Mem. Differential Equations Math. Phys. 21(2000), 134–137

M. Ashordia and N. Kekelia

ON LIAPUNOV STABILITY OF LINEAR SYSTEMS OF GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS

(Reported on March 6, 2000)

Consider the linear system of generalized ordinary differential equations

dx(t) = dA(t) · p(t) · x(t) + df(t),  (1)

where A : [0, +∞[→ ℝ^{n×n} and f : [0, +∞[→ ℝ^{n} are, respectively, real matrix- and vector-functions with locally bounded variation components, and p : [0, +∞[→ ℝ^{n×n} is a matrix-function, locally integrable with respect to A.

In this paper we give some sufficient conditions guaranteeing the stability with respect to, small perturbation in the Liapunov sense, of the system (1).

Before passing to the statement of the basic results, we give some notation and definitions.

ℝ =] − ∞, +∞[ is the set of all real numbers, [a, b] and ]a, b[ are, respectively, closed and open intervals, ℝ_{+} = [0, +∞[.

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

∥x∥ = max_{i=1,...,m} ∑_{i=1}^{n}|x_{ij}|.

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

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 inverse to x matrix and the determinant of x; I_{n} is the identity n × n matrix.

V^{d}_{c} = sup{V^{b}_{a} (x) : c < a < b < d}, where V^{b}_{a} (x) is the sum of total variations on the

closed interval [a, b] of the components x_{ij} (i = 1, …, n; j = 1, …, m) of the matrix-

function x :]c, d[→ ℝ^{n×m}, v(x)(t) = (v(x_{ij})(t))_{i,j=1}^{n}, where v(x_{ij})(t) = ( V^{t}_{−∞}x_{ij}) for

t ∈]c, d[ (i = 1, …, n; j = 1, …, m)^{1}.

x(t−) and x(t+) are the left and the right limits of the matrix-function x :]c, d[→ ℝ^{n×m} at the point t ∈]c, d[, d_{1}x(t) = x(t) − x(t−), d_{2}x(t) = x(t+) − x(t).

BV_{loc}([0, +∞[, ℝ^{n×m}) is the set of all real matrix-functions x : [0, +∞[→ ℝ^{n×m} of bounded variation on every closed interval from [0, +∞[.

If g:[0, +∞[→ℝ is a nondecreasing function, x:[0, +∞[→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),

2000 Mathematics Subject Classification. 34B05.

Key words and phrases. Stability in the Liapunov sense, linear systems of generalized ordinary differential equations.

1 x_{ij} as a constant outside [a, b] is assumed to be continuous. where g_{j} : [0, +∞[→ ℝ (j = 1, 2) are continuous nondecreasing functions such that the function g_{1} − g_{2} is identically equal to the continuous part of g, and ∫_{]s,t[} dg_{j}(τ) is the Lebesgue-Stieltjes 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);

L_{loc}([0, +∞[, ℓ; g) is the set of all real functions x : [0, +∞[→ ℝ μ(g)-measurable (i.e., measurable with respect to measures μ(g_{1}) and μ(g_{2})) and integrable on the closed interval [0, b] for every b ∈ [0, +∞[.

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

If G = (g_{ik})_{i,k}^{ℓ,m} : [0, +∞[→ ℝ^{ℓ×n} is a nondecreasing matrix-function, then L([0, +∞[, ℝ^{n×n}) is the set of all matrix-functions x = (x_{kj})_{k,j}^{n,m} : [0, +∞[→ ℝ^{n×m} such that x_{kj} ∈ L([0, +∞[, , ℝ; g_{ik}) (i = 1, …, ℓ,; k = 1, …, n; j = 1, …, m)

∫_{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 < +∞.

If G_{j} : [0, +∞[→ ℝ^{ℓ×n} (j = 1, 2) are nondecreasing matrix-functions, G ≡ G_{1} − G_{2} and x : [0, +∞[→ 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 < +∞;

L([0, +∞[, ℝ^{n×m}; G) = ∩^{2}_{j=1}L([0, +∞[, ℝ^{n×m}; G_{j}).

