Georgian Mathematical Journal Volume 8 (2001), Number 4, 645–664

ON THE ξ-EXPONENTIALLY ASYMPTOTIC STABILITY OF

LINEAR SYSTEMS OF GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS

M. ASHORDIA AND N. KEKELIA

Abstract. Necessary and sufficient conditions and effective sufficient conditions are established for the so-called ξ-exponentially asymptotic stability of the linear system

dx(t) = dA(t) · x(t) + df(t),

where A : [0, +∞[ → ℝ^{n×n} and f : [0, +∞[ → ℝ^{n} are respectively matrixand vector-functions with bounded variation components, on every closed interval from [0, +∞[ and ξ : [0, +∞[ → [0, +∞[ is a nondecreasing function such that lim_{t→+∞} ξ(t) = +∞.

2000 Mathematics Subject Classification: 34B05.

Key words and phrases: ξ-exponentially asymptotic stablility in the Lyapunov sense, linear generalized ordinary differential equations, Lebesgue– Stiltjes integral.

Let the components of matrix-functions A : [0, +∞[ → ℝ^{n×n} and vectorfunctions f : [0, +∞[ → ℝ^{n} have bounded total variations on every closed segment from [0, +∞[ .

In this paper, sufficient (necessary and sufficient) conditions are given for the so-called ξ-exponentially asymptotic stability in the Lyapunov sense for the linear system of generalized ordinary differential equations

dx(t) = dA(t) · x(t) + df(t).  (1)

The theory of generalized ordinary differential equations enables one to investigate ordinary differential, difference and impulsive equations from the unified standpoint. Quite a number of questions of this theory have been studied sufficiently well ([1]–[3], [5], [6], [8], [10], [11]).

The stability theory has been investigated thoroughly for ordinary differential equations (see [4], [7] and the references therein). As to the questions of stability for impulsive equations and for generalized ordinary differential equations they are studied, e.g., in [3], [9], [10] (see also the references therein).

The following notation and definitions will be used in the paper:

ℝ = ] − ∞, +∞[ , ℝ_{+} = [0, +∞[ , [a, b] and ]a, b[ (a, b ∈ ℝ) are, respectively, closed and open intervals.

Re z is the real part of the complex number z.

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

ℝ^{n×m} is the space of all real n × m-matrices X = (x_{ij})_{i,j=1}^{n,m} with the norm

∥X∥ = max_{j=1,...,m} ∑_{i=1}^{n} |x_{ij}|, |X| = (|x_{ij}|)_{i,j=1}^{n,m},

ℝ_{+}^{n×m} = {X = (x_{ij})_{i,j=1}^{n,m} : x_{ij} ≥ 0 (i = 1, …, n; j = 1, …, m)}.

The components of the matrix-function X are also denoted by [x]_{ij} (i = 1, …, n; j = 1, …, m).

O_{n×m} (or O) is the zero n × m-matrix.

ℝ^{n} = ℝ^{n×1} is the space of all real column n-vectors x = (x_{i})_{i=1}^{n}.

If X ∈ ℝ^{n×n}, then X^{−1} and det(X) are, respectively, the matrix inverse to X and the determinant of X. I_{n} is the identity n × n-matrix; diag(λ_{1}, …, λ_{n}) is the diagonal matrix with diagonal elements λ_{1}, …, λ_{n}.

r(H) is the spectral radius of the matrix H ∈ ℝ^{n×n}.

∨^{+∞}_{0}(X) = sup_{b∈ℝ_{+}} ∨^{b}_{0}(X), where ∨^{b}_{0}(X) is the sum of total variations on [0, b] of the

components x_{ij} (i = 1, …, n; j = 1, …, m) of the matrix-function X : ℝ_{+} →

ℝ^{n×m}; V (X)(t) = (v(x_{ij})(t))_{i,j=1}^{n,m}, where v(x_{ij})(0) = 0 and v(x_{ij})(t) = ∨^{t}_{0}(x_{ij})

for 0 < t < +∞ (i = 1, …, n; j = 1, …, m).

X(t−) and X(t+) are the left and the right limit of the matrix-function X : ℝ_{+} → ℝ^{n×m} at the point t; d_{1}X(t) = X(t)−X(t−), d_{2}X(t) = X(t+)−X(t).

BV_{loc}(ℝ_{+}, ℝ^{n×m}) is the set of all matrix-functions X : ℝ_{+} → ℝ^{n×m} of bounded total variation on every closed segment from ℝ_{+}.

L_{loc}(ℝ_{+}, ℝ^{n×m}) is the set of all matrix-functions X : ℝ_{+} → ℝ^{n×m} such that their components are measurable and integrable functions in the Lebesgue sense on every closed segment from ℝ_{+}.

C̃_{loc}(ℝ_{+}, ℝ^{n×m}) is the set of all matrix-functions X : ℝ_{+} → ℝ^{n×m} such that their components are absolutely continuous functions on every closed segment from ℝ_{+}.

s_{0} : BV_{loc}(ℝ_{+}, ℝ) → BV_{loc}(ℝ_{+}, ℝ) is the operator defined by

s_{0}(x)(t) ≡ x(t) − ∑_{0<τ≤t} d_{1}x(τ) − ∑_{0≤τ<t} d_{2}x(τ).

If g : ℝ_{+} → ℝ is a nondecreasing function x : ℝ_{+} → ℝ and 0 ≤ s < t < +∞, then

∫_{s}^{t} x(τ) dg(τ) = ∫_{]s,t[} x(τ) dg_{1}(τ) − ∫_{]s,t[} x(τ) dg_{2}(τ)

+ ∑_{s<τ≤t} x(τ) d_{1}g(τ) − ∑_{s≤τ<t} x(τ) d_{2}g(τ),

GENERALIZED ODE. ξ-EXPONENTIALLY ASYMPTOTIC STABILITY 647

where g_{j} : ℝ_{+} → ℝ (j = 1, 2) are continuous nondecreasing functions such that g_{1}(t)−g_{2}(t) ≡ s_{0}(g)(t), and ∫_{]s,t[} x(τ) dg_{j}(τ) is the Lebesgue–Stiltjes integral over the open interval ]s, t[ with respect to the measure corresponding to the function

g_{j} (j = 1, 2) (if s = t, then ∫_{s}^{t} x(τ) dg(τ) = 0).

