Mem. Differential Equations Math. Phys. 11(1997), 174–177

N. A. Izobov and S. N. Batan

ON EXACTNESS OF UPPER ESTIMATES OF THE CHARACTERISTIC

EXPONENT OF A LINEAR SYSTEM WITH EXPONENTIALLY DECREASING PERTURBATIONS

(Reported on May 19, 1997)

Consider a linear system

ẋ = A(t)x, x ∈ ℝ^{n}, t ≥ 0,  (1_{A})

with piecewise continuous bounded coefficients and a binormal [1, p. 49] system of solutions X_{A}(t) ordered in increasing exponents. Let λ_{i}(A) be the characteristic exponent of the i-th column of the matrix X(t) and δ_{i} be the characteristic exponent of the i-th row of the matrix X_{A}^{−1}(t). By means of the sums σ_{i}(A) = λ_{i}(A) + δ_{i}(A), we introduce (see [2]) the number σ_{0}(A) = \frac{σ_{m}(A)+σ_{l}(A)}{2} , in which the indices m ∈ \{1, …, n\} and l ∈ {1, …, n}, l ≠ m, are defined by the equalities σ_{m}(A) = max_{i}{σ_{i}(A)}, σ_{l}(A) =

max_{i≠m}{σ_{i}(A)}.

In [2] it is established that the characteristic exponents λ_{1}(A+Q) ≤ ⋯ ≤ λ_{n}(A+Q), of the perturbed system (1_{A+Q}) with a piecewise continuous perturbation Q(·) whose Lyapunov exponent λ[Q] satisfies λ[Q] < −σ_{0}(A), admit the estimates

λ_{k(i)}(A + Q) ≤ λ_{i}(A) + \frac{σ_{k}(A) − σ_{i}(A)}{2} , i = 1, …, n.  (2)

The question of attainability of these estimates arises. The following theorem gives the positive answer to it in a rather general case.

Theorem. For any numbers 2 ≤ n ∈ ℕ, m ∈ {1, …, n}, λ_{1} ≤ ⋯ ≤ λ_{n}, 0 < σ_{1} < σ_{2} and any ε ∈ (0, (σ_{2} −σ_{1})/2) satisfying the additional condition ε < λ_{p} −λ_{m} +(σ_{2} −σ_{1})/2 if there is (the least) p ∈ {1, …, m − 1} for which λ_{m} < λ_{p} + (σ_{2} − σ_{1})/2, there exist:

(i) a system (1_{A}) with infinitely differentiable bounded coefficients such that λ_{i}(A) = λ_{i}, i = 1, …, n, σ_{m}(A) = σ_{2} and σ_{i}(A) = σ_{1} for m ≠ i ∈ {1, …, n}; (ii) an analytical perturbation Q(·) with the Lyapunov exponent λ[Q] < −σ_{0} = − \frac{σ_{1}+σ_{2}}{2} such that the perturbed system (1_{A+Q}) has all different characteristic exponents and: 1) in the absence of the aboves pecified p, λ_{m}(A + Q) = λ_{m} and

λ_{i}(A + Q) = λ_{i} + \frac{σ_{2} − σ_{1}}{2} − \frac{1 + n − i}{n} ε,  (3)

for m ≠ i = 1, …, n; 2) in the presence of such p, λ_{p}(A + Q) = λ_{m}, λ_{i}(A + Q) = λ_{i−1} + \frac{σ_{2}−σ_{1}}{2} − \frac{1+n−i}{n} ε, i = p + 1, …, m, and other exponents, determined by the formula (3).

Proof. We will construct not the system (1_{A}) itself but its fundamental system of solutions X(t) = diag[exp tf_{1}(t), …, exp tf_{n}(t)]. Fix θ > 1 and a rather small ε > 0

1991 Mathematics Subject Classification. 34D10.

Key words and phrases. Charesteric exponent, liner system, perturpation. satisfying the necessary conditions. On the segments ▽_{k}(γ) ≡ [θ^{k+γ}, θ^{k+1}], k ≥ 0, with the determined below number γ ∈ (0, 1), we define the functions f_{i}(t) by

f_{i}(t) = \begin{cases}λ_{i}, & t ∈ ∇_{2k}(γ), \\ −δ_{i}, & t ∈ ▽_{2k+1}(γ), \end{cases} m ≠ i = 1, …, n,

f_{m}(t) = \begin{cases}λ_{m}, & t ∈ ▽_{2k+1}(γ), \\ −δ_{m}, & t ∈ ▽_{2k}(γ). \end{cases}

