Mem. Differential Equations Math. Phys. 15(1998), 157–159

S. A. Mazanik

LAPPO-DANILEVSKIĬ SYSTEMS AND THEIR PLACE AMONG LINEAR SYSTEMS

(Reported on June, 29 1998)

Consider the linear system

Dx = A(t)x, x ∈ ℝ^{n} t ≥ 0, D = d/dt,  (1_{A})

where A(t) is an n × n matrix of real-valued continuous and bounded functions of the real variable t on the non-negative half-line. We say that

i) A(t) is a right Lappo-Danilevskiĭ matrix (A ∈ LD_{r}(s)) if there exists s, s ≥ 0, such that for all t ≥ s

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

ii) A(t) is a left Lappo-Danilevskiĭ matrix (A ∈ LD_{l}(s)) if there exists s, s > 0, such that (2) is fulfilled for all 0 ≤ t ≤ s;

iii) A(t) is a bilateral Lappo-Danilevskiĭ matrix (A ∈ LD_{b}(s)) if there exists s, s ≥ 0, such that (2) is fulfilled for all t ≥ 0.

The corresponding systems (1_{A}) are called right, left or bilateral Lappo-Danilevskiĭ systems (cf. [1, p. 117]). In this paper we present some results on the distribution of the Lappo-Danilevskiĭ systems among linear systems.

Let ρ(A, B) = sup_{t≥0} ∥A(t) − B(t)∥, where ∥.∥ be an arbitrary matrix norm, and let

LD_{r} = ⋃_{s≥0} LD_{r}(s), LD_{l} = ⋃_{s>0} LD_{l}(s), LD_{b} = ⋃_{s≥0} LD_{b}(s). Let, for simplicity, n = 2.

Theorem 1. Among linear differential systems there is a linear system (1_{A}) such that for some ε > 0 the system (1_{A+Q}) is neither a bilateral nor a right Lappo-Danilevskiĭ system for any matrix Q such that ρ(A, A + Q) ≤ ε.

Theorem 2. Among linear differential systems there is a linear system (1_{A}) such that for any s > 0 there exists ε > 0 such that the matrix A+Q ∉ LD_{l}(s) for any matrix Q such that ρ(A, A + Q) ≤ ε.

To prove these theorems it is sufficient to consider the matrix A(t) = (a_{ij}(t)), i, j = 1, 2, where a_{11}(t) = sin ln (t + 1), a_{12}(t) = 1, a_{21}(t) = exp (−t), a_{22}(t) = cos ln (t + 1). (Let the symbol [., .] be used to indicate the Lie brackets, and let [., .]_{ij} be (i, j)-element of the matrix [·].) We have

[A(t) + Q(t), ∫_{s}^{t}(A(u) + Q(u))du] = [A(t), ∫_{s}^{t} A(u)du] + [A(t), ∫_{s}^{t} Q(u)du] +

+[Q(t), ∫_{s}^{t} A(u)du] + [Q(t), ∫_{s}^{t} Q(u)du].

1991 Mathematics Subject Classification. 34A30.

Key words and phrases. Linear differential systems, Lappo–Danilevskiĭ systems.

It is easy to verify that if ρ(A, A + Q) ≤ ε, then for all t ≥ 0, s ≥ 0, and for all sufficiently small ε we have:

|[A(t), ∫_{s}^{t} Q(u)du]_{11}| ≤ 4ε|t − s|, |[A(t), ∫_{s}^{t} Q(u)du]_{12}| ≤ 4ε|t − s|,

|[Q(t), ∫_{s}^{t} A(u)du]_{11}| ≤ 4ε|t − s|, |[Q(t), ∫_{s}^{t} A(u)du]_{12}| ≤ 4ε|t − s|,

|[Q(t), ∫_{s}^{t} Q(u)du]_{11}| ≤ 4ε|t − s|, |[Q(t), ∫_{s}^{t} Q(u)du]_{12}| ≤ 4ε|t − s|.

Therefore, ∀t ≥ 0, s ≥ 0 we have

F_{12}(t, s) = |[A(t) + Q(t), ∫_{s}^{t}(A(u) + Q(u))du]_{12}| ≥ |[A(t), ∫_{s}^{t} A(u)du]_{12}| − 12ε|t − s|.

Set t_{k} = exp (π/2 + 2kπ) − 1, k ∈ ℕ. It follows that F_{12}(t_{k}, s) ≥ |t_{k} − s − (s + 1) cos ln (s + 1)| − 12ε|t_{k} − s|. It is easy to see that for sufficiently large k we have F_{12}(t_{k}, s) > 0, so F_{12}(t, s) ≢ 0. Thus A + Q ∉ LD_{r} and A + Q ∉ LD_{b}.

