Mem. Differential Equations Math. Phys. 15(1998), 150–152

S. A. Mazanik

LAPPO–DANILEVSKI SYSTEMS UNDER LYAPUNOV TRANSFORMATIONS

(Reported on May 18, 1998)

Let A(t) be an n × n matrix of real-valued continuous and bounded functions of real variable t on the non-negative half-line. Consider the linear systems

Dx = A(t)x, t ≥ 0, D = d/dt.  (1_{A})

It is well known that if A is a Lappo-Danulevskiĭ matrix, i.e. there exists s ≥ 0 such that for all t ≥ 0

A(t) ∫_{s}^{t} A(u)du = ∫_{s}^{t} A(u)duA(t),  (2)

then a fundamental solution matrix X_{s}(t) of (1_{A}) (X_{s}(s) = E, E is the identity matrix) can be represented as

X_{s}(t) = exp ∫_{s}^{t} A(u)du.  (3)

This simple representation of the fundamental solution matrix does explain the fact that the class of Lappo-Danulevskiĭ systems is one of the main and interesting class of linear systems. In this paper we consider a problem of reducibility of (1_{A}) to the Lappo-Danulevskiĭ system and to the system (1_{B}) with functional commutative matrix B, where for all s ≥ 0 and t ≥ 0

B(t)B(s) − B(s)B(t) = [B(t), B(s)] = 0.  (4)

(The symbol [., .] is used to indicate the Lie brackets throughout this paper). It is obvious that system (1_{B}) with functional commutative matrix B is a special case of the Lappo-Danulevskiĭ system.

Note that condition (2) is sufficient but not necessary to represent fundamental solution matrix in the form (3) (see [1],[2]). To verify this fact it is sufficient to consider the system (1_{A}) with the matrix A(t) = (a_{ij}(t)), i, j = 1, 2, where a_{11}(t) = a_{33}(t) = −μa(t), a_{31}(t) = −a_{13}(t) = νa(t), a_{21}(t) = b(t), a_{12}(t) = a_{22}(t) = a_{32}(t) = a_{23}(t) = 0, μ ± iν are roots of the equation exp z − z − 1 = 0, a and b are infinitely differentiable non-analytic functions such that

∫_{0}^{s} a(u)du = 1, a(t) = 0 for t ≥ s > 0,

b(t) = \{ \begin{matrix}0, & t ∈ [0, s[, \\ b_{k}(t) ≢ 0, & t ∈ [t_{2k}, t_{2k+1}[, \\ 0 & t ∈ [t_{2k+1}, t_{2k+2}[, & k = 0, 1, ..., \end{matrix}

((t_{k}) is an arbitrary sequence of positive numbers such that t_{k+1} > t_{k} and t_{k} → +∞ as k → +∞.) In this case the fundamental solution matrix X_{0}(t) of (1_{A}) may be represented

1991 Mathematics Subject Classification. 34A30.

Key words and phrases. Linear differential systems, Lappo-Danulevskiĭ systems, Lyapunov transformations. as (3) with s = 0 but [A(t), ∫_{0}^{t} A(u)du] ≢ 0, i.e. A(t) is not a Lappo-Danulevskiĭ matrix. However (4) is the necessary and sufficient condition (see [3]) to represent Cauchy’s matrix of (1_{B}) as K(t, s) = exp ∫_{s}^{t} B(u)du.

It is well known (see, e.g., [4, p. 274]) that any system (1_{A}) is an almost reducible to some diagonal system. It is trivial that any diagonal matrix is a functional commutative matrix. Unfortunately, quite different is the case of linear systems under Lyapunov transformations.

A linear transformation x = L(t)y is a Lyapunov transformation if L(t) is a Lyapunov matrix, i.e.

max{sup_{t≥0} ∥L(t)∥, sup_{t≥0} ∥L^{−1}(t)∥, sup_{t≥0} ∥DL(t)∥} < +∞.

We follow Yu.S.Bogdanov (see [5]) and say that linear system (1_{A}) is asymptotically equivalent to (1_{B}) if there exist a Lyapunov transformation reducing (1_{A}) to (1_{B}).

Theorem 1. There exists a linear system which is not asymptotically equivalent to any system with functional commutative matrix of coefficients.

Theorem 2. There exists a linear system which is not asymptotically equivalent to any Lappo-Danulevskiĭ system.

To prove Theorem 1 and Theorem 2 it is sufficient (see [6,7]) to consider the system (1_{A}) with A(t) = diag{A_{0}(t), A_{1}(t)}, where A_{0}(t) = (a_{ij}(t)), i, j = 1, 2, a_{11}(t) = a_{21}(t) ≡ 0, a_{12}(t) = 1, a_{22}(t) = (t + 1)^{−1}, A_{1}(t) is the (n − 2) × (n − 2) identity matrix, and to use the following lemmas which describe the structure and the zero distribution for the integrals of the Lappo-Danulevskiĭ matrices.