A matrix-function is said to be nondecreasing if each of its components is nondecreasing.

If G = (g_{ik})_{i,k=1}^{ℓ,n} : ℝ_{+} → ℝ^{ℓ×n} is a nondecreasing matrix-function, X = (x_{ik})_{i,k=1}^{n,m} : ℝ_{+} → ℝ^{n×m}, then

∫_{s}^{t} dG(τ) · X(τ) = ( ∑_{k=1}^{n} ∫_{s}^{t} x_{kj}(τ) dg_{ik}(τ))_{i,j=1}^{ℓ,m} for 0 ≤ s ≤ t < +∞,
S_{0}(G)(t) ≡ (s_{0}(g_{ik})(t))_{i,k=1}^{ℓ,n}.

If G_{j} : ℝ_{+} → ℝ^{ℓ×n} (j = 1, 2) are nondecreasing matrix-functions, G(t) ≡ G_{1}(t) − G_{2}(t) and X : ℝ_{+} → ℝ^{n×m}, then

∫_{s}^{t} dG(τ) · X(τ) = ∫_{s}^{t} dG_{1}(τ) · X(τ) − ∫_{s}^{t} dG_{2}(τ) · X(τ) for 0 ≤ s ≤ t < +∞.

A and B : BV_{loc}(ℝ_{+}, ℝ^{n×n}) × BV_{loc}(ℝ_{+}, ℝ^{n×m}) → BV_{loc}(ℝ_{+}, ℝ^{n×m}) are the operators defined, respectively, by

A(X, Y)(t) = Y (t) + ∑_{0<τ≤t} d_{1}X(τ) · (I_{n} − d_{1}X(τ))^{−1} d_{1}Y (τ)

− ∑_{0≤τ<t} d_{2}X(τ) · (I_{n} + d_{2}X(τ))^{−1} d_{2}Y (τ) for t ∈ ℝ_{+}

and

B(X, Y)(t) = X(t)Y (t) − X(0)Y (0) − ∫_{0}^{t} dX(τ) · Y (τ) for t ∈ ℝ_{+}.

L : BV_{loc}^{2}(ℝ_{+}, ℝ^{n×n}) → BV_{loc}(ℝ_{+}, ℝ^{n×n}) is an operator given by

L(X, Y)(t) = ∫_{0}^{t} d(X(τ) + B(X, Y)(τ)) · X^{−1}(τ) for t ∈ ℝ_{+}.

We will use the following properties of these operators (see [2]):

B(X, B(Y, Z))(t) ≡ B(XY, Z)(t),
B(X, ∫_{0} dY (s) · Z(s))(t) ≡ ∫_{0}^{t} dB(X, Y)(s) · Z(s).

Under a solution of the system (1) we understand a vector-function x ∈ BV_{loc}(ℝ_{+}, ℝ^{n}) such that

x(t) = x(s) + ∫_{s}^{t} dA(τ) · x(τ) + f(t) − f(s) (0 ≤ s ≤ t < +∞).

Note that the linear system of ordinary differential equations

\frac{dx}{dt} = P (t)x + q(t) (t ∈ ℝ_{+}),  (2)

where P ∈ L_{loc}(ℝ_{+}, ℝ^{n×n}) and q ∈ L_{loc}(ℝ_{+}, ℝ^{n}), can be rewritten in form (1) if we set

A(t) ≡ ∫_{0}^{t} P (τ) dτ, f(t) ≡ ∫_{0}^{t} q(τ) dτ.

We assume that A ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n}), f ∈ BV_{loc}(ℝ_{+}, ℝ^{n}), A(0) = O_{n×n} and

det (I_{n} + (−1)^{j} d_{j}A(t)) ≠ 0 for t ∈ ℝ_{+} (j = 1, 2).

These conditions guarantee the unique solvability of the Cauchy problem for system (1) (see [11]).

Definition 1. Let ξ : ℝ_{+} → ℝ_{+} be a nondecreasing function such that

lim_{t→+∞} ξ(t) = +∞.  (3)

Then the solution x_{0} of system (1) is called ξ-exponentially asymptotic stable if there exists a positive number η such that for every ε > 0 there exists a positive number δ = δ(ε) such that an arbitrary solution x of system (1), satisfying the inequality ∥x(t_{0}) − x_{0}(t_{0})∥ < δ for some t_{0} ∈ ℝ_{+}, admits the estimate

∥x(t) − x_{0}(t)∥ < ε exp ( − η(ξ(t) − ξ(t_{0}))) for t ≥ t_{0}.

Stability, uniform stability, asymptotic stability and exponentially asymptotic stability are defined just in the same way as for systems of ordinary differential equations, i.e., when A(t) ≡ diag(t, …, t) (see, e.g., [4] or [7]). Note that the exponentially asymptotic stability is a particular case of the ξ-exponentially asymptotic stablility if we assume ξ(t) ≡ t.

Definition 2. System (1) is called stable in this or another sense if every solution of this system is stable in the same sense.

We will use the following propositions.

Proposition 1. System (1) is ξ-exponentially asymptotically stable (uniformly stable) if and only if its corresponding homogeneous system

dx(t) = dA(t) · x(t)  (1_{0})

is ξ-exponentially asymptotically stable (uniformly stable).

Proposition 2. System (1_{0}) is ξ-exponentially asymptotically stable (uniformly stable) if and only if its zero solution is ξ-exponentially asymptotically stable (uniformly stable).

Proposition 3. System (1_{0}) is ξ-exponentially asymptotically stable (uniformly stable) if and only if there exist positive numbers ρ and η such that

∥U(t, s)∥ ≤ ρ exp ( − η(ξ(t) − ξ(s))) for t ≥ s ≥ 0
( ∥U(t, s)∥ ≤ ρ for t ≥ s ≥ 0 ),

where U is the Cauchy matrix of system (1_{0}).

The proofs of these propositions are analogous to those for ordinary differential equations.

Therefore, the ξ-exponentially asymptotic stability (uniform stability) is not the property of a solution of system (1). It is the common property of all solutions and a vector-function f does not influence on this property. Hence the ξ-exponentially asymptotic stability (uniform stability) is the property of the matrix-function A and the following definition is natural.

