Martynyuk
6, 4(2004)

A very large number of definitions of stability exist for the system ( 2.7). Of
course, the various definitions of stability can be broadly classified as those which
deal with the trajectory, or a motion, or the equilibrium of the null solution of free or
unforced systems and those which consider the dynamic response of systems subject
to various classes of forcing functions or inputs. In the following, the equilibrium
state of ( 2.7) can always be set equal to zero by a linear state transformation, so
that the equilibrium state and the null solution to ( 2.7) are considered throughout
as equivalent.

Definition 2.1 The equilibrium state x = 0 of the system ( 2.7) is:

(i) stable iff for every t0 ∈ Ti and every ε > 0 there exists δ(t0,ε) > 0, such that
‖x0‖ < δ(t0,ε) implies

‖x(t; t0,x0)‖ < ε for all t ∈ T0;

(ii) uniformly stable iff both (i) holds and for every ε > 0 the corresponding maximal
δM obeying (i) satisfies

inf [δM (t,ε) t ∈ Ti] > 0;

(iii) stable in the whole iff both (i) holds and

δM (t,ε) → +∞ as ε → +∞ for all t ∈ R;

(iv) uniformly stable in the whole iff both (ii) and (iii) hold;

(v) unstable iff there are t0 ∈ Ti, ε ∈ (0, +∞) and τ ∈ T0, τ > t0, such that for
every δ ∈ (0, +∞) there is x0, ‖x0‖ < δ, for which

‖x(τ; t0,x0)‖ ≥ ε.

Definition 2.2 The equilibrium state x = 0 of the system ( 2.7) is:

(i) attractive iff for every t0 ∈ Ti there exists ∆(t0) > 0 and for every ζ > 0 there
exists τ(t0; x0,ζ) ∈ [0, +∞) such that ‖x0‖ < ∆(t0) implies ‖x(t; t0,x0)‖ < ζ
for all t ∈ (t0 + τ(t0; x0,ζ), +∞);

(ii) x0-uniformly attractive iff both (i) is true and for every t0 ∈ R there
exists ∆(t0) > 0 and for every ζ ∈ (0, +∞) there exists τu[t0, ∆(t0),ζ] ∈
[0, +∞) such that

sup [τm(t0; x0,ζ) x0 ∈ B∆(t0)] = τu(t0,x0,ζ);

(iii) t0-uniformly attractive iff both (i) is true, there is ∆ > 0 and for every (x0,ζ) ∈
B∆ × (0, +∞) there exists τu(R,x0,ζ) ∈ [0, +∞) such that

sup [τm(t0); x0,ζ) t0 ∈ Ti] = τu(Ti,x0,ζ);



6, 4(2004)
Matrix Liapunov’s Functions Method ... 213

(iv) uniformly attractive iff both (ii) and (iii) hold, that is, that (i) is true, there
exists ∆ > 0 and for every ζ ∈ (0, +∞) there is τu(R, ∆,ζ) ∈ [0, +∞) such
that

sup [τm(t0; x0,ζ) (t0,x0) ∈ Ti × B∆] = τ(Ti, ∆,ζ);

The properties (i) – (iv) hold “in the whole” iff (i) is true for every ∆(t0) ∈
(0, +∞) and every t0 ∈ Ti.

Definition 2.3 The equilibrium state x = 0 of the system ( 2.7) is:

(i) asymptotically stable iff it is both stable and attractive;

(ii) equi-asymptotically stable iff it is both stable and x0-uniformly attractive;

(iii) quasi-uniformly asymptotically stable iff it is both uniformly stable and t0-
uniformly attractive;

(iv) uniformly asymptotically stable iff it is both uniformly stable and uniformly
attractive;

(v) The properties (i) – (iv) hold “in the whole” iff both the corresponding stability
of x = 0 and the corresponding attraction of x = 0 hold in the whole;

(vi) exponentially stable iff there are ∆ > 0 and real numbers α ≥ 1 and β > 0
such that ‖x0‖ < ∆ implies

‖x(t; t0,x0)‖ ≤ α‖x0‖ exp[−β(t − t0)], for all t ∈ T0, for all t0 ∈ Ti.

This holds in the whole iff it is true for ∆ = +∞.

Let g : Rn → Rn define the time invariant system

dx

dt
= g(x), (2.8)

where g(0) = 0 and the components of g are smooth functions of the components of
x for x near zero. Every stability property of x = 0 of ( 2.11) is uniform in t0 ∈ R.

Note that the nonperturbed motion equations of the time invariant system can be
reduced to the time invariant system ( 2.11) iff the solution ψ(t) = const. Otherwise,
i.ei̇f ψ(t) 6= const , equations ( 2.11) can be nonstationary.

In the investigation of both system ( 2.2) and ( 2.11) the solution x(t) is assumed
to be definite for all t ∈ T (for all t ∈ T0).

2.2 Classes of Liapunov’s functions

Presently the Liapunov direct method (see Liapunov [1]) in terms of three classes of
auxiliary functions: scalar, vector and matrix ones is intensively applied in qualitative
theory. In this point we shall present the description of the above mentioned classes
of functions.



214 A.A. Martynyuk
6, 4(2004)

2.2.1 Matrix-valued Liapunov function

For the system ( 2.7) we shall consider a continuous matrix-valued function

U(t,x) = [vij (t,x)], i,j = 1, 2, . . . ,m, (2.9)

where vij ∈ C(Tτ ×Rn,R) for all i,j = 1, 2, . . . ,m. We assume that next conditions
are fulfilled

(i) vij (t,x), i,j = 1, 2, . . . ,m, are locally Lipschitzian in x;

(ii) vij (t, 0) = 0 for all t ∈ R+ (t ∈ Tτ ), i,j = 1, 2, . . . ,m;

(iii) vij (t,x) = vji(t,x) in any open connected neighbourhood N of point x = 0 for
all t ∈ R+ (t ∈ Tτ ).

Definition 2.4 All function of the type

v(t,x,α) = αTU(t,x)α, α ∈ Rm, (2.10)

where U ∈ C(Tτ × N , Rm×m), are attributed to the class SL.

Here the vector α can be specified as follows:

(i) α = y ∈ Rm, y 6= 0;

(ii) α = ξ ∈ C(Rn, Rm+ ), ξ(0) = 0;

(iii) α = ψ ∈ C(Tτ × Rn, Rm+ ), ψ(t, 0) = 0;

(iv) α = η ∈ Rm+ , η > 0.

Note that the choice of vector α can influence the property of having a fixed sign of
function ( 2.13) and its total derivative along solutions of system ( 2.7).

2.2.2 Comparison functions

Comparison functions are used as upper or lower estimates of the function v and its
total time derivative. They are usually denoted by ϕ, ϕ : R+ → R+. The main
contributor to the investigation of properties of and use of the comparison functions
is Hahn [2]. What follows is mainly based on his definitions and results.

Definition 2.5 A function ϕ, ϕ : R+ → R+, belongs to

(i) the class K[0,α), 0 < α ≤ +∞, iff both it is defined, continuous and strictly
increasing on [0,α) and ϕ(0) = 0;

(ii) the class K iff (i) holds for α = +∞, K = K[0,+∞);

(iii) the class KR iff both it belongs to the class K and ϕ(ζ) → +∞
as ζ → +∞;



6, 4(2004)
Matrix Liapunov’s Functions Method ... 215

(iv) the class L[0,α) iff both it is defined, continuous and strictly decreasing on [0,α)
and lim [ϕ(ζ) : ζ → +∞] = 0;

(v) the class L iff (iv) holds for α = +∞, L = L[0,+∞).

Let ϕ−1 denote the inverse function of ϕ, ϕ−1[ϕ(ζ)] ≡ ζ.
The next result was established by Hahn [2].

Proposition 2.1

1. If ϕ ∈ K and ψ ∈ K then ϕ(ψ) ∈ K;

2. If ϕ ∈ K and σ ∈ L then ϕ(σ) ∈ L;

3. If ϕ ∈ K[0,α) and ϕ(α) = ξ then ϕ−1 ∈ K[0,ξ);

4. If ϕ ∈ K and lim [ϕ(ζ) : ζ → +∞] = ξ then ϕ−1 is not defined on (ξ, +∞];

5. If ϕ ∈ K[0,α), ψ ∈ K[0,α) and ϕ(ζ) > ψ(ζ) on [0,α) then ϕ−1(ζ) < ψ−1(ζ)
on [0,β], where β = ψ(α).

Definition 2.6 A function ϕ, ϕ : R+ × R+ → R+, belongs to:

(i) the class KK[0;α,β) iff both ϕ(0,ζ) ∈ K[0,α) for every ζ ∈ [0,β) and ϕ(ζ, 0) ∈
K[0,β) for every ζ ∈ [0,α);

(ii) the class KK iff (i) holds for α = β = +∞;

(iii) the class KL[0;α,β) iff both ϕ(0,ζ) ∈ K[0,α) for every ζ ∈ [0,β) and ϕ(ζ, 0) ∈
L[0,β) for every ζ ∈ [0,α);

(iv) the class KL iff (iii) holds for α = β = +∞;

(v) the class CK iff ϕ(t, 0) = 0, ϕ(t,u) ∈ K for every t ∈ R+;

(vi) the class M iff ϕ ∈ C(R+ × Rn,R+), inf ϕ(t,x) = 0, (t,x) ∈ R+ × Rn;

(vii) the class M0 iff ϕ ∈ C(R+ × Rn,R+), inf
x
ϕ(t,x) = 0 for

each t ∈ R+;

(viii) the class Φ iff ϕ ∈ C(K,R+): ϕ(0) = 0, and ϕ(w) is increasing with respect to
cone K.

Definition 2.7 Two functions ϕ1, ϕ2 ∈ K (or ϕ1, ϕ2 ∈ KR) are said to be of the
same order of magnitude if there exist positive constants α, β, such that

αϕ1(ζ) ≤ ϕ2(ζ) ≤ βϕ1(ζ) for all ζ ∈ [0,ζ1] (or for all ζ ∈ [0,∞)).



216 A.A. Martynyuk
6, 4(2004)

2.2.3 Properties of matrix-valued functions

For the functions of the class SL we shall cite some definitions which are applied in
the investigation of dynamics of system ( 2.7).

Definition 2.8 The matrix-valued function U : Tτ × Rn → Rm×m is:

(i) positive semi-definite on Tτ = [τ, +∞), τ ∈ R, iff there are time-invariant
connected neighbourhood N of x = 0, N ⊆ Rn, and vector y ∈ Rm, y 6= 0,
such that

(a) v(t,x,y) is continuous in (t,x) ∈ Tτ × N × Rm;
(b) v(t,x,y) is non-negative on N , v(t,x,y) ≥ 0 for all (t,x,y 6= 0) ∈

Tτ × N × Rm, and
(c) vanishes at the origin: v(t, 0,y) = 0 for all t ∈ Tτ × Rm;
(d) iff the conditions (a) – (c) hold and for every t ∈ Tτ , there is w ∈ N such

that v(t,w,y) > 0, then v is strictly positive semi-definite on Tτ ;

(ii) positive semi-definite on Tτ × G iff (i) holds for N = G;

(iii) positive semi-definite in the whole on Tτ iff (i) holds for N = Rn;

(iv) negative semi-definite (in the whole) on Tτ (on Tτ × N) iff (−v) is positive
semi-definite (in the whole) on Tτ (on Tτ × N) respectively.

The expression “on Tτ ” is omitted iff all corresponding requirements hold for
every τ ∈ R.

Definition 2.9 The matrix-valued function U : Tτ × Rn → Rm×m is:

(i) positive definite on Tτ , τ ∈ R, iff there are a time-invariant connected neigh-
bourhood N of x = 0, N ⊆ Rn and a vector y ∈ Rm, y 6= 0, such that both it
is positive semi-definite on Tτ ×N and there exists a positive definite function w
on N , w : Rn → R+, obeying w(x) ≤ v(t,x,y) for all (t,x,y) ∈ Tτ ×N ×Rm;

(ii) positive definite on Tτ × G iff (i) holds for N = G;

(iii) positive definite in the whole on Tτ iff (i) holds for N = Rn;

(iv) negative definite (in the whole) on Tτ (on Tτ × N × Rm) iff (−v) is positive
definite (in the whole) on Tτ (on Tτ × N × Rm) respectively;

(v) weakly decrescent if there exists a ∆1 > 0 and a function a ∈ CK such that
v(t,x,y) ≤ a(t,‖x‖) as soon as ‖x‖ < ∆1;

(vi) asymptotically decrescent if there exists a ∆2 > 0 and a function b ∈ KL such
that v(t,x,y) ≤ b(t,‖x‖) as soon as ‖x‖ < ∆2.

The expression “on Tτ ” is omitted iff all corresponding requirements hold for
every τ ∈ R.



6, 4(2004)
Matrix Liapunov’s Functions Method ... 217

Proposition 2.2 The matrix-valued function U : R × Rn → Rm×m is positive
definite on Tτ , τ ∈ R, iff it can be written as

yTU(t,x)y = yTU+(t,x)y + a(‖x‖),

where U+(t,x) is a positive semi-definite matrix-valued function and a ∈ K.

Definition 2.10 (cf Grujić, et al. [1]) Set vζ (t) is the largest connected neighborhood
of x = 0 at t ∈ R which can be associated with a function U : R × Rn → Rm×m
so that x ∈ vζ (t) implies v(t,x,y) < ζ, y ∈ Rm.

