M. Ashordia and N. Kekelia

Sukhumi Branch of I. Javakhishvili Tbilisi State University Tbilisi, Georgia

ON THE ASYMPTOTIC STABILITY OF A CLASS OF SYSTEMS OF GENERALIZED LINEAR ORDINARY DIFFERENTIAL EQUATIONS

We study the stability in the Liapunov sense of the system of generalized linear ordinary differential equations

dx(t) = dA(t) · x(t) + df(t),  (1)

where A : R_{+} → R^{n×n}, f : R_{+} → R^{n} are matrix- and vector-functions with locally bounded variation components.

To a considerable extent, the interest to the theory of generalized ordinary differential equations has been stimulated also by the fact that this theory enables one to investigate systems of ordinary differential, difference and impulse equations.

By a solution of system (1) we understand a vector-function x whose components have bounded total variations on every closed segment from

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

Let d_{1}A(t) = A(t) − A(t−), d_{2}A(t) = A(t+) − A(t), S_{0}(A)(t) = A(t) −

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

For the concrete matrix-function A we give some effective conditions for its stability.

Theorem. Let α_{ik} ∈ R (i, k = 1, …, n), and let μ_{i} : R_{+} → R_{+} (1, …, n) be nondecreasing functions such that s_{0}(μ_{i}) ∈ C̃_{loc}(R_{+}; R_{+}) (i = 1, …, n),

lim_{t→+∞} a_{0}(t) = +∞ and σ_{i} = lim inf_{t→+∞} (α_{ii}d_{2}μ_{i}(t)) > −1 (i = 1, …, n),

where a_{0}(t) ≡ ∫_{0}^{t} η_{0}(s)ds + ∑_{0<s≤t} ln(1 − η_{1}(s)) − ∑_{0≤s<t} ln(1 + η_{2}(s)), η_{0}(t) ≡

min{|α_{ii}|(s_{0}(μ_{i})(t))′ : i = 1, …, n}, η_{j}(t) ≡ max {α_{ii}d_{j}μ_{i}(t) : i = 1, …, n} (j = 1, 2). Then the condition

α_{ii} < 0 (i = 1, …, n), r(H) < 1,  (2)

where H = (\frac{1−δ_{ik}}{1+σ_{i}} \frac{|α_{ik}|}{|α_{ii}|})_{i,k=1}^{n}, is sufficient for the matrix-function A(t) = (α_{ik}μ_{i}(t))_{i,k=1}^{n} to be asymptotically stable; and if in addition α_{ik} ≥ 0 (i ≠ k;

i, k = 1, …, n) and ∑_{k=1,i≠k}^{n}α_{ik}d_{1}μ_{i}(t) < min {1 − α_{ii}d_{1}μ_{i}(t), |1 + α_{ii}d_{1}μ_{i}(t)|}

for t ∈ R_{+} (i = 1, …, n), then condition (2) is necessary as well.

Supported by GRDF (Grant No. 3318).