Definition 3. The matrix-function A is called ξ-exponentially asymptotically stable (uniformly stable) if the system (1_{0}) is ξ-exponentially asymptotically stable (uniformly stable).

Theorem 1. Let the matrix-function A_{0} ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n}) be ξ-exponenti-ally asymptotically stable,

det (I_{n} + (−1)^{j} d_{j}A_{0}(t)) ≠ 0 for t ∈ ℝ_{+} (j = 1, 2)  (4)

and

lim_{t→+∞} ⋁_{t}^{ν(ξ)(t)} A(A_{0}, A − A_{0}) = 0,  (5)

where ξ : ℝ_{+} → ℝ_{+} is a nondecreasing function satisfying condition (3),

ν(ξ)(t) = sup {τ ≥ t : ξ(τ) ≤ ξ(t+) + 1}.

Then the matrix-function A is ξ-exponentially asymptotically stable as well.

To prove the theorem we will use the following lemma.

Lemma 1. Let the matrix-function A_{0} ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n}) satisfy condition (4). Let, moreover, the following conditions hold:

(a) the Cauchy matrix U_{0} of the system

dx(t) = dA_{0}(t) · x(t)  (6)

satisfies the inequality

|U_{0}(t, t_{0})| ≤ Ω exp ( − ξ(t) + ξ(t_{0})) (t ≥ t_{0})  (7)

for some t_{0} ∈ ℝ_{+}, where Ω ∈ ℝ_{+}^{n×n}, and ξ is a function from BV_{loc}(ℝ_{+}, ℝ) satisfying (3);

(b) there exists a matrix H ∈ ℝ_{+}^{n×n} such that

r(H) < 1  (8)

and

∫_{t_{0}}^{t} exp (ξ(t) − ξ(τ))|U_{0}(t, τ)| dV (A(A_{0}, A − A_{0}))(τ) < H for t ≥ t_{0}. (9)

Then an arbitrary solution x of system (1) admits an estimate

|x(t)| ≤ R|x(t_{0})| exp ( − ξ(t) + ξ(t_{0})) for t ≥ t_{0},  (10)

where R = (I_{n} − H)^{−1}Ω.

Proof. Let A = (a_{ik})_{i,k=1}^{n}, A_{0} = (a_{0ik})_{i,k=1}^{n}, U_{0} = (u_{0ik})_{i,k=1}^{n}, H = (h_{ik})_{i,k=1}^{n}, and x = (x_{i})_{i=1}^{n} be an arbitrary solution of system (1_{0}).

According to the variation of constants formula and properties of the Cauchy matrix U_{0} (see [11]) we have

x(t) = U_{0}(t, t_{0})x(t_{0}) + ∫_{t_{0}}^{t} U_{0}(t, s) d(A(s) − A_{0}(s)) · x(s)

− ∑_{t_{0}<s≤t} d_{1}U_{0}(t, s) · d_{1}(A(s) − A_{0}(s)) · x(s)

+ ∑_{t_{0}≤s<t} d_{2}U_{0}(t, s) · d_{2}(A(s) − A_{0}(s)) · x(s)

= U_{0}(t, t_{0})x(t_{0}) + ∫_{t_{0}}^{t} U_{0}(t, s) d(A(s) − A_{0}(s)) · x(s)

+ ∑_{t_{0}<s≤t} U_{0}(t, s) d_{1}A(s) · (I_{n} − d_{1}A_{0}(s))^{−1} d_{1}(A(s) − A_{0}(s)) · x(s)

− ∑_{t_{0}≤s<t} U_{0}(t, s) d_{2}A(s) · (I_{n} + d_{2}A_{0}(s))^{−1} d_{2}(A(s) − A_{0}(s)) · x(s).

Therefore

x(t) = U_{0}(t, t_{0})x(t_{0}) + ∫_{t_{0}}^{t} U(t, τ) dA(A_{0}, A − A_{0})(τ) · x(τ) for t ≥ t_{0}. (11)

Let

y_{k}(t) = max { exp (ξ(τ) − ξ(t_{0})) · |x_{k}(τ)| : t_{0} ≤ τ ≤ t},
y(t) = (y_{k}(t))_{k=1}^{n}.

Then

| ∑_{j,k=1}^{n} ∫_{t_{0}}^{t} u_{0ij}(t, τ)x_{k}(τ) d(b_{jk})(τ)| ≤ ∑_{j,k=1}^{n} ∫_{t_{0}}^{t} |u_{0ij}(t, τ)| |x_{k}(τ)| dv(b_{jk})(τ)

≤ ∑_{k,j=1}^{n} ∫_{t_{0}}^{t} exp ( − ξ(τ) + ξ(t_{0}))|u_{0ij}(t, τ)| dv(b_{jk})(τ) · y_{k}(t)

for t ≥ t_{0}, (i = 1, …, n),

where b_{jk}(t) ≡ A(a_{0jk}, a_{jk} − a_{0jk})(t) (j, k = 1, …, n). From this and (11) we have

exp (ξ(t) − ξ(t_{0})) · |x_{i}(t)| ≤ ∑_{k=1}^{n} exp (ξ(t) − ξ(t_{0}))|u_{0ik}(t, t_{0})| |x_{k}(t_{0})|

+ ∑_{k,j=1}^{n} ∫_{t_{0}}^{t} exp (ξ(t) − ξ(τ))|u_{0ij}(t, τ)| dv(b_{jk})(τ) · y_{k}(t)

for t ≥ t_{0}, (i = 1, …, n).

By this, (7) and (9) we obtain

y(t) ≤ Ω|x(t_{0})| + Hy(t) for t ≥ t_{0}.

Hence

(I_{n} − H)y(t) ≤ Ω|x(t_{0})| for t ≥ t_{0}.  (12)

On the other hand, by (8) the matrix I_{n} − H is nonsingular and the matrix (I_{n} − H)^{−1} is nonnegative since H is a nonnegative matrix. From this, (12) and the definition of y we have

y(t) ≤ (I_{n} − H)^{−1}Ω|x(t_{0})| for t ≥ t_{0}

and

|x(t)| ≤ (I_{n} − H)^{−1}Ω|x(t_{0})| exp ( − ξ(t) + ξ(t_{0})) for t ≥ t_{0}.

Therefore estimate (10) is proved.