Definition 2.11 The matrix-valued function U : R × Rn → Rs×s is:

(i) decreasing on Tτ , τ ∈ R, iff there is a time-invariant neighborhood N of
x = 0 and a positive definite function w on N , w : Rn → R+, such that
yTU(t,x)y ≤ w(x) for all (t,x) ∈ Tτ × N ;

(ii) decreasing on Tτ × G iff (i) holds for N = G;

(iii) decreasing in the whole on Tτ iff (i) holds for N = Rn.

The expression “on Tτ ” is omitted iff all corresponding conditions still hold for
every τ ∈ R.

Proposition 2.3 The matrix-valued function U : R × Rn → Rm×m is decreasing
on Tτ , τ ∈ R, iff it can be written as

yTU(t,x)y = yTU−(t,x)y + b(‖x‖), (y 6= 0) ∈ Rm,

where U−(t,x) is a negative semi-definite matrix-valued function and b ∈ K.

Definition 2.12 The matrix-valued function U : R × Rn → Rm×m is:

(i) radially unbounded on Tτ , τ ∈ R, iff ‖x‖→∞ implies yTU(t,x)y → +∞ for
all t ∈ Tτ , y ∈ Rm, y 6= 0;

(ii) radially unbounded, iff ‖x‖ → ∞ implies yTU(t,x)y → +∞ for all t ∈ Tτ for
all τ ∈ R, y ∈ Rm, y 6= 0.

Proposition 2.4 The matrix-valued function U : Tτ × Rn → Rm×m is radially
unbounded in the whole (on Tτ ) iff it can be written as

yTU(t,x)y = yTU+(t,x)y + a(‖x‖) for all x ∈ Rn,

where U+(t,x) is a positive semi-definite matrix-valued function in the whole (on Tτ )
and a ∈ KR.

According to Liapunov function ( 2.17) is applied in motion investigation of system
( 2.7) together with its total derivative along solutions x(t) = x(t; t0,x0) of system



218 A.A. Martynyuk
6, 4(2004)

( 2.7). Assume that each element vij (t,x) of the matrix-valued function ( 2.16) is
definite on the open set Tτ × N , N ⊂ Rn, i.e. vij (t,x) ∈ C(Tτ × N , R).

If γ(t; t0,x0) is a solution of system ( 2.7) with the initial conditions x(t0) = x0,
i.e. γ(t0; t0,x0) = x0, the right-hand upper derivative of function ( 2.17) for α = y,
y ∈ Rm, with respect to t along the solution of ( 2.7) is determined by the formula

D+v(t,x,y) = yTD+U(t,x)y, (2.11)

where D+U(t,x) = [D+vij (t,x)], i,j = 1, 2, . . . ,m, and

D+vij (t,x) = lim sup
{

sup
γ(t,t,x)=x

[vij (t + σ, γ(t + σ, t,x))

− vij (t,x)] σ−1 : σ → 0+
}
, i,j = 1, 2, . . . ,m.

(2.12)

In case when system ( 2.7) has a unique solution for every initial value of x(t0) = x0
((t0,x0) ∈ Tτ × N), the expression ( 2.19) is equivalent to

D+vij (t,x) = lim sup{[vij (t + σ, γ(t + σ, t,x))
− vij (t,x)] σ−1 : σ → 0+}, i,j = 1, 2, . . . ,m.

(2.13)

Further we assume that for all i,j = 1, 2, . . . ,m the functions vij (t,x) are con-
tinuous and locally Lipschitzian in x, i.e. for every point in N there exists a neigh-
bourhood ∆ and a positive number L = L(∆ ) such that

|vij (t,x) − vij (t,y)| ≤ L‖x − y‖, i,j = 1, 2, . . . ,m,

for any (t,x) ∈ Tτ × ∆, (t,y) ∈ Tτ × ∆. Besides, the expression ( 2.12) is equivalent
to

D+vij (t,x) = lim sup {[vij (t + σ, x + σf(t,x))
− vij (t,x)] σ−1 : σ → 0+}, i,j = 1, 2, . . . ,m.

(2.14)

If the matrix-valued function U(t,x) ∈ C1(Tτ × N , Rm×m), i.eȧll its elements
vij (t,x) are functions continuously differentiable in t É x, then the expression ( 2.14)
is equivalent to

Dvij (t,x) =
∂vij
∂t

(t,x) +
n∑

s=1

∂vij
∂xs

(t,x) fs(t,x), (2.15)

where fs(t,x) are components of the vector-function f(t,x) = (f1(t,x), . . . ,fn(t,x))T.
Function ( 2.15) has the Euler derivative ( 2.10) at point (t,x) along solution

x(t; t0,x0) of system ( 2.7) iff

D+v(t,x,y) = D+v(t,x,y) = D
−v(t,x,y)

= D−v(t,x,y) = Dv(t,x,y).
(2.16)

Note that the application of any of the expressions ( 2.12), ( 2.13) or ( 2.15) in
( 2.11) is admissible.



6, 4(2004)
Matrix Liapunov’s Functions Method ... 219

2.2.4 Vector Liapunov function

A vector-valued Liapunov function

V (t,x) = (v1(t,x),v2(t,x), . . . ,vm(t,x))
T (2.17)

can be obtained via matrix-valued function ( 2.9) in several ways.

Definition 2.13 All vector functions of the type

L(t,x,b) = AU(t,x)b, (2.18)

where U ∈ C(Tτ × Rn, Rs×s), A is a constant matrix s × s, and vector b is defined
according to (i) – (iv) similarly to the definition of the vector α, are attributed to the
class VL.

If in two-index system of functions ( 2.9) for all i 6= j the elements vij (t,x) = 0,
then

v(t,x) = diag (v11(t,x), . . . , vmm(t,x))
T,

where vii ∈ C(Tτ × Rn, R), i = 1, 2, . . . ,m, , is a vector-valued function. Besides,
the function ( 2.18) has the components

Lk(t,x,b) =
m∑

i=1

akibivii(t,x), k = 1, 2, . . . ,m.

The methods of application of Liapunov’s vector functions in motion stability
theory are presented in a number of monographs some of which are mentioned in the
end of this section.

2.2.5 Scalar Liapunov function

The simplest type of auxiliary function for system ( 2.7) is the function

v(t,x) ∈ C(T0 × Rn, R+), v(t, 0) = 0, (2.19)

for which

(a) v(t, 0) = 0 for all t ∈ Tτ ;

(b) v(t,x) is locally Lipschitzian in x;

(c) v(t,x) ∈ C(Tτ × N , R).

In stability theory both sign-definite in the sense of Liapunov and semi-definite
functions (see Hahn [2]) are applied. We shall set out some examples.



220 A.A. Martynyuk
6, 4(2004)

Example 2.1

(i) The function
v(t,x) = (1 + sin2 t)x21 + (1 + cos

2 t)x22

is positive definite and decreasing, while the function

v(t,x) = (x21 + x
2
2) sin

2 t

is decreasing and positive semi-definite.

(ii) The function
v(t,x) = x21 + (1 + t)x

2
2

is positive definite but not non-decreasing, while the function

v(t,x) = x21 +
x22

1 + t

is decreasing but not positive definite.

(iii) The function
v(t,x) = (1 + t)(x1 − x2)2

is positive semi-definite and non-decreasing.

Among the variety of the Liapunov functions the quadratic forms

v(t,x) = xTP(t)x, P T(t) = P(t), (2.20)

are of special importance, where P(t) is n × n -matrix with continuous and bounded
elements for all t ∈ Tτ .

Proposition 2.5 For the quadratic form ( 2.20) to be positive definite it is necessary
and sufficient that∣∣∣∣∣∣

p11(t) . . . p1s(t)
. . . . . . . . . . . . . . . . . . .
ps1(t) . . . pss(t)

∣∣∣∣∣∣ > k > 0, s = 1, 2, . . . ,n, (2.21)
for all t ∈ Tτ .

Note that if conditions ( 2.21) are satisfied, the ”quasi-quadratic” form

v(t,x) = xTP(t)x + ψ(t,x), ψ(t, 0) = 0, (2.22)

is positive definite, provided that some constants a,b (a > 0, b ≥ 2) exist, such that
|ψ(t,x)| ≤ arb, where r = (xTx)1/2.

To calculate total derivative of function ( 2.22) along solutions of system ( 2.7)
either ( 2.12) – ( 2.15) is applied for i = j = 1, depending on the assumptions on
system ( 2.7) and function ( 2.22).



6, 4(2004)
Matrix Liapunov’s Functions Method ... 221

It is well known (see Yoshizawa [1]) that if D+v(t,x) ≤ 0 and consequently
D+v(t,x(t)) ≤ 0, then the function v(t,x) is nonincreasing function of t ∈ Tτ .
Further, if D+v(t,x) ≥ 0, then v(t,x(t)) is nondecreasing along any solution of (2.7)
and vice versa.

We shall formulate these observations as follows.

Proposition 2.6 Suppose m(t) = v(t,x(t)) is continuous on (a,b). Then m(t) is
nondecreasing (nonincreasing) on (a,b) if and only if D+m(t) ≥ 0 (≤ 0) for every
t ∈ (a,b), where

D+m(t) = lim sup {[m(t + σ) − m(t)] σ−1 : σ → 0+}.

Further all auxiliary functions allowing the solution of the problem on stability
(instability) of the equilibrium state x = 0 of system ( 2.7) are called the Liapunov
functions. The construction of the Liapunov functions still remains one of the central
problems of stability theory.

2.3 Liapunov’s like theorems in general

Functions ( 2.10), ( 2.18) and ( 2.20) together with their total derivatives ( 2.11)
along solutions of system ( 2.7) allow to establish existence conditions for the motion
properties of system ( 2.7) of various types such as stability, instability, boundedness,
etc. Below we shall set out some results in the direction.

Theorem 2.1 Let the vector-function f in system ( 2.7) be continuous on R × N
(on Tτ × N). If there exist

1. an open connected time-invariant neighborhood G ⊂ N of the point x = 0 ;

2. a matrix-valued function U ∈ C (R × N ,Rm×m) and a vector y ∈ Rm such
that the function v(t,x,y) = yTU(t,x)y is locally Lipschitzian in x for all t ∈ R
(t ∈ Tτ );

3. functions ψi1, ψi2, ψi3 ∈ K, ψ̃i2 ∈ CK, i = 1, 2, . . . ,m;

4. m × m matrices Aj (y), j = 1, 2, 3, Ã2(y) such that

(a) ψT1 (‖x‖)A1(y)ψ1(‖x‖) ≤ v(t,x,y) ≤ ψ̃T2 (t,‖x‖)Ã2(y)ψ̃2(t,‖x‖)
for all (t,x,y) ∈ R × G × Rm (for all (t,x,y) ∈ Tτ × G × Rm);

(b) ψT1 (‖x‖)A1(y)ψ1(‖x‖) ≤ v(t,x,y) ≤ ψT2 (‖x‖)A2(y)ψ2(‖x‖)
for all (t,x,y) ∈ R × G × Rm (for all (t,x,y) ∈ Tτ × G × Rm);

(c) D+v(t,x,y) ≤ ψT3 (‖x‖)A3(y)ψ3(‖x‖)
for all (t,x,y) ∈ R × G × Rm (for all (t,x,y) ∈ Tτ × G × Rm).

Then, if the matrices A1(y), A2(y), Ã2(y), (y 6= 0) ∈ Rm are positive definite and
A3(y) is negative semi-definite, then



222 A.A. Martynyuk
6, 4(2004)

(a) the state x = 0 of system ( 2.7) is stable (on Tτ ), provided condition (4)(a) is
satisfied;

(b) the state x = 0 of system ( 2.7) is uniformly stable (on Tτ ), provided condition
(4)(b) is satisfied.

Corollary 2.1 Let

1. condition (1) of Theorem 2.1 be satisfied;

2. there exist at least one couple of indices (p,q) ∈ [1,m] for which (vpq(t,x) 6=
0) ∈ U(t,x) and function v(t,x,e) = eTU(t,x)e = v(t,x) for all (t,x) ∈ R×G
(for all (t,x) ∈ Tτ × G) satisfy the conditions

(a) ψ1(‖x‖) ≤ v(t,x);
(b) v(t,x) ≤ ψ2(‖x‖);
(c) D+v(t,x)|(2.7) ≤ 0,

where ψ1,ψ2 are some functions of the class K.

Then, the state x = 0 of system ( 2.7) is stable (on Tτ ) under conditions (a) and
(c), and uniformly stable (on Tτ ) under conditions (a) – (c).

Theorem 2.2 Let the vector-function f in system ( 2.7) be continuous on R × Rn
(on Tτ × Rn). If there exist

1. a matrix-valued function U ∈ C (R × Rn,Rm×m) (U ∈ C(Tτ × Rn,Rm×m))
and a vector y ∈ Rm such that the function v(t,x,y) = yTU(t,x)y is locally
Lipschitzian in x for all t ∈ R (t ∈ Tτ );

2. functions ϕ1i, ϕ2i, ϕ3i ∈ KR, ϕ̃2i ∈ CKR, i = 1, 2, . . . ,m;

3. m × m matrices Bj (y), j = 1, 2, 3, B̃2(y) such that

(a) ϕT1 (‖x‖)B1(y)ϕ1(‖x‖) ≤ v(t,x,y) ≤ ϕ̃T2 (t,‖x‖)B̃2(y)ϕ̃2(t,‖x‖) for all
(t,x,y) ∈ R × Rn × Rm (for all (t,x,y) ∈ Tτ × Rn × Rm);

(b) ϕT1 (‖x‖)B1(y)ϕ1(‖x‖) ≤ v(t,x,y) ≤ ϕT2 (‖x‖)B2(y)ϕ2(‖x‖) for all (t,x,y) ∈
R × Rn × Rm (for all (t,x,y) ∈ Tτ × Rn × Rm);

(c) D+v(t,x,y) ≤ ϕT3 (‖x‖)B3(y)ϕ3(‖x‖) for all (t,x,y) ∈ R× Rn × Rm (for
all (t,x,y) ∈ Tτ × Rn × Rm).