On the initial segment [0, 1] let’s assume f_{i}(t) = −δ_{i}, m ≠ i = 1, …, n, f_{m}(t) = λ_{m}. On other intervals Δ_{k}(γ) ≡ (θ^{k}, θ^{k+γ}), k ≥ 0, the functions f_{i}(t), i = 1, …, n, are defined by means of a special infinitely differentiable function

f(t; η_{1}, a; η_{2}, b) = a + (b − a) exp{− ln^{−2}(t/η_{1}) ×
× exp[− ln^{−2}(t/η_{2})]}, η_{1} < t < η_{2},

updating [3] the standard function from [4, p. 54]. The functions f_{i}(t) on the interval Δ_{k}(γ), k ≥ 0, are defined by the equality

f_{i}(t) = f(t; t_{k}, f_{i}(t_{k}); t_{k+γ}, f_{i}(t_{k+γ})), i = 1, …, n,

where t_{α} ≡ θ^{α}. It is easy to see that the system (1_{A}) so constructed has infinitely differentiable coefficients with all derivatives bounded.

We construct an n-th order matrix of perturbation Q(·) as having nonzero elements only in the m-th line, except q_{mm}(t) = 0, t ≥ 0. These elements look like

q_{mi}(t, ε_{i}) = exp(−σ_{0} − ε_{i})t, i ≠ m, t ≥ 0,  (4)

with specially determined below constant ε_{i}, i ≠ m.

We choose γ > 0 involved in the definition of the system (1_{A}) so small that

[2(λ_{n} − λ_{1}) + σ_{2} − σ_{1}](θ^{γ} − 1) < 2ε/n.  (5)

Denote the i-th solutionof the system (1_{A+Q}), by Y_{i}(t, ε_{i}), i ≠ m. Its components are

Y_{ji}(t, ε_{i}) = 0, j ≠ i, m; Y_{ii}(t, ε_{i}) = x_{i}(t) ≡ exp tf_{i}(t);

Y_{mi}(t, ε_{i}) = x_{m}(t)[Y_{mi}(ε_{i}) + ∫_{0}^{t} q_{mi}(τ, ε_{i})x_{i}(τ)x_{m}^{−1}(τ)dτ],

where the constant Y_{mi}(ε_{i}) = 0, if the Lyapunov exponent λ[q_{mi}x_{i}/x_{m}] of the integrand

is not less than zero, and Y_{mi}(ε_{i}) = − ∫_{0}^{+∞}q_{mi}x_{i}x_{m}^{−1}dτ. It is obvious that the m-th

solution of the system (1_{A+Q}) is the vector-function Y_{m}(t) with the unique different from zero component Y_{mm}(t) = x_{m}(t).

For any fixed i ≠ m, let’s establish now the existence of a constant ε_{i} = ε̃_{i} > 0 that the corresponding solution Y_{i}(t, ε̃_{i}) has an exponent λ[Y_{i}] = λ_{i} +(σ_{2} −σ_{1})/2−(1+n−i)ε/n. For this purpose at first we shall establish existence of a constant ε_{i}^{(1)} > 0 such that the inequality λ[Y_{i}] > λ_{i} + (σ_{2} − σ_{1})/2 − ε/n be true. Really, in the case λ_{i} + δ_{m} − σ_{0} > 0, due to the condition (5) a constant ε_{i}^{(1)} > 0 exists such that λ_{i} + δ_{m} − σ_{0} − ε_{i} > 0 and for sequence {t_{2k+1+γ}} the following estimates are fulfilled

λ[Y_{i}] ≥ \overline{lim}_{k→∞}t_{2k+1+γ}^{−1}lnY_{mi}(t_{2k+1+γ}, ε_{i}^{(1)}) ≥ λ_{m} + (λ_{i} + δ_{m} − σ_{0} − ε_{i}^{(1)})θ^{−γ} ≥

≥ λ_{i} + (σ_{2} − σ_{1})/2 − [λ_{n} − λ_{1} + (σ_{2} − σ_{1})/2](1 − θ^{−γ}) − ε_{i}^{(1)}θ^{−γ} >
> λ_{i} + (σ_{2} − σ_{1})/2 − ε/n.  (6_{1})

In the case λ_{i} + δ_{m} − σ_{0} ≤ 0, on the basis of the same condition (5), there exists a constant ε_{i}^{(1)} > 0 such that for a sequence {t_{2k}} the inequalities

λ[Y_{i}] ≥ \overline{lim}_{k→∞}t_{2k}^{−1}ln|Y_{mi}(t_{2k}, ε_{i}^{(1)})| ≥ λ_{m} + \overline{lim}_{k→∞} t_{2k}^{−1} ∫_{t_{2k+γ}}^{t_{2k+γ} +1}q_{mi}x_{i}x_{m}^{−1}dτ =

= λ_{m} + (λ_{i} + δ_{m} − σ_{0} − ε_{i}^{(1)})θ^{γ} ≥ λ_{i} + (σ_{2} − σ_{1})/2 + [λ_{1} − λ_{n}−
−(σ_{2} − σ_{1})/2](θ^{γ} − 1) − ε_{i}^{(1)}θ^{γ} > λ_{i} + (σ_{2} − σ_{1})/2 − ε/n.  (6_{2})