Proof of Theorem 1. By the ξ-exponentially asymptotic stability of the matrixfunction A_{0} and Proposition 3 there exist positive numbers η and ρ_{0} such that the Cauchy matrix U_{0} of system (6) satisfies the estimate

|U_{0}(t, τ)| ≤ R_{0} exp ( − 2η(ξ(t) − ξ(τ))) for t ≥ τ ≥ 0,  (13)

where R_{0} is an n × n matrix whose every component equals ρ_{0}.

Let

ε = (4nρ_{0})^{−1}( exp(η) − 1) exp(−2η).  (14)

652 M. ASHORDIA AND N. KEKELIA

By (5) there exists t^{∗} ∈ ℝ_{+} such that

⋁_{t}^{ν(ξ)(t)} A(A_{0}, A − A_{0}) < ε for t ≥ t^{∗}.  (15)

On the other hand, by (13) we have

∫_{t_{0}}^{t} exp (η(ξ(t) − ξ(τ)))|U_{0}(t, τ)| dV (B)(τ) ≤ J (t) (t ≥ t_{0})  (16)

for every t_{0} ≥ 0, where B(t) ≡ A(A_{0}, A − A_{0})(t) and

J (t) ≡ R_{0} ∫_{t_{0}}^{t} exp ( − η(ξ(t) − ξ(τ))) dV (B)(τ).

Let k(t) be the integer part of ξ(t) − ξ(t_{0}) for every t ≥ t_{0},

T_{i} = {τ ≥ t_{0} : ξ(t_{0}) + i ≤ ξ(τ) < ξ(t_{0}) + i + 1} (i = 0, …, k(t)), where k_{i} = k(t_{i}) (i = 0, …, k(t)), the points t_{0}, t_{1}, …, t_{k(t)} are defined by

t_{0} = sup T_{0}, t_{i} = \begin{cases}t_{i−1} & if T_{i} = ∅ \\ sup T_{i} & if T_{i} ≠ ∅ \end{cases} (i = 1, …, k(t)).

Let us show that

t_{i} ≤ ν(ξ)(t_{i−1}) (i = 1, …, k(t)).  (17)

If T_{i} = ∅, then (17) is evident.

Let now T_{i} ≠ ∅. It is sufficient to show that

T_{i} ⊂ Q_{i},  (18)

where

Q_{i} = {τ : ξ(τ) < ξ(t_{i−1}+) + 1}.

It is easy to verify that

ξ(t_{i−1}+) ≥ ξ(t_{0}) + i.  (19)

Indeed, otherwise there exists δ > 0 such that

ξ(t_{i−1} + s) < ξ(t_{0}) + i for 0 ≤ s ≤ δ.

On the other hand, by the definition of t_{i−1} we have

ξ(t_{0}) + i − 1 ≤ ξ(t_{i−1}−)

and therefore

ξ(t_{0}) + i − 1 ≤ ξ(t_{i−1} + s) < ξ(t_{0}) + i for 0 ≤ s ≤ δ.

But this contradicts the definition of t_{i−1}.

Let τ ∈ T_{i}. Then from (19) and the inequality ξ(τ) < ξ(t_{0}) + i + 1 it follows that ξ(τ) < ξ(t_{i−1}+) + 1, τ_{i} ∈ Q_{i}. Hence (17) is proved.

Let t_{0} ≥ t^{∗}. Then according to (15) and (17) we get

J (t) ≤ R_{0} exp ( − η(ξ(t) − ξ(t_{0}))) ∑_{i=1}^{1+k(t)} ∫_{t_{i−1}}^{t_{i}} exp (η(ξ(τ) − ξ(t_{0}))) dV (B)(τ)

= R_{0} exp ( − η(ξ(t) − ξ(t_{0})))( ∑_{i=1, i=1+k_{i}}^{1+k(t)} ∫_{t_{i−1}}^{t_{i}} exp (η(ξ(τ) − ξ(t_{0}))) dV (B)(τ)

+ ∑_{i=1, i≠1+k_{i}}^{1+k(t)} ∫_{t_{i−1}}^{t_{i}} exp (η(ξ(τ) − ξ(t_{0}))) dV (B)(τ))

≤ R_{0} exp ( − η(ξ(t) − ξ(t_{0})))( ∑_{i=1, i=1+k_{i}}^{1+k(t)} exp(η i)[V (B)(t_{i}) − V (B)(t_{i−1})]

+ ∑_{i=1, i≠1+k_{i}}^{1+k(t)} exp(η i)[V (B)(t_{i}) − V (B)(t_{i−1})]

+ ∑_{i=1, i≠1+k_{i}}^{1+k(t)} exp ((1 + k_{i})η) d_{1}B(t_{i}))

≤ εR_{0} exp ( − η(ξ(t) − ξ(t_{0})))( ∑_{i=1}^{1+k(t)}exp(η i) + ∑_{i=1, i≠1+k_{i}}^{1+k(t)}exp ((1 + k_{i})η))

≤ 2εR_{0} exp ( − η(ξ(t) − ξ(t_{0}))) ∑_{i=1}^{1+k(t)} exp(η i)

= 2εR_{0} exp ( − η(ξ(t) − ξ(t_{0}))) exp(η)( exp ((1 + k(t))η) − 1)( exp(η) − 1)^{−1}

≤ 2εR_{0} exp ( − ηk(t)) exp ((2 + k(t))η)( exp(η) − 1)
= 2εR_{0} exp(2η)( exp(η) − 1)^{−1}.

From (14), (16) and (20) it follows that inequality (9) holds for t_{0} ≥ t^{∗}, where H ∈ ℝ^{n×n} is the matrix whose every component equals \frac{1}{2n}. On the other hand, it can be easily shown that

r(H) < \frac{1}{2} .

Consequently, by Lemma 1 an arbitrary solution x of the system (1_{0}) admits an estimate

∥x(t)∥ ≤ ρ exp ( − η(ξ(t) − ξ(t_{0}))) for t ≥ t_{0} ≥ t^{∗},

where ρ > 0 is a constant independent of t_{0}.

Note that a similar theorem is proved in [7] for the case of ordinary differential equations.