Then, provided that matrices B1(y), B2(y) and B̃2(y) for all (y 6= 0) ∈ Rm are
positive definite and matrix B3(y) is negative semi-definite,

(a) under condition 3(a) the state x = 0 of system ( 2.7) is stable in the whole
(on Tτ );



6, 4(2004)
Matrix Liapunov’s Functions Method ... 223

(b) under condition 3(b) the state x = 0 of system ( 2.7) is uniformly stable in the
whole (on Tτ ).

Corollary 2.2 Let for function v(t,x,e) = v(t,x), mentioned in condition 2 of
Corollary 2.1 for all (t,x) ∈ R×Rn (for all (t,x) ∈ Tτ ×Rn) the following conditions
hold

(a) ϕ1(‖x‖) ≤ v(t,x);

(b) v(t,x) ≤ ϕ2(‖x‖), for some function ϕ2;

(c) D+v(t,x)|(2.7) ≤ 0,

where ϕ1, ϕ2 are of class KR.
Then the state x = 0 of system ( 2.7) is stable in the whole (on Tτ ) under

conditions (a) and (c) and uniformly stable in the whole (on Tτ ) under conditions
(a) – (c).

Theorem 2.3 Let the vector-function f in system ( 2.7) be continuous on R × N
(on Tτ × N). If there exist

1. an open connected time-invariant neighborhood G ⊂ N of the point x = 0 ;

2. a matrix-valued function U ∈ C (R × N , Rm×m) (U ∈ C(Tτ × N ,Rm×m))
and a vector y ∈ Rm such that the function v(t,x,y) = yTU(t,x)y is locally
Lipschitzian in x for all t ∈ R (t ∈ Tτ );

3. functions η1i, η2i, η3i ∈ K, η̃2i ∈ CK, i = 1, 2, . . . ,m;

4. m × m matrices Cj (y), j = 1, 2, 3, C̃2(y) such that

(a) ηT1 (‖x‖)C1(y)η1(‖x‖) ≤ v(t,x,y) ≤ η̃T2 (t,‖x‖)C̃2(y)η̃2(t,‖x‖) for all
(t,x,y) ∈ R × G × Rm (for all (t,x,y) ∈ Tτ × G × Rm);

(b) ηT1 (‖x‖)C1(y)η1(‖x‖) ≤ v(t,x,y) ≤ ηT2 (‖x‖)C2(y)η2(‖x‖) for all (t,x,y) ∈
R × G × Rm (for all (t,x,y) ∈ Tτ × G × Rm);

(c) D∗v(t,x,y) ≤ ηT3 (‖x‖)C3(y)η3(‖x‖) + m (t, η3(‖x‖)) for all (t,x,y) ∈
R×G×Rm (for all (t,x,y) ∈ Tτ ×G×Rm), where function m(t, ·) satisfies
the condition

lim
|m (t,η3(‖x‖)) |

‖η3‖
= 0 as ‖η3‖ → 0

uniformly in t ∈ R (t ∈ Tτ ).

Then, provided the matrices C1(y), C2(y), C̃2(y) are positive definite and matrix
C3(y) (y 6= 0) ∈ Rm is negative definite, then

(a) under condition 4(a) the state x = 0 of the system ( 2.7) is asymptotically stable
(on Tτ );



224 A.A. Martynyuk
6, 4(2004)

(b) under condition 4(b) the state x = 0 of the system ( 2.7) is uniformly asymptot-
ically stable (on Tτ ).

Corollary 2.3 Let

1. condition 1 of Theorem 2.2 be satisfied;

2. for function v(t,x,e) = v(t,x), mentioned in condition 2 of Corollary 2.1 for
all (t,x) ∈ R × G (for all (t,x) ∈ Tτ × G) )

(a) ψ1(‖x‖) ≤ v(t,x) ≤ ψ2(‖x‖);
(b) D+v(t,x)|(2.7) ≤ −ψ3(‖x‖), where ψ1,ψ2,ψ3 are of class K.

Then the state x = 0 of system ( 2.7) is uniformly asymptotically stable (on Tτ ).

Theorem 2.4 Let the vector-function f in system ( 2.7) be continuous on R × Rn
(on Tτ × Rn) and conditions 1 – 3 of Theorem 2.2 be satisfied.

Then, provided that matrices B1(y), B2(y) and B̃2(y) are positive definite and
matrix B3(y) for all (y 6= 0) ∈ Rm is negative definite,

(a) under condition 3(a) of Theorem 2.2 the state x = 0 of system ( 2.7) is asymp-
totically stable in the whole (on Tτ );

(b) under condition 3(b) of Theorem 2.2 the state x = 0 of system ( 2.7) is uniformly
asymptotically stable in the whole (on Tτ ).

Corollary 2.4 For function v(t,x,e) = v(t,x), mentioned in condition 2 of Corollary
2.1 for all (t,x) ∈ R × Rn (for all (t,x) ∈ Tτ × Rn) let

(a) ϕ1(‖x‖) ≤ v(t,x) ≤ ϕ2(‖x‖);

(b) D+v(t,x)|(2.7) ≤ −ψ3(‖x‖),

where ϕ1,ϕ2 are of class KR and ψ3 is of class K.
Then the state x = 0 of system ( 2.7) is uniformly stable in the whole (on Tτ ).

Theorem 2.5 Let the vector-function f in system ( 2.7) be continuous on R × N
(on Tτ × N). If there exist

1. an open connected time-invariant neighborhood G ⊂ N of the point x = 0 ;

2. a matrix-valued function U ∈ C (R × N , Rm×m) and a vector y ∈ Rm such
that the function v(t,x,y) = yTU(t,x)y is locally Lipschitzian in x for all t ∈ R
(t ∈ Tτ );

3. functions σ2i, σ3i ∈ K, i = 1, 2, . . . ,m, a positive real number ∆1 and positive
integer p, m × m matrices F2(y), F3(y) such that



6, 4(2004)
Matrix Liapunov’s Functions Method ... 225

(a) ∆1‖x‖p ≤ v(t,x,y) ≤ σT2 (‖x‖)F2(y)σ2(‖x‖) for all (t,x,y 6= 0) ∈ R × G ×
Rm (for all (t,x,y 6= 0) ∈ Tτ × G × Rm);

(b) D+v(t,x,y) ≤ σT3 (‖x‖)F3(y)σ3(‖x‖) for all (t,x,y 6= 0) ∈ R× G×Rm (for
all (t,x,y 6= 0) ∈ Tτ × G × Rm).

Then, provided that the matrices F2(y), (y 6= 0) ∈ Rm are positive definite, the
matrix F3(y), (y 6= 0) ∈ Rm is negative definite and functions σ2i, σ3i are of the
same magnitude, then the state x = 0 of system ( 2.7) is exponentially stable (on Tτ ).

Corollary 2.5 Let

1. condition (1) of Theorem 2.1 be satisfied;

2. for function v(t,x,e) = v(t,x), mentioned in condition (2) of Corollary 2.1 for
all (t,x) ∈ R × G (for all (t,x) ∈ Tτ × G)

(a) c1‖x‖p ≤ v(t,x) ≤ ϕ1(‖x‖),
(b) D+v(t,x)|(2.7) ≤ −ϕ2(‖x‖).

Then, if the functions ϕ1,ϕ2 are of class K and of the same magnitude, the state
x = 0 of system ( 2.7) is exponentially stable (on Tτ ).

Theorem 2.6 Let the vector-function f in system ( 2.7) be continuous on R × Rn
(on Tτ × Rn). If there exist

1. a matrix-valued function U ∈ C (R × Rn, Rm×m) (U ∈ C(Tτ × Rn,Rm×m))
and a vector y ∈ Rm such that the function v(t,x,y) = yTU(t,x)y is locally
Lipschitzian in x for all t ∈ R (for all t ∈ Tτ );

2. functions ν2i, ν3i ∈ KR, i = 1, 2, . . . ,m, a positive real number ∆2 > 0 and a
positive integer q;

3. m × m matrices H2, H3 such that

(a) ∆2‖x‖q ≤ v(t,x,y) ≤ νT2 (‖x‖)H2(y)ν2(‖x‖) for all (t,x,y 6= 0) ∈ R×Rn ×
Rm (for all (t,x,y) ∈ Tτ × Rn × Rm);

(b) D+v(t,x,y) ≤ νT3 (‖x‖)H3(y)ν3(‖x‖) for all (t,x,y 6= 0) ∈ R × Rn × Rm
(for all (t,x,y 6= 0) ∈ Tτ × Rn × Rm).

Then, if the matrix H2(y) for all (y 6= 0) ∈ Rm is positive definite, the matrix
H3(y) for all (y 6= 0) ∈ Rm is negative definite and functions ν2i, ν3i are of the
same magnitude, the state x = 0 of system ( 2.7) is exponentially stable in the whole
(on Tτ ).

Corollary 2.6 For function v(t,x,e) = v(t,x), mentioned in condition (2) of Corol-
lary 2.1 for all (t,x) ∈ R × Rn (for all (t,x) ∈ Rn × G) let

(a) c2‖x‖q ≤ v(t,x) ≤ ψ1(‖x‖),



226 A.A. Martynyuk
6, 4(2004)

(b) D+v(t,x)|(2.7) ≤ −ψ2(‖x‖),

where ψ1,ψ2 ∈ KR–class and are of the same magnitude.
Then the state x = 0 of system ( 2.7) is exponentially stable in the whole (on

Tτ ).

Theorem 2.7 Let the vector-function f in system ( 2.7) be continuous on R × N
(on Tτ × N). If there exist

1. an open connected time-invariant neighborhood G ⊂ N of the
point x = 0 ;

2. a matrix-valued function U ∈ C1 (R × N , Rm×m) (U ∈ C1(Tτ × N ,Rm×m))
and a vector y ∈ Rm;

3. functions ψ1i,ψ2i,ψ3i ∈ K, i = 1, 2, . . . ,m, m × m matrices A1(y), A2(y),
G(y) and a constant ∆ > 0 such that

(a) ψT1 (‖x‖)A1(y)ψ1(‖x‖) ≤ v(t,x,y) ≤ ψT2 (‖x‖)A2(y)ψ2(‖x‖) for all (t,x,y) ∈
R × G × Rm (for all (t,x,y) ∈ Tτ × G × Rm);

(b) D+v(t,x,y) ≥ ψT3 (‖x‖)G(y)ψ3(‖x‖) for all (t,x,y) ∈ R× G × Rm (for all
(t,x,y) ∈ Tτ × G × Rm);

4. point x = 0 belongs to ∂G;

5. v(t,x,y) = 0 on T0 × (∂G ∩ B∆), where B∆ = {x : ‖x‖ < ∆}.

Then, if matrices A1(y), A2(y) and G(y) for all (y 6= 0) ∈ Rm are positive
definite, the state x = 0 of system ( 2.7) is unstable (on Tτ ).

Corollary 2.7 Let

1. condition (1) of Theorem 2.7 be satisfied;

2. there exist at least one couple of indices (p,q) ∈ [1,m] such that (vpq(t,x) 6=
0) ∈ U(t,x) and a function v(t,x,e) = v(t,x) ∈ C1(R × B∆, R+), B∆ ⊂ G,
such that on T0 × G

(a) 0 < v(t,x) ≤ a < +∞, for some a > 0;
(b) D+v(t,x)|(2.7) ≥ ϕ(v(t,x)) for some function ϕ of class K;
(c) point x = 0 belongs to ∂G;
(d) v(t,x) = 0 on T0 × (∂G ∩ B∆).

Then the state x = 0 of the system ( 2.7) is unstable.
We shall pay our attention to some specific features of the functions applied in

Corollary 2.7.
Function v(t,x) specifies the domain v(t,x) > 0, which is changing for t ∈

Tτ . Clearly this domain may cease its existence before the instability of motion is
discovered.



6, 4(2004)
Matrix Liapunov’s Functions Method ... 227

If the function v(t,x) is positive definite (strictly positive semi-definite), then the
domain v(t,x) > 0 exists for all t ∈ Tτ .

If the function v(t,x) is constant negative, the domain v(t,x) > 0 does not exist.

Example 2.2

(i) Function
v(t,x) = sin tx1x2

is of variable sign and domain v(t,x) > 0 exists but not for all t ∈ Tτ .

(ii) For the function
v(t,x) = (cos t − 2) x21x2

the domain v(t,x) > 0 exists for all t ∈ Tτ .

(iii) For the function

v(t,x) =
(

1
t
− a

)
x1x2 − x22, a > 0,

the domain v(t,x) > 0 exists for all t ≥ t0, and for t0 > 1/a.

Corollary 2.8 Let condition (1) of Theorem 2.7 be satisfied. If there exist t0 ∈ T0,
∆ > 0, (B∆ ⊂ N) and an open set G ⊂ B∆ and the function v(t,x,e) = v(t,x) ∈
C1(T0 × B∆, R), mentioned in Corollary 2.7 such that on T0 × G

(a) 0 < v(t,x) ≤ ϕ1(‖x‖);

(b) D+v(t,x)|(2.7) ≥ ϕ2(‖x‖) for some ϕ1,ϕ2 of class K;

(c) point x = 0 belongs to ∂G;

(d) v(t,x) = 0 on T0 × (∂G ∩ B∆).