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

Under a solution of the system (1) is understood a vector-function x ∈ BV_{loc}([0, +∞[, ℝ^{n}) such that

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

We will assume that f ∈ BV_{loc}([0, +∞[; ℝ^{n}); A ∈ BV_{loc}([0, +∞[, ℝ^{n×n} and p ∈ L_{loc}([0, +∞[, ℝ^{n×n}, A) are such that

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

Let x_{0} ∈ BV_{loc}([0, , +∞[, ℝ^{n}) be a solution of the system (1).

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

lim_{t→+∞} ξ(t) = +∞.

The solution x_{0} of the system (1) is called ξ-exponentially asymptotically 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 the 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_{0})∥ < ε exp(−η(ξ(t) − ξ(t_{0})) for t ≥ t_{0}.

Stability, uniform stability and asymptotic stability of the solution x_{0} are defined just in the same way as for systems of ordinary differential equations (see, e.g., [1] or [2]), i.e., in the case, where A(t) is the diagonal matrix-function with diagonal elements equal to t). Note that exponential asymptotic stability ([1], [2]) is a particular case of ξ-exponential asymptotic stability (ξ(t) ≡ t).

Definition 2. The system (1) is called stable (uniformly stable, asymptotically stable, ξ-exponentially asymptotically stable) if every solution of this system is stable (uniformly stable, asymptotically stable, ξ-exponentially asymptotically stable).

Alongside with the system (1) we consider the corresponding homogeneous system

dx(t) = dA(t) · p(t) · x(t).  (1_{0})

Proposition 1. The system (1) is stable (uniformly stable, asymptotically stable, ξexponentially asymptotically stable) if and only if the zero solution of the system (1_{0}) is stable (uniformly stable, asymptotically stable, ξ-exponentially asymptotically stable).

Proposition 2. The system (1) is stable (uniformly stable, asymptotically stable, ξexponentially asymptotically stable) if and only if some solution of that system is stable (uniformly stable, asymptotically stable, ξ-exponentially asymptotically stable).

Therefore the stability (in all senses) of the system (1) is the property of the matrixfunctions A and p.

Definition 3. A pair (A, p) of matrix-functions A ∈ BV_{loc}([0, +∞[, ℝ^{n×n}) and p ∈ ([0, +∞[, ℝ^{n×n} A) satisfying the condition (2) is called stable (uniformly stable, asymptotically stable, ξ-exponentially asymptotically stable) if the system (1) is stable, (uniformly stable, asymptotically stable, ξ-exponentially asymptotically stable).

Now we formulate the basic lemma which will be applied in proving theorems below.

Lemma 1. Let the condition (2) hold. Moreover, let the matrix-functions A_{0} ∈ BV_{loc}([0, +∞[, ℝ^{n+×n}) and p_{0} ∈ L_{loc}([0, +∞[, ℝ^{n+×n}, A_{0}) be such that the following conditions are valid:

(a) det(I_{n} + (−1)^{j}djA_{0}(t) · p(t)) ≠ 0 for t ∈ ℝ_{+} (j = 1, 2);  (3)

(b) for some t_{0} ∈ ℝ_{+}, the Cauchy matrix u_{0} of the system

dx(t) = dA_{0}(t) · p_{0}(t) · x(t)

satisfies the inequality

|u(t, t_{0})| ≤ Ωe^{−ξ(t)+ξ(t_{0})} for t ≥ t_{0},

where Ω ∈ ℝ_{+}^{n×n}, and ξ : ℝ_{+} → ℝ_{+} is a nondecreasing function satisfying (3); (c) there exists a matrix H ∈ ℝ_{+}^{n×n} such that r(H) < 1 and