Corollary 1. Let the components a_{ik} (i, k = 1, …, n) of the matrix-function A satisfy the conditions

1 + (−1)^{j} d_{j}a_{ii}(t) ≠ 0 for t ∈ ℝ_{+} (i = 1, …, n; j = 1, 2),  (20)
lim_{t→+∞} ⋁_{t}^{ν(ξ)(t)} A(a_{ii}, a_{ik}) = 0 (i, k = 1, …, n),  (21)

and

a_{ii}(t) − a_{ii}(τ) ≤ −η(ξ(t) − ξ(τ)) for t ≥ τ ≥ 0 (i = 1, …, n), (22)

where η > 0, ξ : ℝ_{+} → ℝ_{+} is a nondecreasing function satisfying condition (3), and ν(ξ) : ℝ_{+} → ℝ_{+} is the function defined as in Theorem 1. Then the matrix-function A is ξ-exponentially asymptotically stable.

Proof. Corollary 1 follows from Theorem 1 if we assume that

A_{0}(t) ≡ diag (a_{11}(t), …, a_{nn}(t)).

Indeed, by the definition of the operator A we have

[A(A_{0}, A − A_{0})(t)]_{ik} = a_{ik}(t) + ∑_{0<τ≤t} \frac{d_{1}a_{ii}(τ)}{1 − d_{1}a_{ii}(τ)} d_{1}a_{ik}(τ)

− ∑_{0≤τ<t} \frac{d_{2}a_{ii}(τ)}{1 + d_{2}a_{ii}(τ)} d_{2}a_{ik}(τ) = A(a_{ii}, a_{ik})(t)

for t ∈ ℝ_{+} (i ≠ k; i, k = 1, …, n)

and

[A(A_{0}, A − A_{0})(t)]_{ii} = 0 for t ∈ ℝ_{+} (i = 1, …, n).

Therefore, by (21) and (22) the matrix-function A is ξ-exponentially asymptotically stable.

Corollary 2. Let the matrix-function P ∈ L_{loc}(ℝ_{+}, ℝ^{n×n}) be ξ-exponentially asymptotically stable and

lim_{t→+∞} ⋁_{t}^{ξ(t)+1}(A − A_{0}) = 0 for t ∈ ℝ_{+},

where A_{0}(t) ≡ ∫_{0}^{t} P (τ) dτ, ξ : ℝ_{+} → ℝ_{+} is a continuous nondecreasing func-

tion satisfying condition (3). Then the matrix-function A is ξ-exponentially asymptotically stable as well.

Proof. Corollary 2 immediately follows from Theorem 1 if we observe that

A(A_{0}, A − A_{0})(t) = A(t) − A_{0}(t) (t ∈ ℝ_{+})

in this case and, moreover,

ν(ξ)(t) = ξ(t) + 1 (t ∈ ℝ_{+})

because ξ is a nondecreasing continuous function.

Theorem 2. The matrix-function A is ξ-exponentially asymptotically stable if and only if there exist a positive number η and a nonsingular matrix-function H ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n}) such that

sup {∥H^{−1}(t)H(s)∥ : t ≥ s ≥ 0} < +∞  (23)

and

⋁_{0}^{+∞} B_{η}(H, A) < +∞,  (24)

where

B_{η}(H, A)(t) ≡ ∫_{0}^{t} exp ( − ηξ(τ)) d[ exp (ηξ(τ))H(τ)

+ exp (ηξ(τ))H(τ)A(τ) − ∫_{0}^{τ} d( exp (ηξ(s))H(s)) · A(s)].  (25)

Proof. Let U and U^{∗} be the Cauchy matrices of systems (1_{0}) and

dy(t) = dA^{∗}(t) · y(t),

respectively, where A^{∗}(t) = L(exp(ηξ(·))H, A)(t). Then by the definition of the operator L and by the equality

U(t, s) = exp ( − η(ξ(t) − ξ(s)))H^{−1}(t)U^{∗}(t, s)H(s) for t, s ∈ ℝ_{+}

we obtain that

exp (η(ξ(τ) − ξ(s)))U(t, s) = H^{−1}(t)H(s)

+H^{−1}(t) ∫_{s}^{t} exp (η(ξ(τ) − ξ(s))) dB_{η}(H, A)(τ) · U(τ, s) for t, s ∈ ℝ_{+}.

Hence

W (t, s) = H^{−1}(t)H(s) + H^{−1}(t)d_{1}B_{η}(H, A)(t) · W (t, s)
+ H^{−1}(t) ∫_{s}^{t} dG(τ) · W (τ, s) for t, s ∈ ℝ_{+},  (26)

656 M. ASHORDIA AND N. KEKELIA

where

W (t, s) = exp (η(ξ(t) − ξ(s)))U(t, s), G(t) = B_{η}(H, A)(t−).

On the other hand by (23), (24) and by the equalities

det (I_{n} + (−1)^{j} d_{j}A^{∗}(t)) = exp ((−1)^{j}nη d_{j}ξ(t)) det (H(t) + (−1)^{j} d_{j}H(t))

× det (I_{n} + (−1)^{j} d_{j}A(t)) det (H^{−1}(t)) for t ∈ ℝ_{+} (j = 1, 2) and

I_{n} + (−1)^{j}H^{−1}(t) d_{j}B_{η}(H, A)(t)
= H^{−1}(t)(I_{n} + (−1)^{j} d_{j}A^{∗}(t))H(t) for t ∈ ℝ_{+} (j = 1, 2)

there exists a positive number r_{0} such that

det (I_{n} + (−1)^{j}H^{−1}(t) d_{j}B_{η}(H, A)(t)) ≠ 0 for t ∈ ℝ_{+} (j = 1, 2) (27)

and

∥(I_{n} + (−1)^{j}H^{−1}(t) d_{j}B_{η}(H, A)(t))^{−1}∥ < r_{0} for t ∈ ℝ_{+} (j = 1, 2). (28)

From (26), by (23), (27) and (28) we get

∥W (t, s)∥ ≤ r_{0}(ρ + ρ_{1} ∫_{s}^{t} ∥W (τ, s)∥ d∥V (G)(τ)∥) for t ≥ s ≥ 0,

where

ρ = sup {∥H^{−1}(t)H(s)∥ : t ≥ s}, ρ_{1} = ρ∥H^{−1}(0)∥.

Hence, according to the Gronwall inequality ([11])

∥W (t, s)∥ ≤ M < +∞ for t ≥ s ≥ 0,

where

M = r_{0} exp (r_{0}ρ_{1} ⋁_{0}^{+∞} B_{η}(H, A)).