Then the state x = 0 of ( 2.7) is unstable.

Corollary 2.9 If in Corollary 2.8 condition (b) is replaced by
(b′) D+v(t,x)|(2.7) ≥ kv(t,x) + w(t,x)

on T0 ×G, where k > 0 and function w(t,x) ≥ 0 is continuous on T0 ×G, then the

state x = 0 of system ( 2.7) is unstable.

3 Formulas of Liapunov Matrix-Valued Functions

The two-index system of functions ( 2.9) being suitable for construction of the Lya-
punov functions allows to involve more wide classes of functions as compared with
those ussually applied in motion stability theory. For example, the bilinear forms



228 A.A. Martynyuk
6, 4(2004)

prove to be natural non-diagonal elements of matrix-valued functions. Another pe-
culiar feature of the approach being of importance is the fact that the application of
the matrix-valued function in the investigation of multidimensional systems enables
to allow for the interconnections between the subsystems in their natural form, i.eṅot
necessarily as the destabilizing factor. Finally, for the determination of the property
of having a fixed sign of the total derivative of auxiliary function along solutions of the
system under consideration it is not necessary to encorporate the estimation functions
with the quasimonotonicity property. Naturally, the awkwardness of calculations in
this case is the price.

3.1 A class of large-scale systems

We consider a system with finite number of degrees of freedom whose motion is
described by the equations ( 3.1)

dxi
dt

= fi(xi) + gi(t,x1, . . . ,xm), i = 1, 2, . . . ,m (3.1

where xi ∈ Rni , t ∈ Tτ , Tτ = [τ, +∞), fi ∈ C(Rni,Rni ), gi ∈ C(Tτ × R\∞ × ··· ×
R\m,R\〉).

Introduce the designation

Gi(t,x) = gi(t,x1, . . . ,xm) −
m∑

j=1, j 6=i

gij (t,xi,xj ), (3.2)

where gij (t,xi,xj ) = gi(t, 0, . . . ,xi, . . . ,xj, . . . , 0) for all i 6= j; i, j = 1, 2, . . . ,m.
Taking into consideration ( 3.2) system ( 3.1) is rewritten as

dxi
dt

= fi(xi) +
m∑

j=1, j 6=i

gij (t,xi,xj ) + Gi(t,x). (3.3)

Actually equations ( 3.3) describe the class of large-scale nonlinear nonautonomously
connected systems. It is of interest to extend the method of matrix Liapunov functions
to this class of equations in view of the new method of construction of nondiagonal
elements of matrix-valued functions.

3.2 Formulae for non-diagonal elements of matrix-valued func-
tion

In order to extend the method of matrix Liapunov functions to systems ( 3.3) it is
necessary to estimate variation of matrix-valued function elements and their total
derivatives along solutions of the corresponding systems. Such estimates are provided
by the assumptions below.

Assumption 3.1 There exist open connected neighborhoods Ni ⊆ Rni of the equi-
libriums state xi = 0, functions vii ∈ C1(Rni,R+), the comparison functions ϕi1,
ϕi2 and ψi of class K(KR) and real numbers c¯ii

> 0, c̄ii > 0 and γii such that



6, 4(2004)
Matrix Liapunov’s Functions Method ... 229

1. vii(xi) = 0 for all (xi = 0) ∈ Ni;

2. c
¯ii
ϕ2i1(‖xi‖) ≤ vii(xi) ≤ c̄iiϕ

2
i2(‖xi‖);

3. (Dxivii(xi))
Tfi(xi) ≤ γiiψ2i (‖xi‖) for all xi ∈ Ni, i = 1, 2, . . . ,m.

It is clear that under conditions of Assumption 3.1 the equilibrium states xi = 0
of nonlinear isolated subsystems

dxi
dt

= fi(xi), i = 1, 2, . . . ,m (3.4)

are

(a) uniformly asymptotically stable in the whole, if γii < 0 and
(ϕi1, ϕi2, ψi) ∈ KR-class;

(b) stable, if γii = 0 and (ϕi1, ϕi2) ∈ K-class;

(c) unstable, if γii > 0 and (ϕi1, ϕi2, ψi) ∈ K-class.

The approach proposed in this section takes large scale systems ( 3.3) into consid-
eration, subsystems ( 3.4) having various dynamical properties specified by conditions
of Assumption 3.1

Assumption 3.2 There exist open connected neighborhoods Ni ⊆ Rni of the equi-
librium states xi = 0, functions vij ∈ C1,1,1(Tτ ×R\〉×R\|,R), comparison functions
ϕi1, ϕi2 ∈ K(KR), positive constants (η1, . . . ,ηm)T ∈ Rm, ηi > 0 and arbitrary con-
stants c

¯ij
, c̄ij , i, j = 1, 2, . . . ,m, i 6= j such that

1. vij (t,xi,xj ) = 0 for all (xi,xj ) = 0 ∈ Ni × Nj , t ∈ Tτ , i, j = 1, 2, . . . ,m,
(i 6= j);

2. c
¯ij
ϕi1(‖xi‖)ϕj1(‖xj‖) ≤ vij (t,xi,xj ) ≤ c̄ijϕi2(‖xi‖)ϕj2(‖xj‖)

for all (t,xi,xj ) ∈ Tτ × N〉 × N|, i 6= j;

3. Dtvij (t,xi,xj ) + (Dxivij (t,xi,xj ))
Tfi(xi) + (Dxj vij (t,xi,xj ))

Tfj (xj ) +
ηi
2ηj

(Dxivii(xi))
Tgij (t,xi,xj ) +

ηj
2ηi

(Dxj vjj (xj ))
Tgji(t,xi,xj ) = 0; ( 3.5)

It is easy to notice that first order partial equations ( 3.5) are a somewhat variation
of the classical Liapunov equation proposed for determination of auxiliary function
in the theory of his direct method of motion stability investigation. In a particular
case these equations are transformed into the systems of algebraic equations whose
solutions can be constructed analytically.

Assumption 3.3 There exist open connected neighbourhoods Ni ⊆ Rni of the
equilibrium states xi = 0, comparison functions ψ ∈ K(KR), i = 1, 2, . . . ,m, real
numbers α1ij , α

2
ij , α

3
ij , ν

1
ki, ν

1
kij , µ

1
kij and µ

2
kij , i, j, k = 1, 2, . . . ,m, such that



230 A.A. Martynyuk
6, 4(2004)

1. (Dxivii(xi))
TGi(t,x) ≤ ψi(‖xi‖)

m∑
k=1

ν1kiψ(‖xk‖) + R1(ψ)

for all (t,xi,xj ) ∈ Tτ × N〉 × N|;

2. (Dxivij (t, ·))Tgij (t,xi,xj ) ≤ α1ijψ
2
i (‖xi‖)+α

2
ijψi(‖xi‖)ψj (‖xj‖)+α

3
ijψ

2
j (‖xj‖)+

R2(ψ) for all (t,xi,xj ) ∈ Tτ × N〉 × N|;

3. (Dxivij (t, ·))TGi(t,x) ≤ ψj (‖xj‖)
m∑

k=1

ν2ijkψk(‖xk‖) + R3(ψ)

for all (t,xi,xj ) ∈ Tτ × N〉 × N|;

4. (Dxivij (t, ·))Tgik(t,xi,xk) ≤ ψj (‖xj‖)(µ1ijkψk(‖xk‖) +

µ2ijkψi(‖xi‖)) + R4(ψ) for all (t,xi,xj ) ∈ Tτ × N〉 × N|.

Here Rs(ψ) are polynomials in ψ = (ψ1(‖x1‖, . . . ,ψm(‖xm‖)) in a power higher
than three, Rs(0) = 0, s = 1, . . . , 4.

Under conditions (2) of Assumptions 3.1 and 3.2 it is easy to establish for function

v(t,x,η) = ηTU(t,x)η =
m∑

i,j=1

vij (t, ·)ηiηj (3.6)

the bilateral estimate

uT1 H
TC

¯
Hu1 ≤ v(t,x,η) ≤ uT2 H

TC̄Hu2, (3.7)

where
u1 = (ϕ11(‖x1‖, . . . ,ϕm1(‖xm‖))T,

u2 = (ϕ12(‖x1‖, . . . ,ϕm2(‖xm‖))T

which holds true for all (t,x) ∈ Tτ × N , N = N1 × ··· × Nm.
Based on conditions (3) of Assumptions 3.1, 3.2 and conditions (1) – (4) of As-

sumption 3.3 it is easy to establish the inequality estimating the auxiliary function
variation along solutions of system ( 3.3). This estimate reads

Dv(t,x,η)
∣∣
(2.1)

≤ uT3 Mu3, (3.8)

where u3 = (ψ1(‖x1‖), . . . ,ψm(‖xm‖) and holds for all (t,x) ∈ Tτ × N .
Elements σij of matrix M in the inequality ( 3.8) have the following structure

σii = η2i γii + η
2
i νii +

m∑
k=1, k 6=i

(ηkηiν2kii + η
2
i ν

2
kii) + 2

m∑
j=1, j 6=i

ηiηj (α1ij + α
3
ji);

σij = 12 (η
2
i ν

1
ji + η

2
j ν

1
ij ) +

m∑
k=1, k 6=j

ηkηjν
2
kij +

m∑
k=1, k 6=i

ηiηjν
2
kij + ηiηj (α

2
ij + α

2
ji)

+
m∑

k=1, k 6=i,k 6=j
(ηkηjµ1kji + ηiηjµ

2
ijk + ηiηkµ

1
kij + ηiηjµ

2
jik),

i = 1, 2, . . . ,m, i 6= j.



6, 4(2004)
Matrix Liapunov’s Functions Method ... 231

3.3 Theorems on stability

Sufficient criteria of various types of stability of the equilibrium state x = 0 of system
( 3.3) are formulated in terms of the sign definiteness of matrices C

¯
, C̄ and M from

estimates ( 3.7), ( 3.8). We shall show that the following assertion is valid.

Theorem 3.1 Assume that the perturbed motion equations are such that all conditions
of Assumptions 3.1 – 3.3 are fulfilled and moreover

1. matrices C and C in estimate ( 3.7) are positive definite;

2. matrix M in inequality ( 3.8) is negative semi-definite (negative definite).

Then the equilibrium state x = 0 of system ( 3.3) is uniformly stable (uniformly
asymptotically stable).

If, additionally, in conditions of Assumptions 3.1 – 3.3 all estimates are satisfied
for Ni = Rni , Rk(ψ) = 0, k = 1, . . . , 4 and comparison functions (ϕi1,ϕi2) ∈ KR-
class, then the equilibrium state of system ( 3.3) is uniformly stable in the whole
(uniformly asymptotically stable in the whole).

Proof If all conditions of Assumptions 3.1 – 3.2 are satisfied, then it is possible
for system ( 3.3) to construct function v(t,x,η) which together with total derivative
Dv(t,x,η) satisfies the inequalities ( 3.7) and ( 3.8). Condition (1) of Theorem 4.1
implies that function v(t,x,η) is positive definite and decreasing for all t ∈ Tτ . Under
condition (2) of Theorem 4.1 function Dv(t,x,η) is negative semi-definite (definite).
Therefore all conditions of Theorem 2.3.1, 2.3.3 from Martynyuk [/] are fulfilled. The
proof of the second part of Theorem 4.1 is based on Theorem 2.3.4 from the same
monograph.

An example of non-linear systems Consider the non-linear system

dxi
dt

= aiixi +
n∑

j=1,j 6=i

aij (xj )xj, (3.9)

where xi ∈ R, aii < 0 for i = 1, 2, . . . ,n.
We assume on functions aij (x) as follows.

Assumption 3.4 There exist constants ∆ > 0, ε > 0 and Q > 0 such that

1. aij (x) ∈ C(R \ (−ε,ε), R)

2. |aij (x)| < Q|τ|γij +∆ for all τ ∈ (−ε,ε) i,j = 1, 2, . . . ,n, i 6= j,

where γji = −(aii + ajj )/aii, γij = −(aii + ajj )/ajj .
For each scalar subsystem

dxi
dt

= aiixi, i = 1, 2, . . . ,n, (3.10)



232 A.A. Martynyuk
6, 4(2004)

we take an auxiliary function in the form vii = x2i . Non-diagonal elements of matrix-
valued function U(x) are found as pseudo-quadratic forms vij (xi,xj ) = p(xi,xj )xixj .
Basing on equation ( 3.5) of Assumption 3.2 for η = (1, 1, . . . , 1)T we get

aiixi
∂vij
∂xi

+ ajjxj
∂vij
∂xj

= −[aij (xj ) + aji(xi)]xixj. (3.11)

In view that the partial derivatives of functions vij (xi,xj ) are

∂vij
∂xi

= pij (xi,xj )xj +
∂pij
∂xi

xixj,

∂vij
∂xj

= pij (xi,xj )xi +
∂pij
∂xj

xixj,

we find from equations ( 3.11)

aiixi
∂pij
∂xi

+ ajjxj
∂pij
∂xj

+ (aii + ajj )pij (xi,xj ) = −aij (xj ) − aji(xi). (3.12)

Further function pij (xi,xj ) is found as a sum of two functions pij (xi,xj ) = q1(xi) +
q2(xj ). Besides equation ( 3.12) becomes

aiixi
dq1
dxi

+(aii +ajj )q1(xi)+aji(xi) = −ajjxj
dq2
dxj

−(aii +ajj )q2(xj )−aij (xj ). (3.13)

The right-hand part of ( 3.13) depends on xi, while the left-hand part of ( 3.13)
depends on xj , therefore the right-hand and the left-hand parts equal to a constant
which is set equal to zero

aiixi
dq1
dxi

+ (aii + ajj )q1(xi) + aji(xi) = 0,

ajjxj
dq2
dxj

+ (aii + ajj )q2(xj ) + aij (xj ) = 0.
(3.14)

The corresponding homogeneous equations

aiixi
dq̃1
dxi

+ (aii + ajj )q̃1(xi) = 0,

ajjxj
dq̃2
dxj

+ (aii + ajj )q̃2(xj ) = 0
(3.15)

have general solutions

lclq̃1(xi) = C1|xi|γji, q̃2(xj ) = C2|xj|γij

respectively. To find partial solutions to equations ( 3.14) the method of variation of
a constant is applied. If these solutions are presented as

q1(xi) = C1(xi)|xi|γji, q2(xj ) = C2(xj )|xj|γij



6, 4(2004)
Matrix Liapunov’s Functions Method ... 233

with the initial conditions C1(0) = C2(0) = 0, it is easy to find that

q1(xi) = −|xi|γji
xi∫
0

aji(τ) sign τ
aii|τ|1+γji

dτ,

q2(xj ) = −|xj|γij
xj∫
0

aij (τ) sign τ
ajj|τ|1+γij

dτ,

(3.16)

where

sign τ ,



−1, for τ < 0,
∈ [−1, 1], for τ = 0,

1, for τ > 0.