Similarly one can show that

F_{11}(t, s) = |[A(t) + Q(t), ∫_{s}^{t}(A(u) + Q(u))du]_{11}| ≥
≥ | exp (−s) − exp (−t) − (t − s) exp (−t)| − 12ε|t − s|.

Since F_{11}(0, s) ≥ | exp (−s) − 1 + s| − 12εs and | exp (−s) − 1 + s| > 0 for all s > 0, we see that F_{11}(t, s) ≢ 0, i.e., A + Q ∉ LD_{l}(s).

Theorem 3. For any Lappo-Danilevskiĭ system (1_{A})and for any ε there exists a system (1_{B}) such that ρ(A, B) ≤ ε but (1_{B}) is not a Lappo-Danilevskiĭ system.

Indeed, if a_{12} and a_{21} are constant, then we can set b_{12}(t) = a_{12}(t) + αa_{12}(t) + φ(t), b_{21}(t) = a_{21}(t) + β + φ(t), where φ is a continuous function, 0 ≤ α ≤ ε, 0 ≤ β ≤ ε, and b_{11}(t) = a_{11}(t), b_{22}(t) = a_{22}(t). If we choose α and β such that a_{12} − a_{21} + αa_{12} − β ≠ 0 (the existence of such α and β is obvious), then one can show that B ∉ {LD_{b} ⋃ LD_{r} ⋃ LD_{l}}. If a_{21} ≢ const or a_{12} ≢ const, then we set b_{12}(t) = α + a_{12}(t), where 0 ≤ α ≤ ε, and b_{ij}(t) = a_{ij}(t) for all i, j = 1, 2, (i, j) ≠ (1, 2), or b_{21}(t) = β+a_{21}(t), where 0 ≤ β ≤ ε, and b_{ij}(t) = a_{ij}(t) for all i, j = 1, 2, (i, j) ≠ (2, 1), respectively. For both these cases one can show that B ∉ {LD_{b} ⋃ LD_{r} ⋃ LD_{l}}.

Theorem 4. Let A_{i} ∈ LD_{α}(s_{i}), i ∈ ℕ, α ∈ {b, r}, and ρ(A, A_{i}) → 0 as i → +∞. If there exists M such that s_{i} ≤ M < +∞ for all i ∈ ℕ, then A is a bilateral or right Lappo-Danilevskiĭ matrix.

Indeed, since the sequence (s_{i}) is bounded, there exists a subsequence (s_{i_{k}}) such that s_{i_{k}} → s ≥ 0 as i_{k} → +∞. Without loss of generality, s_{i} → s as i → +∞. So for the corresponding values of t we have [A_{i}(t), ∫_{s}^{t} A_{i}(u)du] = [A_{i}(t), ∫_{s_{i}}^{t} A_{i}(u)du] +

[A_{i}(t), ∫_{s}^{s_{i}} A_{i}(u)du] = [A_{i}(t), ∫_{s}^{s_{i}} A_{i}(u)du]. Since A_{i} is uniformly bounded on [0, +∞[, we have [A_{i}(t), ∫_{s}^{s_{i}} A_{i}(u)du] → 0 as i → +∞. On the other hand, the sequence A_{i}

is uniformly convergent on the non-negative half-line Therefore [A_{i}(t), ∫_{s}^{t} A_{i}(u)du] →

[A(t), ∫_{s}^{t} A(u)du] as i → +∞. So for the corresponding values of t we have [A(t),

∫_{s}^{t} A(u)du] ≡ 0, i.e., A is a bilateral or right Lappo-Danilevskiĭ matrix.

Similarly one can prove

Theorem 5. Let A_{i} ∈ LD_{l}(s_{i}), i ∈ ℕ, and ρ(A, A_{i}) → 0 as i → +∞. If there exist m, M such that 0 < m ≤ s_{i} ≤ M < +∞ for all i ∈ ℕ, then A is a left Lappo-Danilevskiĭ matrix.

Theorem 6. There exists a sequence A_{i}, A_{i} ∈ LD_{r}(s_{i}), i ∈ ℕ, ρ(A, A_{s_{i}}) → 0 and s_{i} → +∞ as i → +∞, such that A ∉ LD_{r}.