Therefore

∥U(t, s)∥ ≤ M exp ( − η(ξ(t) − ξ(s))) for t ≥ s ≥ 0,

i.e., the matrix-function A is ξ-exponentially asymptotically stable.

Let us show the necessity. Let the matrix-function A is ξ-exponentially asymptotically stable. Then there exist positive numbers η and ρ such that

∥Z(t)Z^{−1}(s)∥ ≤ ρ exp ( − η(ξ(t) − ξ(s))) for t ≥ s ≥ 0,  (29)

where Z (Z(0) = I_{n}) is the fundamental matrix of system (1_{0}).

Let

H(t) ≡ exp ( − ηξ(t))Z^{−1}(t).

Then according to (25), (29) and the equality

Z^{−1}(t) = I_{n} − Z^{−1}(t)A(t) + ∫_{0}^{t} dZ^{−1}(τ) · A(τ) for t ∈ ℝ_{+}  (30)

(see [11]) we have

∥H^{−1}(t)H(s)∥ = ∥Z(t)Z^{−1}(s)∥ exp (η(ξ(t) − ξ(s))) ≤ ρ for t ≥ s ≥ 0

and

B_{η}(H, A)(t) = B_{η}( exp(−ηξ)Z^{−1}, A)(t) = 0 for t ∈ ℝ_{+}.

Therefore conditions (23) and (24) are fulfilled.

Remark 1. If in Theorem 2 the function ξ : ℝ_{+} → ℝ_{+} is continuous, then condition (24) can be rewritten as

∥ ∫_{0}^{+∞}dV (I(H, A) + η diag(ξ, …, ξ))(t) · |H(t)|∥ < +∞.

Corollary 3. Let the matrix-function Q ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n}) be uniformly stable and

det (I_{n} + (−1)^{j} d_{j}Q(t)) ≠ 0 for t ∈ ℝ_{+} (j = 1, 2).  (31)

Let, moreover, there exist a positive number η such that

∥ ∫_{0}^{+∞}|Z^{−1}(t)| dV (G_{η}(ξ, Q, A))(t)∥ < +∞  (32)

where Z (Z(0) = I_{n}) is the fundamental matrix of the system

dz(t) = dQ(t) · z(t),  (33)

and

G_{η}(ξ, Q, A)(t) ≡ A(Q, A − Q)(t) + ηs_{0}(ξ)(t) · I_{n}

+ ∑_{0<τ≤t} exp ( − ηξ(τ)) d_{1} exp (ηξ(τ)) · (I_{n} − d_{1}Q(τ))^{−1}(I_{n} − d_{1}A(τ))

+ ∑_{0≤τ<t} exp ( − ηξ(τ)) d_{2} exp (ηξ(τ)) · (I_{n} + d_{2}Q(τ))^{−1}(I_{n} + d_{2}A(τ)). (34)

Then the matrix-function A is ξ-exponentially asymptotically stable.

Proof. Let B_{η}(H, A) be the matrix-function defined by (25), where H(t) ≡ Z^{−1}(t). Using the formula of integration by parts ([11]), the properties of the operator B given above and equality (30), we conclude that

B_{η}(H, A)(t)

= ∫_{0}^{t} exp ( − ηξ(τ)) d( exp(ηξ(τ))Z^{−1}(τ) + B( exp(ηξ)Z^{−1}, A)(τ))

= ∫_{0}^{t} exp ( − ηξ(τ)) d( exp(ηξ(τ))Z^{−1}(τ))

+ ∫_{0}^{t} exp ( − ηξ(τ)) dB( exp(ηξ)I_{n}, B( exp(ηξ)Z^{−1}, A)(τ))

= ∫_{0}^{t} exp ( − ηξ(τ)) d( exp(ηξ(τ))Z^{−1}(τ))

+ ∫_{0}^{t} exp ( − ηξ(τ)) dB( exp(ηξ)I_{n}, B(Z^{−1}, A))(τ) for t ∈ ℝ_{+};  (35)

∫_{0}^{t} exp ( − ηξ(τ)) d( exp(ηξ(τ))Z^{−1}(τ))

= ∫_{0}^{t} Z^{−1}(τ) d(ηs_{0}(ξ)(τ)I_{n} − A(Q, Q)(τ))

+ ∑_{0<s≤τ} exp ( − ηξ(s)) d_{1} exp (ηξ(s)) · (I_{n} − d_{1}Q(s))^{−1}

+ ∑_{0≤s<τ} exp ( − ηξ(s)) d_{2} exp (ηξ(s)) · (I_{n} + d_{2}Q(s))^{−1} for t ∈ ℝ_{+}; (36)

B(Z^{−1}, A)(t) ≡ ∫_{0}^{t} Z^{−1}(τ) dA(τ) − ∑_{0<τ≤t} d_{1}Z^{−1}(τ) · d_{1}A(τ)

+ ∑_{0≤τ<t} d_{2}Z^{−1}(τ) · d_{2}A(τ) = ∫_{0}^{t} Z^{−1}(τ) dA(Q, A − Q)(τ) for t ∈ ℝ_{+}, (37)

B( exp(ηξ)I_{n}, B(Z^{−1}, A))(t)

= ∫_{0}^{t} Z^{−1}(τ) dB( exp(ηξ)I_{n}, A(Q, A))(τ) for t ∈ ℝ_{+}  (38)

and

∫_{0}^{t} exp ( − ηξ(τ)) dB( exp(ηξ)I_{n}, A(Q, A))(τ)

= A(Q, A)(t) − ∑_{0<τ≤t} exp ( − ηξ(τ)) d_{1} exp (ηξ(τ)) · (I_{n} − d_{1}Q(τ))^{−1} d_{1}A(τ)

+ ∑_{0≤τ<t} exp ( − ηξ(τ)) d_{2} exp (ηξ(τ)) · (I_{n} + d_{2}Q(τ))^{−1} d_{2}A(τ) (39)
for t ∈ ℝ_{+}.