In view of the assumption on functions aij (x) it is easy to show that the functions
q1(xi) and q2(xj ) are determined over the whole numerical axis and are differentiable
there. Thus, we can choose

pij (xi,xj ) = −|xi|γji
xi∫
0

aji(τ) sign τ
aii|τ|1+γji

dτ − |xj|γij
xj∫
0

aij (τ) sign τ
ajj|τ|1+γij

dτ (3.17)

and setting pii = 1 we present function v(x,η) as

v(x,η) = ηTU(x)η = xTP(x)x, (3.18)

where P(x) = [pij (xi,xj )], i,j = 1, 2, . . . ,n.

Calculating the corresponding total derivatives of the components of matrix-valued
function U(x) we find

Dv(x,η)
∣∣∣
(3.9)

= xTS(x)x, (3.19)

where S(x) = [σij (x)]ni,j=1 is a matrix whose elements have the following structure



234 A.A. Martynyuk
6, 4(2004)

σii = 2aii + 2
n∑

j=1,j 6=i

(
aii
a2jj

|xj|γij
xj∫
0

aij (τ) sign τ
|τ|1+γij

dτ

−|xi|γji
xi∫
0

aji(τ) sign τ
aii|τ|1+γji

dτ −
aij (xj )
ajj

)
aji(xi),

i = 1, 2, . . . ,n,

σij (x) =
n∑

k=1, k 6=i,k 6=j

[(
aii
a2kk

|xk|γik
xk∫
0

aik(τ) sign τ
|τ|1+γik

dτ

−|xi|γki
xi∫
0

aki(τ ) sign τ

aii|τ|1+γki
dτ − aik(xk)

akk

)
akj (xj )

+
(
ajj
a2kk

|xk|γjk
xk∫
0

ajk(τ) sign τ
|τ|1+γjk

dτ

−|xj|γkj
xj∫
0

akj (τ) sign τ
ajj|τ|1+γkj

dτ −
ajk(xk)
akk

)
aki(xi)

]
,

i 6= j, i,j = 1, 2, . . . ,n.
Using Theorem 3.1 and estimates ( 3.18) and ( 3.19) one can formulate the suffi-

cient conditions of stability, asynptotic stability and asymptotic stability in the whole
of system ( 3.9).

Theorem 3.2 Let system of equations ( 3.9) be such that

1. matrix P(x) is positive definite;

2. matrix S(x) is negative semi-definite (negative definite).

Then the equilibrium state x = 0 of system ( 3.9) is stable (asymptotically stable).
If in addition to conditions (1) and (2) one more condition is satisfied, namely

1. there exist constants r > 0, ε > 0 and L > 0 such that

‖P−1(x)‖ >
L

‖x‖2−ε
for ‖x‖ > r,

then the equilibrium state x = 0 of syste ( 3.9) is asymptotically stable in the whole.

4 4 On polystability of motion analysis

Consider the nonlinear system of differential equations

dx

dt
= A(t,x)x + B(t,y)y + F(t,x,y),

dy

dt
= D(x)x + G(t,x,y),

(4.1)



6, 4(2004)
Matrix Liapunov’s Functions Method ... 235

where x ∈ Rn1 , y ∈ Rn2 . Assume that functions A(x), B(t,y), D(x), F(t,x,y) and
G(t,x,y) are definite and continuous in the domain

D = {(t,x,y)|t ≥ 0, ‖x‖ ≤ h, ‖y‖ ≤ h},

and functions F(t,x,y) and G(t,x,y) satisfy the inequalities

‖F‖ ≤ c1(x,y)‖x‖γ1, ‖G‖ ≤ c2‖x‖γ2 for all (t,x,y) ∈ D.

Here function c1(x,y) → 0 as ‖x‖ + ‖y‖ → 0, F(t, 0, 0) = 0, G(t, 0, 0) = 0 for all
t ∈ J+t . According to [10, 11] we introduce the following definition.

Definition 4.1 The equilibrium state x = 0 of system ( 3.9) is called

1. x-polystable, iff it is stable and asymptotically x-stable;

2. uniformly x-polystable, if it is uniformly stable and uniformly
asymptotically x-stable;

Assumption 4.1 The pseudo-linear system

dx

dt
= A(t,x)x (4.2)

satisfies the following conditions

1. the equilibrium state x = 0 of system ( 4.2) is uniformly asymptotically stable;

2. there exists a function v(t,x) continuously differentiable in the domain H =
{(t,x) : t ≥ 0, ‖x‖ ≤ h}, positive definite and such that

c(‖x‖)‖x‖2 ≤ v(t,x) ≤ c̄(‖x‖)‖x‖2,
dv

dt

∣∣∣
(4.2)

≤ −α(‖x‖)‖x‖x2,∥∥∥∥∂v∂x
∥∥∥∥ ≤ ρ‖x‖α, ρ > 0 α > 0,

where c, c̄,α ∈ C(R+,R+).

Consider a pseudo-linear approximation of system ( 4.1)

dx

dt
= A(x)x + B(t,y)y,

dy

dt
= D(x)x,

and construct a matrix-valued function U(t,x,y). The diagonal elements of this
function are taken as

v11(x) = v(t,x), v22(y) = y
Ty.



236 A.A. Martynyuk
6, 4(2004)

To construct the non-diagonal elements v12(t,x,y) of the matrix-valued function we
consider the equation

Dtv12 + (Dxv12)
TA(t,x)x = −

η1
2η2

(Dxv(x))
TB(t,y)y −

η2
η1
yTD(x)x (4.3)

for some η = (η1,η2)T. Applying function U(t,x,y) and vector η we construct a
scalar function v(t,x,y) = ηTU(t,x,y)η.

Theorem 4.1 Assume that the perturbed motion equations are such that

1. all conditions of Assumption 4.1 are satisfied;

2. equation ( 4.3) has a solution in the form of a continuously differentiable func-
tion v12(t,x,y) admitting the estimates

c12(x,y)‖x‖‖y‖ ≤ v12(t,x,y) ≤ c̄12(x,y)‖x‖‖y‖

‖Dxv12‖ ≤ ρ1‖x‖α1‖y‖β1 ; ‖Dxv12‖ ≤ ρ2‖x‖α2‖y‖β2, ρ1,ρ2 > 0,

where c12 ∈ C(Rn1 × Rn2, R), c̄12 ∈ C(Rn1 × Rn2, R);

3. matrices

C(x,y) =
(

c11(x) c12(x,y)
c12(x,y) 1

)
, c11(x) = c(‖x‖),

C(x,y) =
(

c̄11(x) c̄12(x,y)
c̄12(x,y) 1

)
, c̄11(x) = c̄(‖x‖),

satisfy in the domain D = {(x,y) : ‖x‖ ≤ h, ‖y‖ ≤ h} the generalized Silvester
conditions;

4. there exists a constant κ > 0 such that

−a(‖x‖)η21 + sup
‖x‖=1

(Dyv12)TD(x)x + xTDT(x)Dyv12
‖x‖2

< −κ

for all (t,x,y) ∈ D;

5. sup
‖y‖=1

(Dxv12)TB(t,y)y + yTBT(t,y)Dxv12
‖y‖2

≤ 0

for all t ≥ 0 and ‖x‖ ≤ h;

6. α + γ1 ≥ 2, α1 + γ1 ≥ 2, α2 + γ2 ≥ 2, β1 ≥ 0, β2 > 0, β1 > 0.



6, 4(2004)
Matrix Liapunov’s Functions Method ... 237

Then the equilibrium state x = y = 0 of system ( 4.1) is unoformly x-polystable

Proof Conditions (2) from Assumption 4.1 and Theorem 4.1 for the components of
matrix-valued function U(t,x,y) allow to estimate the scalar function v(t,x,y,η) =
ηTU(t,x,y)η as

uTHTC(x,y)Hu ≤ v(t,x,y,η) ≤ uTHTC(x,y)Hu,

where u = (‖x‖,‖y‖)T, H = diag (η1,η2). Under condition (3) the function v(t,x,y,η)
is positive definite and decreasent. We shall estimate the total derivative of function
v(t,x,y,η) along solutions of system ( 4.1) taking unto account conditions (4) and
(5)

dv

dt

∣∣∣
(4.1)

≤ −κ‖x‖2 + η21(Dxv)TF(t,x,y) + η22yTG(t,x,y)

+ 2η1η2(Dxv12)TF(t,x,y) + 2η1η2(Dyv12)TG(t,x,y)

≤ −κ‖x‖2 + ρη21‖x‖αc1(x,y)‖x‖γ1 + η22ρ2c2‖x‖α2‖y‖β2‖x‖γ2

+2η1η2ρ1c1(x,y)‖x‖α1‖y‖β1‖x‖γ1 + 2η1η2ρ2c2‖x‖α2‖y‖β2‖x‖γ2

≤ −κ‖x‖2 + c(x,y)‖x‖2,

where the function c(x,y) → 0 as ‖x‖+‖y‖ → 0. Therefore there exists a magnitude
h1 ≤ h such that c(x,y) < κ/2 for ‖x‖ + ‖y‖ ≤ h1. Thus in the domain D̃ =
{(t,x,y) : t ≥ 0, ‖x‖+‖y‖ ≤ h1} the derivative of function v(t,x,y) along solutions
of system ( 4.1) is estimated by the inequality

dv

dt

∣∣∣
(4.1)

≤ −
κ
2
‖x‖2.

In terms of Theorem 2.5.2 from Martynyuk [9] we conclude on uniform
asymptotic stability of the equilibrium state x = y = 0 of system ( 4.1). The asser-
tion on uniform asymptotic x-stability follows from Theorem 2.6.1 by Martynyuk [9]
(see also Theorem 6.1 from Rumyantsev and Oziraner [1]).

5 Large-Scale Linear Systems

Linear systems of perturbed motion equations are of an essential interest in the de-
scription of various phenomena in physical and technical systems. General theory of
such systems is developed well because in some cases such systems can be integrated
precisely. On the other hand systems of the type are the first approximation of quasi-
linear equations in the investigation of which the information on the properties of the
first approximation system is encorporated. For this class of systems of equations
the construction of the Liapunov functions remains in the focus of attention of many
researchers.



238 A.A. Martynyuk
6, 4(2004)

5.1 Non-autonomous linear systems

Consider a large-scale system whose motion is described by the equations

dxi
dt

= Aiixi +
m∑

j=1,j 6=i

Aij (t)xj, i = 1, 2, . . . ,m. (5.1)

Here the state vectors xi ∈ Rni and Aii ∈ Rni×ni are constant matrices for all
i = 1, 2, . . . ,m; Aij (t) ∈ C(R,Rni×nj ), i,j = 1, 2, . . . ,m, i 6= j, n =

m∑
i=1

ni.

For the independent subsystems

dxi
dt

= Aiixi, i = 1, 2, . . . ,m, (5.2)

the auxiliary functions vii(xi) are constructed as the quadratic forms

vii(xi) = x
T
i Piixi, i = 1, 2, 3, (5.3)

whose constant matrices Pii are determined by the algebraic Liapunov equations

ATiiPii + PiiAii = −Gii, i = 1, 2, . . . ,m, (5.4)

where Gii are pre-assigned matrices of constant sign. For the construction of non-
diagonal elements vij (t,xi,xj ) of the matrix-valued function U(t,x) we apply equa-
tion ( 3.5). Note that for the system ( 5.1)

fi(xi) = Aiixi, fj (xj ) = Ajjxj,
gij (t,xi,xj ) = Aij (t)xj, gji(t,xi,xj ) = Aji(t)xj, Gi(t,x) = 0.