Lemma 1. Let the scalar functions φ, ψ be continuous on the half-interval [a, c[, a < c ≤ +∞, and ∫_{a}^{t} ψ(u)du ≠ 0, ∀ t ∈]b, c[, for some b, a ≤ b < c. If

φ(t) ∫_{a}^{t} ψ(u)du = ψ(t) ∫_{a}^{t} φ(u)du, ∀ t ∈ [a, c[,

then there exists δ such that φ(t) = δψ(t), ∀ t ∈ [b, c[.

Lemma 2. Let the scalar functions φ, ψ be continuous on [a, +∞[ and

φ(t) ∫_{a}^{t} ψ(u)du = ψ(t) ∫_{a}^{t} φ(u)du, ∀ t ≥ a.

If sup A = sup B = +∞, where A = {α ≥ a | ∫_{a}^{α} φ(u)du = 0} and B = {β ≥

a | ∫_{a}^{β} ψ(u)du = 0}, then sup{A ⋂ B} = +∞.

However, the system mentioned above is a regular system and it can be reduced (Basov-Grobman-Bogdanov’s criterion, [8, p. 77]) to the system with functional commutative coefficients by generalized Lyapunov transformation x = L(t)y with the matrix L such that

\overline{lim}_{t→+∞}t^{−1} ln ∥L(t)∥ = \overline{lim}_{t→+∞}t^{−1} ln ∥L^{−1}(t)∥ = 0.

But even if we expand the set of transformations up to the set of generalized Lyapunov’s transformations there are statements which are similar to Theorem 1 and Theorem 2.

Theorem 3. There exists a linear system which is not generalized asymptotically equivalent to any system with functional commutative matrix of coefficients.

Theorem 4. There exists a linear system which is not generalized asymptotically equivalent to any Lappo-Danulevskiĭ system.

The proofs of these theorems are based on Lemma 1, Lemma 2 and on the following lemmas.

Lemma 3. Let the scalar functions φ_{i}, i = 1, 2, 3, be continuous on [a, +∞[ and

φ_{i}(t) ∫_{a}^{t} φ_{j}(u)du = φ_{j}(t) ∫_{a}^{t} φ_{i}(u)du, ∀ t ≥ a, ∀ i, j = 1, 2, 3, i ≠ j.

If sup A_{1} = sup A_{2} = sup A_{3} = +∞, then sup{A_{1} ⋂ A_{2} ⋂ A_{3}} = +∞, where A_{i} = {α ≥ a | ∫_{a}^{α} φ_{i}(u)du = 0}, i = 1, 2, 3.

Lemma 4. If L(t) is a generalized Lyapunov matrix, then

lim_{t→+∞} t^{−1} ln ∥ det L(t)∥ = lim_{t→+∞} t^{−1} ln ∥ det L^{−1}(t)∥ = 0.

Lemma 5. If f(t) = 2 + sin (μ ln t) + μ cos (μ ln t), where μ > 0, then

\underline{lim}_{t→+∞} ∫_{s}^{t}(exp ∫_{s}^{τ} f(σ)dσ)dτ ≥ 3 exp (−2π/μ).

To prove Theorem 3 and Theorem 4 it is sufficient to consider the system (1_{A}) with A(t) = (a_{ij}(t)), i, j = 1, 2, a_{12}(t) = 1, a_{21}(t) = 0, a_{22}(t) − a_{11}(t) = 2 + sin (μ ln(t + 1)) + μ cos (μ ln(t + 1)), where μ > 0, 3 exp (−2π/μ) > 2. In this case (1_{A}) is generalized asymptotically equivalent neither to a system with functional commutative matrix of coefficients nor to a Lappo-Danulevskiĭ system.

References

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2. J. F. P. Martin, On the exponential representation of solutions of linear differential equations. J. Differential Equations (1968), No. 4, 257 — 279.

3. V. A. Vinokurov, Explicit solution of a linear ordinary differential equation and the main property of the exponential function. J. Differential Equations 33(1997), No. 3, 298–304.

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5. Yu. S. Bogdanov, Asymptotically equivalent linear differential systems. J. Differential Equations 1(1965), No. 6, 541–549.

6. L. A. Mazanik and S. A.Mazanik, On irreducibility of linear differential systems to systems with functional commutative matrices of coefficients. (Russian) Vestnik Belorusskogo gosudarstvennogo universiteta. Ser. 1 (1997), No. 3, 42–46.

7. S. A. Mazanik, On irreducibility of linear differential systems to Lappo-Danulevskiĭ systems. (Russian) Dokl. Akad. Nauk Belarusi 41(1997), No. 6, 30–33.

8. N. A. Izobov, Linear systems of ordinary differential equations. (Russian) Itogi Nauki i Tekhniki, Mat. Anal. 12(1974), 71–146.

Author’s address:

Belarussian State University

4, F.Skorina Ave., Minsk 220050

Belarus