From (35), by (36)–(39) we get

B_{η}(H, A)(t) = ∫_{0}^{t} exp ( − ηξ(τ)) d( exp(ηξ(τ)) · Z^{−1}(τ))

+ ∫_{0}^{t} Z^{−1}(τ) d( ∫_{0}^{τ} exp ( − ηξ(s)) dB( exp(ηξ)I_{n}, A(Q, A))(s))

= ∫_{0}^{t} Z^{−1}(τ) dG_{η}(ξ, Q, A)(τ) for t ∈ ℝ_{+}

and

⋁_{0}^{+∞} B_{η}(H, A) ≤ ∥ ∫_{0}^{+∞}|Z^{−1}(t)| dV (G_{η}(ξ, Q, A))(t)∥.

Therefore from (32) and the fact that the matrix-function Q is ξ-exponen-tially asymptotically stable, it follows that the conditions of Theorem 2 are fulfilled.

Remark 2. In Corollary 3 if the function ξ : ℝ_{+} → ℝ_{+} is continuous, then

G_{η}(ξ, Q, A)(t) = A(Q, A − Q)(t) + ηξ(t)I_{n} for t ∈ ℝ_{+}.

Corollary 4. Let the matrix-function Q ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n}), satisfying condition (31), be ξ-exponentially asymptotically stable and

⋁_{0}^{+∞} B(Z^{−1}, A − Q) < +∞,  (40)

where Z (Z(0) = I_{n}) is the fundamental matrix of system (33). Then the matrix-function A is ξ-exponentially asymptotically stable as well.

Proof. Since Q is ξ-exponentially asymptotically stable there exists a positive number η such that the estimate (29) holds.

Let now B_{η}(H, A) be the matrix-function defined by (25), where

H(t) ≡ exp ( − ηξ(t))Z^{−1}(t).

Using equality (30) for the matrix-function Q we conclude that

Z^{−1}(t) = I_{n} + B(Z^{−1}, −Q)(t) for t ∈ ℝ_{+}

and

B_{η}(H, A)(t) = ∫_{0}^{t} exp ( − ηξ(τ)) dB(Z^{−1}, A − Q)(τ) for t ∈ ℝ_{+}.

By this and (40), condition (24) holds. Therefore, the conditions of Theorem 2 are fulfilled.

Remark 3. By the equality

B(Z^{−1}, A − Q)(t) = ∫_{0}^{t} Z^{−1}(τ) d(A(τ) − Q(τ)) for t ∈ ℝ_{+}

the condition

∥ ∫_{0}^{+∞}|Z^{−1}(t)| dV (A(Q, A − Q))(t)∥ < +∞

guarantees the fulfilment of condition (40) in Corollary 4. On the other hand,

lim_{η→0+} G_{η}(ξ, Q, A)(t) = A(Q, A − Q)(t) for t ∈ ℝ_{+},

where G_{η}(ξ, Q, A)(t) is defined by (34). Consequently, Corollary 3 is true in the limit case (η = 0), too, if we require the ξ-exponentially asymptotic stability of Q instead of the uniform stability.

Corollary 5. Let Q ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n}) be a continuous matrix-function satisfying the Lappo-Danilevskiǐ condition

∫_{0}^{t} Q(τ) dQ(τ) = ∫_{0}^{t} dQ(τ) · Q(τ) for t ∈ ℝ_{+}.

Let, moreover, there exist a nonnegative number η such that

∥ ∫_{0}^{+∞}| exp(−Q(t))| dV (A − Q + ηξI_{n})(t)∥ < +∞,

where ξ : ℝ_{+} → ℝ_{+} is a continuous function satisfying condition (3). Then:

(a) the uniform stability of the matrix-function Q guarantees the ξ-exponen-tially asymptotic stability of the matrix-function A for η > 0;

(b) the ξ-exponentially asymptotic stability of Q guarantees the ξ-exponenti-ally asymptotic stability of A for η = 0.

Proof. The corollary follows immediatelly from Corollaries 3 and 4 and Remark 3 if we note that

Z(t) = exp(Q(t)) for t ∈ ℝ_{+}

and in this case

G_{η}(ξ, Q, A)(t) = A(t) − Q(t) + ηξ(t)I_{n} for t ∈ ℝ_{+}.

Corollary 6. Let there exist a nonnegative number η such that the components a_{ik} (i, k = 1, …, n) of the matrix-function A satisfy conditions (20),

s_{0}(a_{ii})(t) − s_{0}(a_{ii})(τ) − ∑_{τ<s≤t} ln |1 − d_{1}a_{ii}(s)| + ∑_{τ≤s<t} ln |1 + d_{2}a_{ii}(s)|

≤ −η(s_{0}(ξ)(t) − s_{0}(ξ)(τ)) − μ(ξ(t) − ξ(τ)) for t ≥ τ ≥ 0  (41)
(i = 1, …, n),
(−1)^{j} ∑_{0≤t<+∞} |z_{i}^{−1}(t)|( exp ((−1)^{j} d_{j}ξ(τ)) − 1) < +∞  (42)
(j = 1, 2; i = 1, …, n)

and

∫_{0}^{+∞}|z_{i}^{−1}(t)| dv(g_{ik})(t) < +∞ (i ≠ k; i, k = 1, …, n),  (43)

where μ = 0 if η > 0, μ > 0 if η = 0,

z_{i}(t) ≡ exp (s_{0}(a_{ii})(t) + ηs_{0}(ξ)(t))

× ∏_{0<τ≤t} (1 − d_{1}a_{ii}(τ))^{−1} ∏_{0≤τ<t} (1 + d_{2}a_{ii}(τ)) (i = 1, …, n),

g_{ik}(t) ≡ s_{0}(a_{ik})(t) + ∑_{0<τ≤t} exp ( − η d_{1}ξ(τ)) d_{1}a_{ik}(τ) · (1 − d_{1}a_{ii}(τ))^{−1}

+ ∑_{0≤τ<t} exp (η d_{2}ξ(τ)) d_{2}a_{ik}(τ) · (1 + d_{2}a_{ii}(τ))^{−1}

(i ≠ k; i, k = 1, …, n),

and ξ : ℝ_{+} → ℝ_{+} is a nondecreasing function satisfying condition (3). Then the matrix-function A is ξ-exponentially asymptotically stable.

Proof. For η > 0 the corollary follows from Corollary 3 if we assume that

Q(t) ≡ diag (a_{11}(t) + ηs_{0}(ξ)(t), …, a_{nn}(t) + ηs_{0}(ξ)(t)).