Let’s establish now the existence of a constant ε_{i}^{(2)} > ε_{i}^{(1)} such that λ[Y_{mi}(·, ε_{i}^{(2)})] < λ_{i} + (σ_{2} − σ_{1})/2 − ε. In the first place, choose this constant ε_{i}^{(2)} so large, that the λ_{i} + δ_{m} − σ_{0} − ε_{i}^{(2)} < 0. Then, due to the Lyapunov lemma concerning the exponent of the integral, we have

λ[Y_{mi}(·, ε_{i}^{(2)})] ≤ λ_{m} + λ_{i} + δ_{m} − σ_{0} − ε_{i}^{(2)} < λ_{i} + (σ_{2} − σ_{1})/2 − ε  (7)

under the second additional condition ε_{i}^{(2)} > ε on ε_{i}^{(2)}. Thus, the inequality λ[Y_{i}(·, ε_{i}^{(2)})] < λ_{i} + (σ_{2} − σ_{1})/2 − ε becomes obvious. Due to established in [5] continuous dependence of the exponent λ[Y_{i}(·, ε_{i}]) on the parameter ε_{i} > 0, from (6_{1}), (6_{2}) and (7) it follows the existence of the required ε̃_{i} ∈ (ε_{i}^{(1)}, ε_{i}^{(2)}).

For the completion of the proof of the theorem it is necessary to order the solutions Y_{i}(t, ε̃_{i}), i ≠ m, and Y_{m}(t) according to increase of their exponents. In the first case λ_{m} ≥ λ_{m−1} +(σ_{2} −σ_{1})/2, mentioned in the formulation of the theorem, we have obvious inequalities λ[Y_{1}] < ⋯ < λ[Y_{m−1}] < λ[Y_{m}] = λ_{m} < λ[Y_{m+1}] < ⋯ < λ[Y_{n}] and so all characteristic exponents of the perturbed system (1_{A+Q}) are different. In the second case of existence of (the least) p ∈ {1, …, m − 1} for which λ_{m} < λ_{p} + (σ_{2} − σ_{1})/2, the fundamental system, ordered in decreasing of the exponents, looks like Y (t) = [Y_{1}(t), …, Y_{p−1}(t), Y_{m}(t), Y_{p}(t), …, Y_{m−1}(t), Y_{m+1}(t), …, Y_{n}(t)], and the exponents of its solutions are all various, because of the choice, in this case, of the number ε > 0, and for the obtained exponents λ[Y_{i}] the inequalities

λ[Y_{p−1}] = λ_{p−1} + (σ_{2} − σ_{1})/2 − (2 + n − p)ε/n <
< λ_{p−1} + (σ_{2} − σ_{1})/2 ≤ λ_{m} = λ[Y_{m}] < λ_{p} + (σ_{2} − σ_{1})/2 − ε ≤
≤ λ[Y_{p}] = λ_{p} + (σ_{2} − σ_{1})/2 − (1 + n − p)ε/n <
< ⋯ < λ[Y_{m−1}] < λ[Y_{m+1}] < ⋯ < λ[Y_{n}]

are true. Thus, the fundamental matrix Y (t) is normal and the system (1_{A+Q}), in this case, has all different characteristic exponents specified in the formulation of the theorem. ■

Remark. For the characteristic exponents of the systems (1_{A}) and (1_{A+Q}) constructed in the proof of the theorem, the attainability of the estimates (2) is shown by the inequalities λ_{k(i)}(A + Q) ≥ λ_{i}(A) + (σ_{2} − σ_{1})/2 − ε, i = 1, …, n, which are valid for the permutation

k(i) = \begin{cases}i, & if i = 1, …, p − 1, …, m + 1, …, n, \\ p, & if i = m, \\ i + 1, & if i = p, …, m − 1. \end{cases}

References

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2. N. A. Izobov, On stability in the first approximation. (Russian) Differentsial’nye Uravnenija 2(1966), No. 7, 898–907.

3. N. A. Izobov and A. V. Filiptsov, On the unimprovability of matching conditions for lower Perron exponents of linear differential systems. (Russian) Differentsial’nye Uravnenija 31(1995), No. 8, 1300–1309.

4. B. Gelbaum and J. Olmsted, Counterexamples in Analysis. Holden-Day, San Francisco-London-Amsterdam,1964.

5. N. A. Izobov, On lower exponent of two-dimensional linear system with Perron perturbations. (Russian) Differentsial’nye Uravnenija 33(1997), No. 5, 623–631.

Author’s addresses:

N. A. Izobov Institute of Mathematics Belorussian Academy of Sciences 11, Surganova St., Minsk 220072 Belarus

S. N. Batan Institute of Mathematics Belorussian Academy of Sciences 11, Surganova St., Minsk 220072 Belarus