Suppose that at least one of the matrices Aij or Aji is not equal to constant. Then
we determine function vij (t,xi,xj ) as

vij (t,xi,xj ) = vji(t,xj,xi) = x
T
i Pij (t)xj, (5.5)

where Pij ∈ C1(R,Rni×nj ).
Since for the bilinear forms ( 5.5)

Dtvij (t,xi,xj ) = xi
dPijdt

xj, Dxivij (t,xi,xj ) = x
T
j Pij (t)

T,

Dxj vij (t,xi,xj ) = x
T
i Pij (t)

the equation ( 3.5) becomes

xTi

(
dPij
dt

+ ATiiPij + PijAjj +
ηi
ηj
PiiAij (t) +

ηj
ηi
ATji(t)Pjj

)
xj = 0.



6, 4(2004)
Matrix Liapunov’s Functions Method ... 239

For determination of matrices Pij this correlation yields a system of matrix differential
equations

dPij
dt

+ ATiiPij + PijAjj = −
ηi
ηj
PiiAij (t) −

ηj
ηi
ATji(t)Pjj

i, j = 1, 2, . . . ,m, i 6= j.
(5.6)

Note that equations ( 5.6) can be solved in the explicit form. To this end we consider
a linear operator (general information on linear operators can be found, for example,
in Daletskii and Krene [1])

Fij : R
ni×nj → Rni×nj , FijX = ATiiX + XAjj.

Equation ( 5.6) can be represented as

dPij
dt

+ FijPij = −
ηi
ηj
PiiAij (t) −

ηj
ηi
ATji(t)Pii, i 6= j.

Consider the homogeneous equations

dPij
dt

+ FijPij = 0, (5.7)

whose general solution is presented as

Pij (t) = exp{−Fijt}Cij,

where Cij is a constant ni × nj matrix and exp{−Fijt} =
∞∑

k=0

(−1)kF kijt
k

k!
is an

operator exponent.
To find the solution of equation ( 5.6) the method of variation of a constant is

applied. Solution of equation ( 5.6) is presented in the form

Pij (t) = exp{−Fijt}Cij (t), (5.8)

where Cij ∈ C1(R,Rn1×n2 ) and Cij (0) = 0. Substituting by ( 5.8) into ( 5.6) yields

dCij
dt

= − exp{Fijt}
(
ηi
ηj
PiiAij +

ηj
ηi
ATjiPii

)
, i 6= j.

Integrating the last correlation from 0 to t we determine a partial solution of equation
( 5.6)

Pij (t) = −
t∫

0

exp{−Fij (t − τ)}
(
ηi
ηj
PiiAij (τ) +

ηj
ηi
ATji(τ)Pjj

)
dτ, i 6= j. (5.9)

We establish estimates for the function

v(t,x,η) = ηTU(t,x)η =
m∑

i,j=1

vij (t, .)ηiηj,



240 A.A. Martynyuk
6, 4(2004)

where

U(t,x) =


 v11(x1) · · · v1m(t,x1,xm)... . . . ...

v1m(t,x1,xm) · · · vmm(xm)


 .

Introduce the designations c̄ii = λM (Pii) and cii = λm(Pii) and assuming
sup
t≥0

‖Pij (t)‖ < ∞ denote c̄ij = sup
t≥0

‖Pij (t)‖, cij = −c̄ij .

Since for the forms ( 5.3) and ( 5.5) the estimates

λm(Pii)‖xi‖2 ≤ vii(xi) ≤ λM (Pii)‖xi‖2, xi ∈ Rni ;

−c̄ij‖xi‖‖xj‖ ≤ vij (t,xi,xj ) ≤ c̄ij‖xi‖‖xj‖, (xi,xj ) ∈ Rni × Rnj ,
(5.10)

are valid, for the function v(t,x,η) = ηTU(t,x)η

wTHTCHw ≤ v(t,x,η) ≤ wTHTCHw for all x ∈ Rn, (5.11)

where w = (‖x1‖, . . . ,‖xm‖)T, H = diag (η1,η2, . . . ,ηm), C = [c̄ij ]mi,j=1,
C = [cij ]

m
i,j=1.

In order to estimate the derivative of function v(t,x,η) along solutions of system
( 5.1) we calculate the constants from Assumption 3.3

α1ij = α
2
ij = 0, α

3
ij (t) = λM (A

T
ij (t)Pij (t) + P

T
ij (t)Aij (t)),

ν1ki = ν
2
ijk = 0, ν

1
ijk(t) = λ

1/2
M [(P

T
ij (t)Aik(t))(P

T
ij (t)Aik(t))], µ

2
ijk = 0.

Therefore the elements σij of matrix M(t) in estimate ( 3.8) for system ( 5.1) have
the structure

σii(t) = −η2i λm(Gii) + 2
m∑

j=1,j 6=i

ηiηjα
3
ij, i = 1, . . . ,m,

σij (t) =
m∑

k=1,k 6=i,k 6=j

(ηkηjν
1
ijk + ηiηkν

1
kij ), i,j = 1, . . . ,m, i 6= j.

Consequently, the variation of function Dv(t,x,η) along solutions of system ( 5.1) is
estimated by the inequality

Dv(t,x,η)
∣∣∣
(5.1)

≤ wTM(t)w (5.12)

for all (x1, . . . ,xm) ∈ Rn1 × ··· × Rnm .

Remark 5.1 In the partial case when matrices Aij and Aji do not depent on t it is
reasonable to choose Pij (t) = const. Then equation ( 5.6) becomes

AiiPij + PijAjj = −
ηi
ηj
PiiAij −

ηj
ηi
ATjiPjj (5.13)



6, 4(2004)
Matrix Liapunov’s Functions Method ... 241

or in the operator form

FijPij = −
ηi
ηj
PiiAij −

ηj
ηi
ATjiPjj.

Therefore for the equation ( 5.13) to have a unique solution it is necessary and suffi-
cient that the operator Fij be nondegenerate.

It is known (see Daletskii and Krene [1]) that the set of eigenvalues of the operator
Fij consists of the numbers λk(Aii) + λl(Ajj ), where λk(·) is an eigenvalue of the
corresponding matrix. Basing on these speculations one can formulate the following
result.

For the equation ( 5.13) to have a unique solution it is necessary and sufficient
that

λk(Aii) + λl(Ajj ) 6= 0 for all k, l,

and this solution can be presented as

Pij = −F−1ij
(
ηi
ηj
PiiAij +

ηj
ηi
ATjiPjj

)
.

This result is summed up as follows.

Theorem 5.1 Assume that for system ( 5.1) the following conditions are satisfied

1. the sign-definite matrices Pii, i = 1, 2, 3, are the solution of algebraic equations
( 5.4);

2. the bounded matrices Pij (t) for all i,j = 1, 2, . . . ,m, i 6= j, are the solution of
matrix differential equations ( 5.6);

3. matrices C É C in estimate ( 5.11) are positive definite;

4. matrix M(t) in estimate ( 5.12) is negative semi-definite (negative definite).

Then the equilibrium state x = 0 of system ( 5.1) is uniformly stable in the whole
(uniformly asymptotically stable in the whole).

5.2 Time invariant linear systems

Assume that in the system

dx1
dt

= A11x1 + A12x2 + A13x3,
dx2
dt

= A21x1 + A22x2 + A23x3,
dx3
dt

= A31x1 + A32x2 + A33x3,

(5.14)

the state vectors xi ∈ Rni , i = 1, 2, 3, and Aij ∈ Rni×nj are constant matrices for
all i,j = 1, 2, 3.



242 A.A. Martynyuk
6, 4(2004)

For the independent systems

dxi
dt

= Aiixi, i = 1, 2, 3, (5.15)

we construct auxiliary functions vii(xi) as the quadratic forms

vii(xi) = x
T
i Piixi, i = 1, 2, 3, (5.16)

whose matrices Pii are determined by

ATiiPii + PiiAii = −Gii, i = 1, 2, 3, (5.17)

where Gii are prescribed matrices of definite sign.
In order to construct non-diagonal elements vij (xi,xj ) of matrix-valued function

U(x) we employ equation ( 3.5). Note that for system ( 5.14)

fi(xi) = Aiixi, fj (xj ) = Ajjxj,

gij (xi,xj ) = Aijxj, Gi(t,x) = 0, i = 1, 2, 3.

Since for the bilinear forms

vij (xi,xj ) = vji(xj,xi) = x
T
i Pijxj (5.18)

the correlations

Dxivij (xi,xj ) = x
T
j P

T
ij , Dxj vij (xi,xj ) = x

T
i Pij

are true, equation ( 3.5) becomes

xTi

(
ATiiPij + PijAjj +

ηi
ηj
PiiAij +

ηj
ηi
ATjiPii

)
xj = 0.

¿From this correlation for determining matrices Pij we get the system of algebraic
equations

AiiPij + PijAjj = −
ηi
ηj
PiiAij −

ηj
ηi
ATjiPii,

i 6= j, i,j = 1, 2, 3.
(5.19)

Since for ( 5.16) and ( 5.18) the estimates

vii(xi) ≥ λm(Pii)‖xi‖2, xi ∈ Rni ;

vij (xi,xj ) ≥ −λ
1/2
M (PijP

T
ij )‖xi‖‖xj‖, (xi,xj ) ∈ R

ni × Rnj ,

hold true, for function v(x,η) = ηTU(x)η the inequality

wTHTCHw ≤ v(x,η) (5.20)



6, 4(2004)
Matrix Liapunov’s Functions Method ... 243

is satisfied for all x ∈ Rn, w = (‖x1‖,‖x2‖,‖x3‖)T and the matrix

C =




λm(P11) −λ
1/2
M (P12P

T
12) −λ

1/2
M (P13P

T
13)

−λ1/2M (P12P
T
12) λm(P22) −λ

1/2
M (P23P

T
23)

−λ1/2M (P13P
T
13) −λ

1/2
M (P23P

T
23) λm(P33)


 .

For system ( 5.14) the constants from Assumption 3.3 are:

α1ij = α
2
ij = 0; α

3
ij = λM (A

T
ijPij + P

T
ij Aij ),

ν1ki = ν
2
ijk = 0; ν

1
ijk = λ

1/2
M [(P

T
ij Aik)(P

T
ij Aik)], µ

2
ijk = 0.

Therefore the elements σij of matrix M in ( 5.12) for system ( 5.14) have the structure

σii = −η2i λm(Gii) + 2
3∑

j=1, j 6=i
ηiηjα

3
ij, i = 1, 2, 3,

σij =
3∑

k=1, k 6=i,k 6=j
(ηkηjν1ijk + ηiηkν

1
kij ), i,j = 1, 2, 3, i 6= j.

Consequently, the function Dv(x,η) variation along solutions of system ( 5.14) is
estimated by the inequality

Dv(x,η)
∣∣
(5.14)

≤ wTMw (5.21)

for all (x1,x2,x3) ∈ Rn1 × Rn2 × Rn3 .
We summarize our presentation as follows.

Corollary 5.1 Assume for system ( 5.14) the folowing conditions are satisfied:

1. algebraic equations ( 5.17) have the sign-definite matrices Pii, i = 1, 2, 3, as
their solutions;

2. algebraic equations ( 5.19) have constant matrices Pij , for all i,j = 1, 2, 3,
i 6= j, as their solutions;

3. matrix C in ( 5.20) is positive definite;

4. matrix M in ( 5.21) is negative semi-definite (negative definite).

Then the equilibrium state x = 0 of system ( 5.14) is uniformly stable (uniformly
asymptotically stable).

This corollary follows from Theorem 3.1 and hence its proof is obvious.



244 A.A. Martynyuk
6, 4(2004)

Example 5.3 We study the motion of two non-autonomously connected oscillators
whose behaviour is described by the equations

dx1
dt

= γ1x2 + v cos ωty1 − v sin ωty2,
dx2
dt

= −γ1x1 + v sin ωty1 + v cos ωty2,
dy1
dt

= γ2y2 + v cos ωtx1 + v sin ωtx2,
dy2
dt

= −γ2y2 + v cos ωtx2 − v sin ωtx1,

(5.22)

where γ1, γ2, v, ω, ω + γ1 − γ2 6= 0 are some constants.
For the independent subsystems

dx1
dt

= γ1x2,
dx2
dt

= −γ1x1

dy1
dt

= γ2y2,
dy2
dt

= −γ2y1

(5.23)

the auxiliary functions vii, i = 1, 2, are taken in the form

v11(x) = xTx, x = (x1,x2)T,
v22(y) = yTy, y = (y1,y2)T.

(5.24)

We use the equation ( 3.5) (see Assumption 3.2) to determine the non-diagonal
element v12(x,y) of the matrix-valued function U(t,x,y) = [vij (·)], i,j = 1, 2. To
this end set η = (1, 1)T and v12(x,y) = xTP12y, where P12 ∈ C1(Tτ,R∈×∈). For
the equation

dP12
dt

+
(

0 −γ1
γ1 0

)
P12 + P12

(
0 γ2
−γ2 0

)
+2v

(
cos ωt − sin ωt
sin ωt cos ωt

)
= 0,

(5.25)

the matrix

P12 = −
2v

ω + γ1 − γ2

(
sin ωt cos ωt
− cos ωt sin ωt

)
is a partial solution bounded for all t ∈ Tτ .

Thus, for the function v(t,x,y) = ηTU(t,x,y)η it is easy to establish the estimate
of ( 3.7) type with matrices C and C in the form

C =
(
c11 c12
c12 c22

)
, C =

(
c̄11 c̄12
c̄12 c̄22

)
,

where c̄11 = c11 = 1, c̄22 = c22 = 1, c̄12 = −c12 =
|2v|

|ω + γ1 − γ2|
. Besides, the vector

uT1 = (‖x‖,‖y‖) = uT2 , since the system ( 5.22) is linear.