To prove this statement, it is sufficient to consider a sequence A_{k}(t) = a_{ijk}(t), i, j = 1, 2, such that a_{11k}(t) = a_{22k}(t) = g(t)with g continuous and bounded, and a_{21k}(t) = exp (−t), a_{12k} = f_{k}(t), where

f_{k} = \begin{cases}(1 − exp (−t)) exp (−t), & 0 ≤ t ≤ k, \\ (1 − exp (−k)) exp (−t), & t > k. \end{cases}

Theorem 7. There exists a sequence A_{i}, A_{i} ∈ LD_{l}(s_{i}), i ∈ ℕ, ρ(A, A_{s_{i}}) → 0 and s_{i} → +0 as i → +∞, such that A ∉ LD_{l}.

To prove this statement, it is sufficient to consider a sequence A_{k}(t) = a_{ijk}(t), i, j = 1, 2, such that a_{11k}(t) = a_{22k}(t) = g(t) with g continuous and bounded, a_{21k}(t) = exp (−t), a_{12k} = f_{k}(t), where

f_{k} = \begin{cases}exp (−k^{−1} − t)), & 0 ≤ t ≤ k^{−1}, \\ exp (−2t), & t > k^{−1}. \end{cases}

Theorem 8. Let A_{i} ∈ LD_{b}(s_{i}), i ∈ ℕ. If ρ(A, A_{s_{i}}) → 0 as i → +∞, then A is a bilateral Lappo-Danilevskiĭ matrix.

Theorem 9. Let A_{i} ∈ LD_{l}(s_{i}), i ∈ ℕ. If there exists m such that 0 < m ≤ s_{i} for all i ∈ ℕ, then A is a left Lappo-Danilevskiĭ matrix.

The proofs of Theorem 8 and Theorem 9 are based on the following lemmas.

Lemma 1. Let continuous scalar functions f and g satisfy f(t) ∫_{s}^{t} g(u)du = g(t) ×

∫_{s}^{t} f(u)du for some s ≥ 0 and for all t, t ∈]b, c[⊂ [0, +∞[. If ∫_{s}^{t} g(u)du ≠ 0 for all t, t ∈

]b, c[, then there exists a number λ such that ∫_{s}^{t} f(u)du = λ ∫_{s}^{t} g(u)du and f(t) = λg(t) ∀ t ∈ [b, c].

Let Z(g; s) = {t ≥ 0 | ∫_{s}^{t} g(u)du = 0}, N(g; s) = {t ∈ Z(g; s) | g(t) ≠ 0}. Denote by R(g; s) the subset of Z(g; s)\N(g; s) with the following property: ∀t_{0} ∈ R(g; s) ∀δ > 0 ∃t_{δ}, t_{0} < t_{δ} ≤ t_{0} + δ, t_{δ} ∉ Z(g; s). Denote by L(g; s) the subset of Z(g; s)\N(g; s) with the following property: ∀t_{0} ∈ L(g; s) ∀δ, 0 < δ ≤ t_{0}, ∃t_{δ}, t_{0} − δ ≤ t_{δ} < t_{0}, t_{δ} ∉ Z(g; s).

Lemma 2. Let continuous scalar functions f and g satisfy f(t) ∫_{s}^{t} g(u)du = g(t) ×

∫_{s}^{t} f(u)du, for some s ≥ 0 and for all t ≥ 0. Then N(g; s) ⋃ R(g; s) ⋃ L(g; s) ⊂ Z(f; s).

Lemma 3. Let a sequence of continuous scalar functions (g_{i}) uniformly over [0, +∞[ converge to a function g. Then for any σ ∈]0, +∞[, g(σ) ≠ 0, there exist positive ε and ν such that for all i ≥ ν the inequalities g_{i}(t) ≠ 0, g(t) ≠ 0 hold for all t ∈ [σ − ε, σ + ε].

Lemma 4. Let sequences of continuous scalar functions (g_{i}) (f_{i}) uniformly over [0, +∞[ converge to functions g and f respectively. If for any i ∈ ℕ there is s_{i} such that for all t ≥ 0 we have f_{i}(t) ∫_{s_{i}}^{t} g_{i}(u)du = g_{i}(t) ∫_{s_{i}}^{t} f_{i}(u)du, then there exists s ≥ 0 such

that the equality f(t) ∫_{s}^{t} g(u)du = g(t) ∫_{s}^{t} f(u)du holds for all t ≥ 0.

References

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

Author’s address:

Department of Higher Mathematics

Belarussian State University

4, F.Skorina Ave., Minsk 220050

Belarus