Mem. Differential Equations Math. Phys. 21(2000), 142–145

N. Kekelia

ON RELATION BETWEEN STABILITY AND CORRECTNESS OF LINEAR SYSTEMS OF GENERALIZED ORDINARY DIFFERENTIAL

EQUATIONS

(Reported on April 17, 2000)

Consider the problem

dx(t) = dA(t) · p(t) · x(t) + df(t),  (1)
x(t_{0}) = c_{0},  (2)

where A : ℝ_{+} → ℝ^{n×n} and f : ℝ_{+} → ℝ^{n} are, respectively, the real matrix- and vectorfunctions with locally bounded variation components, p : ℝ_{+} → ℝ^{n×n} is a matrixfunction locally integrable with respect to A, c_{0} ∈ ℝ^{n} and t_{0} ∈ ℝ_{+}.

Along with the problem (10), (2) let us consider the problem

dx(t) = d_{A}(t) · _{p}(t) · x(t) + d_{f}(t),  (3)
x(_{t_{0}}) = _{c_{0}},  (4)

where _{A} : ℝ_{+} → ℝ^{n×n} and _{f}: ℝ_{+} → ℝ^{n} are, respectively, real matrix– and vectorfunctions with locally bounded variation components, _{p} : ℝ_{+} → ℝ^{n×n} is a matrixfunction locally integrable with respect to _{A}, _{c_{0}} ∈ ℝ^{n} and _{t_{0}} ∈ ℝ_{+}.

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

ℝ =] − ∞, +∞[ is the set of 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_{j=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 a 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 x^{b}_{a} is the sum of total variations on a

closed interval [a, b] of 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).

2000 Mathematics Subject Classification. 34B05.

Key words and phrases. Stability, correctness, linear systems of generalized ordinary differential equations.

1 x_{ij} as a constant outside [a, b] is assumed to be continuous.

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

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),

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

A matrix-function t_{0} is said to be nondecreasing if each of its components is such. If G = (g_{ik})_{i,k=1}^{ℓ,n} : ℝ_{+} → ℝ^{ℓ×n} is a nondecreasing matrix-functions, then

L(ℝ_{+}, ℝ^{n×n} : G) is the set of all matrix-functions x = (x_{kj})_{k,j=1}^{n,m} : ℝ_{+} → ℝ^{n×m} such that x_{kj} ∈ L(ℝ_{+}, 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} : ℝ_{+} → ℝ^{ℓ×n} (j = 1, 2) are nondecreasing matrix-functions, G ≡ G_{1} − G_{2} 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 < +∞;

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

Under a solution of the system (1) is understood a vector-function x ∈ BV_{loc}(ℝ_{+}, 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}(ℝ_{+}, ℝ^{n}), A ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n}) and p ∈ L_{loc}(ℝ_{+}, ℝ^{n×n}, A) are such that

det(I_{n} + (−1)^{j} djA(t) · p(t)) ≠ 0 for t ∈ ℝ_{+} (j = 1, 2).  (5)

Then the problem (1), (2) has a unique solution (see [1]).

Definition 1. The problem (1), (2) is said to be correct if for every arbitrarily small ε > 0 and arbitrarily large ρ > 0 there exists δ > 0 such that for any _{t_{0}} ∈ ℝ_{+}, _{c_{0}} ∈ ℝ^{n},

_{A} ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n}, _{f} ∈ BV_{loc}(ℝ_{+}, ℝ^{n}) and _{p} ∈ L_{loc}(ℝ_{+}, ℝ^{n×n}, A) satisfying the

conditions

|t_{0} − _{t_{0}}| < δ, ∥c_{0} − _{c_{0}}∥ < δ,  (6)

∥M(t) − _{M}(t)∥ < δ ∥f(t) − _{f}(t)∥ < δ, V^{+∞}_{0}(M − _{M}) < ρ

and

det(I_{n} + (−1)^{j}dj_{A}(t) · _{p}(t)) ≠ 0 for t ∈ ℝ_{+} (j = 1, 2)  (7)

with

M(t) = ∫_{0}^{t} dA(τ) · p(τ), _{M}(t) = ∫_{0}^{t} d_{A}(τ) · _{p}(τ),  (8)

the inequality

∥x(t) − y(t)∥ < ε for t ∈ ℝ_{+}  (9)

holds, where x and y are the solutions of the problems (1), (2) and (3), (4), respectively.

Definition 2. The problem (1), (2) is said to be weakly correct if for arbitrary ε > 0 there exists δ > 0 such that for any _{t_{0}} ∈ ℝ^{n} and _{c_{0}} ∈ ℝ^{n}, _{A} ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n}), _{f}∈ BV_{loc}(ℝ_{+}, ℝ^{n}) and _{p}∈ L_{loc}(ℝ_{+}, ℝ^{n×n}, A) satisfying the conditions (6), (7) and

V^{+∞}_{0}(M − _{M}) < δ, V^{+∞}_{0}(f − _{f}) < δ,

where the matrix-functions M and _{M} are defined by (8), the inequality (9) holds, where x and y are the solutions of the problems (1), (2) and (3), (4), respectively.

Definition 3. Let ξ : ℝ_{+} →ℝ_{+} be a nondecreasing function such that lim_{t→+∞} ξ(t) = +∞. A solution x of the system (1) is said to be ξ-exponentially asymptotically stable if there exists a positive number η such that for every ε > 0 there exists δ = dl(ε) > 0 such that an arbitrary solution y of the system (1) the satisfying the inequality

∥x(t_{0}) − y(t_{0})∥ < δ

for some t_{0} ∈ ℝ_{+}, admits the estimate

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

The uniform stability of the solution x is defined just in the same way as for systems of ordinary differential equations (see, e.g., [2] or [3]).

Definition 4. The system (1) is said to be uniformly stable (ξ-exponentially asymptotically stable) if every solution of that system is uniformly stable (ξ-exponentially asymptotically stable).

Definition 5. A pair (A, p) of matrix-functions A ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n} and p ∈ L_{loc}(ℝ_{+}, ℝ^{n×n}, A) satisfying the condition (5) is said to be uniformly stable (ξ-expon-entially asymptotically stable) if the system

dx(t) = dA(t) · p(t) · x(t)

is uniformly stable (ξ-exponentially asymptotically stable).

Theorem 1. Let A ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n}), f ∈ BV_{loc}(ℝ_{+}, ℝ^{n}), p ∈ L_{loc}(ℝ_{+}, ℝ^{n×n}, A), and let the condition (5) hold. Moreover, let the pair (A, p) be ξ-exponentially asymptotically stable and the conditions

lim sup_{t→+∞} V^{ν(ξ)(t)}_{t}(B) < +∞,

and

lim_{t→+∞} V^{ν(ξ)(t)}_{t}(_{B}) = 0

hold, where

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

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

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

_{B}(A, p, f)(t) = 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).

Then the problem (1), (2) is correct.

Theorem 2. Let A ∈ BV_{loc}(ℝ_{+}, ℝ^{n×n}), f ∈ BV_{loc}(ℝ_{+}, ℝ^{n}), p ∈ L_{loc}(ℝ_{+}, ℝ^{n×n}, A), and let the condition (5) hold. Let, moreover, the pair (A, p) be uniformly stable. Then the problem (1), (2) is weakly correct.

References

1. S. Schwabik, M. Tvrdý and O. Vejvoda, Differential and integral equations. Boundary value problems and adjoints. Academia, Praha, 1979.

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

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

Author’s address:

Sukhumi Branch of

Tbilisi State University

12, Djikia St., 380086 Tbilisi

Georgia