Indeed, by the definition of the operator A we have

[A(Q, A − Q)(t)]_{ik} = a_{ik}(t) + ∑_{0<τ≤t} d_{1}a_{ii}(τ) · (1 − d_{1}a_{ii}(τ))^{−1}d_{1}a_{ik}(τ)

− ∑_{0≤τ<t} d_{2}a_{ii}(τ) · (1 + d_{2}a_{ii}(τ))^{−1}d_{2}a_{ik}(τ) for t ∈ ℝ_{+} (i ≠ k; i, k = 1, …, n),

[A(Q, A − Q)(t)]_{ii} = −ηs_{0}(ξ)(t) for t ∈ ℝ_{+} (i = 1, …, n).

From these relations, using (34) we obtain

[G_{η}(ξ, Q, A)(t)]_{ik} = a_{ik}(t) + ∑_{0<τ≤t} d_{1}a_{ii}(τ) · (1 − d_{1}a_{ii}(τ))^{−1}d_{1}a_{ik}(τ)

− ∑_{0≤τ<t} d_{2}a_{ii}(τ) · (1 + d_{2}a_{ii}(τ))^{−1}d_{2}a_{ik}(τ)

− ∑_{0<τ≤t} d_{1}a_{ik}(τ) · (1 − d_{1}a_{ii}(τ))^{−1}(1 − exp ( − ηd_{1}ξ(τ)))

− ∑_{0≤τ<t} d_{2}a_{ik}(τ) · (1 + d_{2}a_{ii}(τ))^{−1}(1 − exp (ηd_{2}ξ(τ)))

= g_{ik}(t) for t ∈ ℝ_{+} (i ≠ k; i, k = 1, …, n)

and

[G_{η}(ξ, Q, A)(t)]_{ii} = ∑_{0<τ≤t} (1 − exp ( − ηd_{1}ξ(τ)))

+ ∑_{0≤τ<t} (1 − exp (ηd_{2}ξ(τ))) for t ∈ ℝ_{+} (i = 1, …, n).

On the other hand, the matrix-function Z(t) = diag(z_{1}(t), …, z_{n}(t)) is the fundamental matrix of the system (33), satisfying the condition Z(0) = I_{n}. Therefore, by (41)–(43) the conditions of Corollary 3 are valid. For η = 0 the corollary follows from Corollary 4 and Remark 3.

Remark 4. If, in addition to (20), the components a_{ik} (i, k = 1, …, n) of the matrix-function A satisfy the condition

(−1)^{j} d_{j}a_{ik}(t) · (1 + (−1)^{j} d_{j}a_{ii}(t))^{−1} ≥ 0 for t ∈ ℝ_{+}
(i ≠ k; i, k = 1, …, n; j = 1, 2),

then we can assume without loss of generality that η > 0 and μ = 0 in Corollary 6.

Corollary 7. Let there exist a nonnegative number η such that

∫_{0}^{+∞}t^{n_{ℓ}−1} exp(−t Re λ_{ℓ}) dv(b_{ik})(t) < +∞ (ℓ = 1, …, m; i, k = 1, …, n),

where b_{ik}(t) ≡ a_{ik}(t)−p_{ik}t+ηξ(t) (i, k = 1, …, n), ξ : ℝ_{+} → ℝ_{+} is a nondecreasing function satisfying condition (3), and λ_{1}, …, λ_{m} (λ_{i} ≠ λ_{j} for i ≠ j) are the characteristic values of the matrix P = (p_{ik})_{i,k=1}^{n} with multiplicities n_{1}, …, n_{m}, respectively. Then:

(a) if η > 0, Re λ_{ℓ} ≤ 0 (ℓ = 1, …, m), and, in addition, n_{ℓ} = 1 for Re λ_{ℓ} = 0, then A is ξ-exponentially asymptotically stable;

(b) if η = 0 and Re λ_{ℓ} < 0 (ℓ = 1, …, m), then A is exponentially asymptotically stable;

(c) if η = 0 and P is ξ-exponentially asymptotically stable, then A is ξexponentially asymptotically stable as well.

Proof. The corollary immediatelly follows from Corollary 5 if we assume Q(t) ≡ P t and derive the matrix-function exp(−P t) by the standard way using the Jordan canonical form of the matrix P.

Consider now system (2), where P ∈ L_{loc}(ℝ_{+}, ℝ^{n×n}), q ∈ L_{loc}(ℝ_{+}, ℝ^{n}).

Theorem 2 and Corollary 6 have the following form for this system.

Theorem 2′. Let ξ : ℝ_{+} → ℝ_{+} be an absolutely continuous nondecreasing function satisfying condition (3). Then the matrix-function P ∈ L_{loc}(ℝ_{+}, ℝ^{n×n}) is ξ-exponentially asymptotically stable if and only if there exist a positive number η and nonsingular matrix-function H ∈ C̃_{loc}(ℝ_{+}, ℝ^{n×n}) such that

sup {∥H^{−1}(t)H(s)∥ : t ≥ s ≥ 0} < +∞

and

∫_{0}^{+∞}∥H′(t) + H(t)(P (t) + ηξ′(t)I_{n})∥ dt < +∞.

Corollary 6′. Let there exist a positive number η such that the components p_{ik} ∈ L_{loc}(ℝ_{+}, ℝ) (i, k = 1, …, n) of the matrix-function P satisfy the conditions

p_{ii}(t) ≤ −η for t ≥ t^{∗} (i = 1, …, n)

and

∫_{t^{∗}}^{+∞}exp ( − ∫_{t_{∗}}^{t} (p_{ii}(τ) + η) dτ ) |p_{ik}(t)| dt < +∞ (i ≠ k; i, k = 1, …, n)

for some t^{∗} ∈ ℝ_{+}. Then the matrix-function P is exponentially asymptotically stable.

Acknowledgement

This work was supported by a research grant in the framework of the Bilateral S&T Cooperation between the Hellenic Republic and Georgia.

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(Received 13.03.2001)

Authors’ addresses:

M. Ashordia I. Vekua Institute of Applied Mathematics Tbilisi State University 2, University St., Tbilisi 380043 Georgia

Sukhumi Branch of Tbilisi State University 12, Jikia St., Tbilisi 380086 Georgia

N. Kekelia Sukhumi Branch of Tbilisi State University 12, Jikia St., Tbilisi 380086 Georgia