∫_{t_{0}}^{t} e^{ξ(t)−ξ(τ)}|u(t, τ)|dV (B)(τ) < H for t ≥ t_{0},

where

B(A, p, A_{0}, p_{0})(t) ≡ ∫_{0}^{t} dA(τ) · p(τ) − ∫_{0}^{t} dA_{0}(τ) · p_{0}(t) +

+ ∑_{0<τ≤t} d_{1}A_{0}(τ) · p_{0}(τ)(I_{n} − d_{1}A_{0}(τ) · p_{0}))^{−1}(d_{1}A(τ) · p(τ) − d_{1}A_{0}(τ) · p_{0}(τ)) −

− ∑_{0≤τ<t} d_{2}A_{0}(τ) · p_{0}(τ)(I_{n} + d_{2}A_{0}(τ) · p_{0}(τ))^{−1}(d_{2}A(τ) · p(τ) − d_{2}A_{0}(τ) · p_{0}(τ)).

Then an arbitrary solution x of the system (1_{0}) admits the estimate

|x(t)| ≤ Q|x(t_{0})|e^{−ξ(t)+ξ(t_{0})} for t ≥ t_{0},

where Q(I_{n} − H)^{−1}Ω.

Theorem 1. Let the conditions (2) and (4) hold, where A, A_{0}∈BV_{loc}([0, +∞[, ℝ^{n×n}), p ∈ L_{loc}([0, +∞[, ℝ^{n×n}; A) and p_{0} ∈ L_{loc}([0, +∞[, ℝ^{n×n}; A_{0}). Moreover, let the pair (A_{0}, p_{0}) be uniformly stable and

V^{+∞}_{0}(B) < +∞,  (4)

where the matrix-function B(A, p; A_{0}, p_{0}) : ℝ_{+} → ℝ^{n×n} is defined by (5). Then the pair (A, p) is uniformly stable as well.

Theorem 2. Let the conditions (2) and (4) hold, where A, A_{0}∈BV_{loc}([0, +∞[, ℝ^{n×n}), p ∈ L_{loc}([0, +∞[, ℝ^{n×n}; A) and p_{0} ∈ L_{loc}([0, +∞[, ℝ^{n×n}; A_{0}). Moreover, let the pair (A_{0}, p_{0}) be ξ-exponentially asymptotically stable and the condition

lim_{t→+∞} V^{ν(ξ)(t)}_{t}B = 0

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

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

and the matrix-function B(A, p; A_{0}, p_{0}) : ℝ_{+} → ℝ^{n×n} is defined by (5). Then the pair (A, p) is ξ-exponentially asymptotically stable as well.

Remark 1. If the pair (A, p) is ξ-exponentially asymptotically stable and the conditions (2) and

lim_{t→+∞} V^{ν(ξ)(t)}_{t}_{B} = 0,

hold, where

_{B}(A, p, f) ≡ f(t) + ∑_{0<τ≤t} d_{1}A(τ) · p(τ)(I_{n} − d_{1}A(τ) · p(τ))^{−1} · d_{1}f(τ) −

− ∑_{0≤τ<t} d_{2}A(τ) · p(τ)(I_{n} + d_{2}A(τ) · p(τ))^{−1} · d_{2}f(τ)

and the function ν(ξ) : ℝ_{+} → ℝ_{+} is defined as in Theorem 2, then an arbitrary solution x of the system (1) satisfies the condition

lim_{t→+∞} ∥x(t)∥ = 0.

Analogous results were obtained in [2] for linear systems of ordinary differential equations.

References

1. B. P. Demidovich, Lectures on mathematical theory of stability. (Russian) Nauka, Moscow, 1967.

2. I. T. Kiguradze, Initial and boundary value problems for systems of ordinary differential equations. I. (Russian) Metsniereba, Tbilisi, 1997.

Authors’ addresses:

M. Ashordia

I. Vekua Institute of Applied Mathematics

Tbilisi State University

2, University St., Tbilisi 380043

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