6, 4(2004)
Matrix Liapunov’s Functions Method ... 245

For system ( 5.22) the estimate ( 5.12) becomes

Dv(t,x,y)
∣∣
(5.1)

= 0

for all (x,y) ∈ R2 × R2 because M = 0.
Due to Theorem 3.1 the motion stability conditions for system ( 5.22) are estab-

lished basing on the analysis of matrices C and C property of having fixed sign.
It is easy to verify that the matrices C and C are positive definite, if

1 −
4v2

(ω + γ1 − γ2)2
> 0.

Consequently, the motion of nonautonomously connected oscillators is uniformly sta-
ble in the whole, if

|v| <
1
2
|ω + γ1 − γ2|.

5.3 Discussion and Numerical Example

To start to illustrate the possibilities of the proposed method of Liapunov function
construction we consider a system of two connected equations that was studied earlier
by the Bellman-Bailey approach (see Barbashin [1], Voronov and Matrosov [1], etc.).

Partial case of system ( 6.14) is the system

dx1
dt

= Ax1 + C12x2
dx2
dt

= Bx2 + C21x1,
(6.1)

where x1 ∈ Rn1 , x2 ∈ Rn2 , and A, B, C12 and C21 are constant matrices of corre-
sponding dimensions. For independent subsystems

dx1
dt

= Ax1,
dx2
dt

= Bx2
(6.2)

the functions v11(x1) and v22(x2) are constructed as the quadratic forms

v11 = x
T
1 P11x1, v22 = x

T
2 P22x2, (6.3)

where P11 and P22 are sign-definite matrices.
Function v12 = v21 is searched for as a bilinear form v12 = xT1 P12x2 whose matrix

is determined by the equation

ATP12 + P12B = −
η1
η2
P11C12 −

η2
η1
CT21P22, η1 > 0, η2 > 0. (6.4)

According to Lankaster [1] equation ( 6.4) has a unique solution, provided that ma-
trices A and −B have no common eigenvalues.



246 A.A. Martynyuk
6, 4(2004)

Matrix C in (/././) for system ( 6.4) reads

C =


 λm(P11) −λ

1/2
M (P12P

T
12)

−λ1/2M (P12P
T
12) λm(P22)


 . (6.5)

Here λm(·) are minimal eigenvalues of matrices P11, P22, and λ
1/2
M (·) is the norm of

matrix P12P T12.
Estimate ( 5.15) for function Dv(x,η) by virtue of system ( 6.1) is

Dv(x,η)|(6.1) ≤ wTΞw, (6.6)

where w = (‖x1‖, ‖x2‖)T, Ξ = [σij ], i,j = 1, 2;

σ11 = λ1η21 + η1η2α22,

Cσ22 = λ2η22 + η1η2β22,

V σ12 = σ21 = 0.

The notations are
λ1 = λM (ATP11 + P11A),

λ2 = λM (BTP22 + P22B),

α22 = λM (CT12P12 + P
T
12C12),

β22 = λM (CT21P
T
12 + P12C21),

λ(·) is a maximal eigenvalue of matrix (·). Partial case of Assumption 3.1 is as follows.

Corollary 6.1 For system ( 6.1) let functions vij (·), i,j = 1, 2, be constructed so
that matrix C for system ( 6.1) is positive definite and matrix Ξ in inequality (/././)
is negative definite. Then the equilibrium state x = 0 of system ( 6.1) is uniformly
asymptotically stable.

We consider the numerical example. Let the matrices from system ( 6.1) be of the
form

A =
(
−2 1

3 −2

)
, B =

(
−4 1

2 −1

)
, (6.7)

C12 =
(
−0.5 −0.5

0.8 −0.7

)
, C21 =

(
1 0.5

−0.6 −0.3

)
. (6.8)

Functions vii for subsystems

ẋ = Ax, x = (x1,x2)T,
ẏ = Bx, y = (y1,y2)T



6, 4(2004)
Matrix Liapunov’s Functions Method ... 247

are taken as the quadratic forms

v11 = 75x21 + x1x2 + 5x
2
2,

v22 = 0.35y21 + 0.9y1y2 + 0.95y
2
2.

(6.9)

Let η = (1, 1)T. Then λ1 = λ2 = −1,

P12 =
(
−0.011 0.021
−0.05 −0.022

)
,

α22 = 0.03, β22 = −0.002.

It is easy to verify that σ11 < 0 and σ22 < 0, and hence all conditions of Corollary 6.1
are fulfilled in view that

λ
1/2
M (P12P

T
12) ≤ (λm(P11)λm(P22))

1/2,

for the values of λ1/2M (P12P
T
12) = 0.06, λm(P11) = 08, λm(P22) = 0.115. This implies

uniform asymptotic stability in the whole of the equilibrium state of system ( 6.1)
with matrices ( 6.7) and ( 6.8).

Let us show now that stability of system ( 6.1) with matrices ( 6.7) and ( 6.8) can
not be studied in terms of the Bailey [1] theorem.

We recall that in this theorem the conditions of exponential stability of the equi-
librium state are

1. for subsystems ( 6.2) functions ( 6.3) must exist satisfying the estimates

(a) ci1‖xi‖2 ≤ vi(t,xi) ≤ ci2‖xi‖2,
(b) Dvi(t,xi) ≤ −ci3‖xi‖2,
(c) ‖∂vi/∂xi‖ ≤ ci4‖xi‖ for xi ∈ Rni ,

where cij are some positive constants, i = 1, 2, j = 1, 2, 3, 4;

2. the norms of matrices Cij in system (/././) must satisfy the inequality (see
Voronov and Matrosov [1], p. 106)

‖C12‖‖C21‖ <
(
c11c21
c12c22

)1/2(
c13c23
c14c24

)
. (6.10)

We note that this inequality is refined as compared with the one obtained firstly by
Bailey [1].

The constants c11, . . . ,c24 for functions (/././) and system ( 6.1) with matrices
(/././) and (/././) take the values

c11 = 1.08, c21 = 0.115, c12 = 2.14, c22 = 2.14,

c22 = 1.135 , c13 = c23 = 1, c14 = 4.83, c24 = 2.4.



248 A.A. Martynyuk
6, 4(2004)

Condition (/././) requires that

‖C12‖‖C21‖ < 0.0184 (6.11)

whereas for system (/././), (/././), and (/././) we have

‖C12‖‖C21‖ = 75.

Thus, the Bailey theorem turns out to be nonapplicable to this system and the con-
dition ( 6.11) is “super-sufficient” for the property of stability.

6 Problems for Investigations

7.1 To obtain existence conditions for solutions to system ( 3.5) which satisfy bilinear
estimates (condition (2) of Assumption 3.2) or other similar conditions allowing to
establish algebraic conditions of sign-definiteness and decrease (radial unboundedness)
of function ( 3.6).
7.2 To construct an algorithm of approximate solution of system ( 3.5) in terms of
the method of perturbed nonlinear mechanics.
7.3 To obtain criterion for exponential stability of system ( 3.3) in terms of function
( 3.6) provided that the independent subsystems ( 3.4) are not exponentially stable.
7.4 To investigate other than stability in the sense of Liapunov dynamical properties of
system ( 3.3) or its partial cases such as stability, boundedness, uniform boundedness
in terms of two measures.
7.5 In terms of Liapunov function ( 3.6) to construct algorithms for estimation of
domains of stability, attraction and asymptotic stability of system ( 3.3) and its
partial cases in the phase space or/and in the parameter space.

Hint. For the initial definitions of the corresponding domains of stability, attrac-
tion and asymptotic stability see Grujic, et al. [1], Krasovskii [1], and Martynyuk
[7].
7.6 For the class of autonomous systems

dx

dt
= X(x(t)), x(t0) = x0, (7.1)

where x ∈ Rn, X ∈ C1(Rn, Rn), X(0) = 0, admitting decomposition to ( 3.3) form,
to establish conditions of global asymptotic stability under condition that the origin
for system ( 7.1) is an asymptotic attractor.

7 Brief Outline of the References and Remarks

Section 2 Nonlinear dynamics of continuous systems is a traditional domain of inten-
sive investigations starting with the works by Galilei, Newton, Euler, Lagrange, etc.
The problem of motion stability arises whenever the engineering or physical problem



6, 4(2004)
Matrix Liapunov’s Functions Method ... 249

is formulated as a mathematical problem of qualitative analysis of equations. Poincaré
and Liapunov laid a background for the method of auxiliary functions for continuous
systems which allow not to integrate the motion equations for their qualitative analy-
sis. The ideas of Poincare and Liapunov were further developed and applied in many
branches of modern natural sciences.

The results of Liapunov [1], Chetaev [1], Persidskii [1], Malkin [1], Ascoli [1], Bar-
basin and Krasovskii [1], Massera [1], and Zubov [1], were base for the Definitions 2.1 –
2.3 (ad hoc see Grujić et al. [1], pp. 8 – 12) and cfṘao Mohana Rao [1], Yoshizawa [1],
Rouche et al. [1], Antosiewicz [1], Lakshmikantham and Leela [1], Hahn [2], etc. For
the Definitions 2.4 – 2.7, and 2.13 see Hahn [2], and Martynyuk [9]. Definitions 2.8 –
2.12 are based on some results by Liapunov [1], Hahn [2], Barbashin and Krasovskii [1]
(see and cfḊjordjevic [1], Grujić [2], Martynyuk [3 – 6]). The proofs of Proposalls 2.1 –
2.5 are in Hahn [2], Kuz’min [1], Martynyuk [9], Zubov [2], etc.

Theorems 2.1 – 2.7 are set out according to Martynyuk [10] (see also Martynyuk [13]).
For the proof of Corollary 2.1 see Liapunov [1], and Chetaev [1]; for the proof of
Corollary 2.2 see Barbashin and Krasovskii [1]; for the proof of Corollary 2.3 – 2.4 see
Liapunov [1], Massera [1], Yoshizava [1], Halanay [1], etc; for the proof of Corollary
2.5 – 2.6 see He and Wang [1], Krasovskii [1], and Hahn [2]; for the proof of Corollary
2.8 see Chetaev [1], Rouche, et al. [1]; and for the proof of Corollary 2.9 – 2.10 see
Liapunov [1], and Rouche, et al. [1].

Further results obtained via the Liapunov’s methods can be found in Burton [1],
Galperin [1], Gruyitch [1], Rama Mohana Rao [1], Coppel [1], Cesari [1], Lakshmikan-
tham and Leela [2], Martynyuk [14], Sivasundaram [1], Vincent [1], Vorotnikov [1],
Zubov [3] (see also CD ROM by Kramer and Hofmann [1] for references), etc.

Section 3 The problem of constructing the Liapunov functions for nonlinear nonau-
tonomous system of general type remains still unsolved though its more than one-
hundred existence. Meanwhile the efforts of many mathematicians and mechanical
scientists have resulted in the efficient approach of constructing the appropriate aux-
iliary functions for specific classes of systems of equations with reference to many
applications.

The approach proposed in this section is based on the idea of matrix-valued func-
tion as an appropriate medium for Liapunov function construction. This approach
has been developed since 1984 and some of the obtained results are published and
summarized by Martynyuk [9, 12], and Kats and Martynyuk [1].

Actually, the problem of constructing the Liapunov functions for the class of non-
linear systems of ( 3.3) type is reduced to the solution of systems of first order partial
equations ( 3.5) which are more simple than the Liapunov equation for the initial
system proposed by in 1892 in his famous dissertation paper.

This section is based on some results by Martynyuk and Slyn’ko [1, 2, 3], and
Slyn’ko [2]. Besides, some results by Djordjevic [1, 2], Hahn [1], Krasovskii [1],
Lankaster [1], etcȧre used.

Section 4 The phenomenon of motion polystability has been investigated in nonlinear
dynamics since 1987. As noticed by Aminov and Sirazetdinov [1], and Martynyuk [16]



250 A.A. Martynyuk
6, 4(2004)

this phenomenon was discovered while developing the notion of stability with respect
to a part of variables. In monographs by Martynyuk [9, 12] some results are pre-
sented obtained in the development of the theory of motion polystability including
sufficient conditions for exponential polystability in the first approximation (see also
Martynyuk [14, 15], and Slyn’ko [1]). This section encorporated the results by Mar-
tynyuk and Slyn’ko [3].

Section 5 Linear nonautonomous system of ( 5.1) type or autonomous system (
5.14) is of essential interest in context with the problem of constructing the Liapunov
function since this allows to investigate stability of the equilibrium state of some
quasilinear systems. In spite of the seeming simplicity of linear systems the problem
of constructing the appropriate Liapunov function remains open in this case es well
(see, e.gḂarbashin [1], Zhang [1], etc.).

In this section for the above-mentioned systems we adopt the algorithm of Lia-
punov function construction presented in Section 3. Since in this case systems of
equations ( 3.5) turns to be linear differential or algebraic, their exact solutions can
sometimes be found. The section is based on the results by Martynyuk and Slyn’ko
[1 – 3].

Section 6 The Bellman–Bailey approach (see Bellman [1] and Bailey [1]) to stability
investigation of large-scale systems has been developed considerably in many papers.
In monographs by Barbashin [1], Michel and Miller [1], Siljak [1], Grujic, Martynyuk
and Ribbens-Pavella [1], Voronov and Matrosov [1], etcȧlongside the original results
the results of many investigations of dynamics of linear and nonlinear systems in terms
of vector Liapunov functions are summarized. An essential deficiency of this approach
is the supersufficiency of stability conditions for the systems of motion equations under
consideration (see Piontkovskii and Rutkovskaya [1], and Martynyuk and Slyn’ko [1]).

The application of the matrix-valued Liapunov function for the same classes of
systems of equations provides wider conditions of stability. The reasons for this were
scrutinized by Martynyuk [9,/,12]. In this section by the example of linear system it
is shown how supersufficient the stability conditions obtained via the Bellman–Bailey
approach are as compared with those obtained via the application of matrix-valued
function.

Section 7 The problems set out in this section are addressed first of all to the young
researchers in the area of motion stability theory and its application. Solution of
any of the problems will be not only a subject of a significant paper but an essential
contribution to the development of the method of matrix Liapunov functions which
was called by ProfV̇. Lakshmikantham (in Moscow, 2001) one of three outstanding
achivements of stability theory in the 20th century.

Received: March 2003. Revised: June 2003.



6, 4(2004)
Matrix Liapunov’s Functions Method ... 251

References

Aminov, A.B., and Sirazetdinov, T.K.

1. The method of Liapunov functions in problems of multistability of motion, Prikl.
Mat. Mekh. 51 (1987) 553-558. [Russian]

Antosiewicz, H. A.

1. A survey of Lyapunov’s second method. Contributions to the Theory of Nonli-
near Oscillations (4) (1958) 141 – 166.

Ascoli, G.

1. Osservazioni sopora alcune questioni di stabilita Atti AccadṄazL̇incei Rend
ClṠciḞisṀatṄat. 9 (1950) 129 – 134.

Bailey, F.N.

1. The application of Liapunov’s second method to interconnected systems. J.
SocİndȦpplṀath. SerȦ 3 (1965) 443 – 462.

Barbashin, Ye.A.

1. The Liapunov Functions. Nauka: Moscow, 1970. [Russian]

Barbashin, Ye.A., and Krasovskii, N.N.

1. On the stability of motion in the large. DoklȦkadṄauk SSSR 86 (1952) 453 –
456.

Bellman, R.

1. Vector Liapunov functions. SIAM J. Contr. SerȦ 1 (1962) 32-34.

Bhatia, N. P. and Szegö, G. P.

1. Stability Theory of Dynamical Systems. Berlin: Springer-Verlag, 2002.

Birkhoff, G. G.

1. Dynamical Systems. Moscow-Leningrad: GITTL, 194 [Russian].

Borne, P., Dambrine, M., Perruquetti, W., Richard, J.P.

1. Vector Lyapunov functions: Time-varing, ordinary and functional differential
equations. In: Advances in Stability Theory at the End of XXth Century
(Ed.: A.A. Martynyuk), London: Taylor and Francis, 2002, 89 –112.

Burton, T.A.

1. Stability and Periodic Solutions of Ordinary and Functional Differential Equa-
tions. Orlando: Academic Press, 1985.



252 A.A. Martynyuk
6, 4(2004)

Bylov, B.F., Vinograd, R.E., Grobman, D.M., and Nemytskii, V.V.

1. Lyapunov Characteristic Exponents Theory and Its Applications to Stability Is-
sues. Moscow: Nauka, 1966. [Russian]

Cesari, L.

1. Asymptotic Behaviour and Stability Problems in Ordinary Differential Equa-
tions. 2nd edn. Berlin: Springer-Verlag, 1963.

Chetaev, N. G.

1. Stability of Motion. Moscow: Nauka, 1990. [Russian]

Coppel, W. A.

1. Stability and Asymptotic Behaviour of Differential Equations. Boston: Heath,
1965.

Daletskii Yu.L., Krein M.G.

1. Stability of solutions of Differential Equations in the Banach Space. Moscow:
Nauka, 1986. [Russian]

Djordjević, M. Z.

1. Stability analysis of large scale systems whose subsystems may be unstable. Large
Scale Systems 5 (1983) 252 – 262.

2. Stability analysis of interconnected systems with possibly unstable subsystems.
Systems and Control Letters. 3 (3) (1983) 165 –169.

Galperin, E.A.

1. Some generalizations of Lyapunov’s approach to stability and control. Nonlinear
Dynamics and Systems Theory 2 (1) (2002) 1 – 23.

Gruyitch, Lj.T.

1. Consistent Lyapunov methodology: non-differentiable non-linear systems. Non-
linear Dynamics and Systems Theory 1 (1) (2001) 1 – 22.

2. On large-scale systems stability. Proc1̇2th World Congress IMACS, Vol1̇, 1988,
224 – 229.

Grujić, Lj. T., Martynyuk, A. A. and Ribbens-Pavella, M.

1. Large-Scale Systems Stability under Structural and Singular Perturbations. Berlin:
Springer-Verlag, 1987.

Hahn, W.

1. Stability of Motion. Berlin: Springer-Verlag, 1967.



6, 4(2004)
Matrix Liapunov’s Functions Method ... 253

Hale, J. K.

1. Asymptotic Behaviour of Dissipative Systems. MathṠurveys and Monographs,
Providence: AmerṀathṠoc., Vol. 25, 1988.

Iooss, G., and Joseph, D.D.

1. Elementary Stability and Bifurcation Theory. New York: Springer- Verlag, 1990.

Kats, I.Ya., and Martynyuk, A.A.

1. Stability and Stabilization of Nonlinear Systems with Random Structures. Lon-
don and New York: Taylor and Francis, 2002.

Kramer and Hofman G. E. I.

1. Zentralblatt MATH CD-ROM, Volumes 926 – 962, Berlin: Springer, 2000.

Krasovskii N.N.

1. Stability of Motion. Stanford, California: Stanford UnivṖress, 1963.

Kuz’min, P.A.

1. Small Oscillations and Stability of Motion. Moscow: Nauka, 1973. [Russian]

Lakshmikantham, V. and Leela, S.

1. Differential and Integral Inequalities. Theory and Applications. New York:
Academic Press, 1969.

2. Fuzzy differential systems and the new concept of stability. Nonlinear dynamics
and Systems Theory 1 (2) (2001) 111 – 119.

Lakshmikanthan, V., Leela, S. and Martynyuk, A. A.

1. Stability Analysis of Nonlinear Systems. New York: Marcel Dekker, Inc., 1989.

Lakshmikantham V., Matrosov V.M. and Sivasundaram S.

1. Vector Lyapunov Function and Stability Analysis of Nonlinear Systems. Dor-
drecht, etc.: Kluwer Academic Publishers, 199

Lankaster, P.

1. Matrix Theory. Moscow: Nauka, 1978. [Russian]

Lefschetz, S.

1. Differential Equations: Geometric Theory. New York: Interscience Publishers,
1957.



254 A.A. Martynyuk
6, 4(2004)

Lyapunov, A. M.

1. The General Problem of Stability of Motion. Kharkov, Kharkov Mathematical
Society, 1892. [Russian] (see also French translation in AnnḞac Toulouse 9
(1907) 203 – 474, and English translation in: Annȯf Mathematical Study, No. 17,
Princeton: Princeton University Press, 1949, and London: Taylor and Francis,
1992)

Malkin, I.G.

1. To the question of inverse Liapunov theorem on asymptotic stability. PriklṀatṀekh.
18(2) (1954) 129 –138. [Russian]

Martynyuk, A. A.

1. Stability of Motion of Complex Systems. Kiev: Naukova Dumka, 1975. [Russian]

2. The Lyapunov matrix function. NonlinȦnal. 8 (1984) 1223 –1226.

3. Extension of the state space of dynamical systems and the problem of stability.
Colloquia Mathematica Societaties Janas Bolyai 47. Differential Equations:
Qualitative Theory, Szeged (hungary), 1984, 711 – 749.

4. The matrix-function of Liapunov and hybrid systems stability. Int. Appl. Mech.
21 (4) (1985) 89-96.

5. The Lyapunov matrix function and stability of hybrid systems. NonlinȦnal. 10
(1986) 1449 –1457.

6. Stability analysis of nonlinear systems via Liapunov matrix function. PriklṀekh.
27(8) (1991) 3 –15. [Russian]

7. Matrix method of comparison in the theory of the stability of motion. IntȦpplṀech.
29 (1993) 861 – 867.

8. Stability Analysis: Nonlinear Mechanics Equations. New York: Gordon and
Breach Science Publishers, 1995.

9. Stability by Liapunov’s Matrix Function Method with Applications. New York:
Marcel Dekker, Inc., 1998.

10. Stability and Liapunov’s matrix functions method in dynamical systems. PriklṀekh.
34(10) (1998) 144 –152. [English]

11. A survey of some classical and modern developments of stability theory. Non-
linȦnal. 40 (2000) 483 – 496.

12. Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov
Matrix Functions. New York: Marcel Dekker, Inc., 2002.



6, 4(2004)
Matrix Liapunov’s Functions Method ... 255

13. Matrix Liapunov functions and stability analysis of dynamical systems. In: Ad-
vances in Stability Theory at the End of XXth Century (Ed.: A.A. Martynyuk),
London: Taylor and Francis, 2003, 135 –15

14. On polystability with respect a part of variables. DoklȦkadṄauk Russia 324 (1)
(1992) 39 – 4

15. A theorem on polystability. DoklȦkadṄauk SSSR 318 (4) (1991) 808 – 81

Martynyuk, A.A. (Ed.)

1. Advances in Stability Theory at the End of the 20th Century. London and New
York: Taylor and Francis, 2003.

Martynyuk, A. A. and Gutowski, R.

1. Integral Inequalities and Stability of Motion. Kiev: Naukova Dumka, 1979.
[Russian]

Martynyuk, A.A., Lakshmikantham, V., and Leela, S.

1. Stability of Motion: The Method of Integral Inequalities. Kiev: Naukova Dumka,
1989. [Russian]

Martynyuk, A.A., and Slyn’ko, V.I.

1. Solution of the problem of constructing Liapunov matrux function for a class
of large scale system. Nonlinear Dynamics and Systems Theory, 1 (2) (2001)
193 – 203.

2. On stability of large-scale system with non-autonomous connected subsystems.
IntȦpplṀech. 37 (10) (2001) 1341 – 135

3. [3] On stability of motion of large-scale system. DiffU̇ravn. 39 (6) (2003) (to
appear).

Massera, J. L.

1. Contributions to stability theory. Annȯf Math. 64 (1956) 182 –206.

Michel, A. N. and Miller, R. K.

1. Qualitative Analysis of Large Scale Dynamical Systems. New York: Academic
Press, 1977.

Michel, A. N., Wang, K. and Hu, B.

1. Qualitative Theory of Dynamical Systems. The Role of Stability Preserving
Mappings. New York: Marcel Dekker, Inc., 200



256 A.A. Martynyuk
6, 4(2004)

Persidskii, K.P.

1. Stability Theory of Solutions of Differential Equations. Theory Probability.
Selected works in two volumes, VolȦlma-Ata: Nauka, 1976. [Russian]

Piontkovskii, A. A., and Rutkovskaya, L. D.

1. Investiagation of certain stability theory problems by the vector Lyapunov func-
tion method. Avtomatika i Telemekhanika 10 (1967) 23 – 3 [Russian]

Rama Mohana Rao, M.

1. Ordinary Differential Equations. Theory and Applications. New Delhi – Madras:
Affiliated East-West Press Pvt Ltd, 1980.

Rouche, N., Habets, P., and Laloy, M.

1. Stability Theory by Liapunov’s Direct Method. New York: Springer-Verlag,
1977.

Rumiantsev, V.V., and Oziraner, A.S.

1. Stability and stabilization of motion with respect to a part of variables. Moscow:
Nauka, 1987. [Russian]

Sell, G .R.

1. References on dynamical systems. University of Minnesota, Preprint 91–129,
1993.

Šiljak, D.D.

1. Large-Scale Dynamic Systems: Stability and Structure. New York: North Hol-
land, 1978.

2. Decentralized Control of Complex Systems. Boston: Academic Press, Inc., 199

Sivasundaram, S.

1. Stability of dynamic systems on the time scale. Nonlinear Dynamics and Sys-
tems Theory 2 (2) (2002) 185 – 202.

Skowronski, J. M.

1. Nonlinear Liapunov Dynamics. Singapore, etc.: World Scientific, 1990.

Slyn’ko, V.I.

1. To the problem on polystability of motion. IntȦpplṀech. 37 (12) (2001) 1624 –
1628.

2. On constructing of non-diagonal elements of the Liapunov’s matrix function.
DopṄatsȦkadṄauk of Ukr. (4) (2001) 58 – 62.



6, 4(2004)
Matrix Liapunov’s Functions Method ... 257

Vincent, T.L.

1. Chaotic control systems. Nonlinear Dynamics and Systems Theory 1 (2) (2001)
205 – 218.

Voronov, A. A., and Matrosov, V.M. (Eds.)

1. Method of Vector Lyapunov Functions in Stability Theory. Moscow: Nauka,
1987. [Russian]

Vorotnikov, V. I.

1. Stability of dynamical systems with respect to part of variables. Moscow: Nauka,
199 [Russian]

Zhang, Bo

1. Formulas of Liapunov functions for systems of linear ordinary and delay differ-
ential equations. FunkcialĖkvac. 44 (2) (2001) 253 – 278.

Zubov, V. I.

1. Methods of A. M. Liapunov and its Applications. Leningrad: IzdatL̇eningradĠosU̇niversitet,
1957. [Russian]

2. Mathematical Methods of Investigations of Automatic Control Systems. Lenin-
grad: Mashinostroeniye, 1979. [Russian]

3. Control Processes and Stability. St-Petersberg: St-Petersberg GosU̇niversitet,
1999. [Russian]