USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 225–241 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Exponential dichotomies for linear systems with impulsive effects ^{∗}

Raúl Naulin

Abstract

In this paper we give conditions for the existence of a dichotomy for the impulsive equation

μ(t, ε)x′ = A(t)x, t ≠ t_{k},
x(t_{k}^{+}) = C_{k}x(t_{k}^{−}) ,

where μ(t, ε) is a positive function such that lim μ(t, ε) = 0 in some sense. The results are expressed in terms of the properties of the eigenvalues of matrices A(t), the properties of the eigenvalues of matrices {C_{k}} and the location of the impulsive times {t_{k}} in [0, ∞).

1 Introduction

In this paper we study the dichotomic properties of the impulsive system

μ(t, ε)x′(t) = A(t)x(t), t ≠ t_{k}, J = [0, ∞),  (1)
x(t_{k}^{+}) = C_{k}x(t_{k}^{−}), k ∈ ℕ = {1, 2, 3, …} ,

where x(t_{k}^{±}) = lim_{t→t_{k}^{±}} x(t). The function A(·) and the sequence {C_{k}} have properties to be specified later. The function μ(t, ε) depends on a parameter ε, in general, belonging to a metric space E. We will assume that μ(t, ε), for each fixed ε, is continuous. The cases we are interested in most are μ(t, ε) = ε > 0, μ(t, ε) = μ(t), such that lim_{t→∞} μ(t) = 0 and μ(t, ε) = 1. In what follows, for technical purposes we shall suppose that

0 < μ(t, ε) ≤ 1, ∀(t, ε) ∈ J × E.  (2)

For ordinary differential equations, the singular perturbed case (μ(t, ε) = ε > 0) has been intensively studied in [7, 15]; the regular case (μ(t, ε) = 1) has been considered in [6]; the general setting of the problem (1), when μ(t, ε) = μ(t), lim_{t→∞} μ(t) = 0 was studied in [13].

∗Mathematics Subject Classifications: 34A05, 34E05.

Key words: Impulsive linear systems, singularly perturbed impulsive systems,

dichotomies, splitting of impulsive systems.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published January 8, 2001.

Supported by Proyecto UDO-CI-5-1003-0936/00.

The aim of this paper is to give a set of algebraic conditions of existence of a (μ_{1}, μ_{2})-dichotomy [4], meaning by this conditions involving the properties of the functions of eigenvalues of matrices A(t), the eigenvalues of matrices belonging to the sequence {C_{k}}, and the location of the impulsive times {t_{k}}.

2 Notations and basic hypotheses

In this paper V stands for the field of complex numbers. We will assume that a fixed norm ∥ · ∥ on the space V^{n} is defined. For a matrix A ∈ V^{n×n}, ∥A∥ will denote the corresponding functional matrix norm. If m and n are integral numbers, then the set \{m, m + 1, m + 2, …, n\} will be denoted by \overline{m, n}. The symbol {t_{k}} identifies a strictly increasing sequence of positive numbers, satisfying lim_{k→∞} t_{k} = ∞. The solutions of all considered impulsive systems are uniformly continuous on each interval J_{k} = (t_{k−1}, t_{k}]. Further notations;

- For a bounded function f, we denote ∥f∥_{∞} = sup{∥f(t)∥ : t ∈ J},

- For an absolutely integrable function f, we denote ∥f∥_{1} = ∫_{0}^{∞} ∥f(t)∥dt,

- For a bounded sequence {C_{k}}, we denote ∥{C_{k}}∥_{∞} = sup{∥C_{k}∥ : k ∈ ℕ},

- For a summable sequence {C_{k}}, we denote ∥{C_{k}}∥_{1} = ∑_{k=1}^{∞} ∥C_{k}∥, -C({t_{k}}) = {f : J → V^{n} : f is uniformly continuous on all intervals J_{k}}, -BC({t_{k}}) = {f ∈ C({t_{k}}) : f is bounded}.

- The function i[s, t) will denote the number of impulsive times contained in the interval [s, t) if t > s; if s ≤ t_{k} < t_{k+1} < ⋯ < t_{h} < t, we define

∑_{[s,t)} C_{i} = C_{k} + C_{k+2} + ⋯ + C_{h}, ∑_{[t,t)} C_{i} = 0,

∏_{[s,t)} C_{i} = C_{h}C_{h−1} ⋯ C_{k}, ∏_{[t,t)} C_{i} = I .

We will denote by X(t) = X(t, ε) the fundamental matrix of the impulsive system (1). By this we mean a function X : J → V^{n×n} uniformly continuous, of class C^{1} on each interval J_{k}, such that X(0^{+}) = I and X satisfies (1). The definition and basic properties of function X(t, ε), for each fixed ε, are described in [2, 8].

Below, we list the basic hypotheses H1-H5 we will use.

H1: The function A is bounded and piecewise uniformly continuous on J with respect to {t_{k}}. This last means: For any ρ > 0, there exists a number δ(ρ) > 0, such that ∥A(t) − A(s)∥ < ρ, if |t − s| < δ, t, s ∈ J_{k} for all k ∈ N.

H2: There exist numbers p ≥ 0 and q > 1, such that

|i[s, t) − p(t − s)| ≤ q, s ≤ t.

H3: {C_{k}}_{k=1}^{∞} is a bounded sequence of invertible matrices.

H4: There exists a positive number γ, such that for any k, all eigenvalues μ_{k} of the matrix C_{k} satisfy the condition γ|μ_{k}| ≥ 1.

Definition 1 We shall say that {λ_{1}, λ_{2}, …, λ_{n}}, the eigenvalues of matrix A, are ordered by real parts (respectively, ordered by norms) iff

Reλ_{1} ≤ Reλ_{2} ≤ … ≤ Reλ_{n}, (respectively |λ_{1}| ≤ |λ_{2}| ≤ … ≤ |λ_{n}|).

In the sequel, we will assume that {λ_{1}(t), λ_{2}(t), …, λ_{n}(t)} the eigenvalues of matrix A(t) are ordered by real parts, and {μ_{1}(k), μ_{2}(k), …, μ_{n}(k)} the eigenvalues of matrix C_{k} are ordered by norms.

We will consider the following piecewise constant function

u_{m} : J → ℝ, u_{m}(t) = \frac{ln |μ_{m}(k)|}{t_{k} − t_{k−1}}, if t ∈ J_{k}.  (3)

In order to alleviate the writing, let us denote for m ∈ \overline{1, n − 1}

α_{m}(t, ε) = \frac{Re(λ_{m}(t) − λ_{m+1}(t))}{μ(t, ε)} + u_{m}(t) − u_{m+1}(t).

The following hypothesis is a slight modification of a condition of splitting used in [9].

H5: There exists a positive constant M such that the function

\begin{aligned}U_{m}(t, ε) & = & ∫_{0}^{t} \frac{1}{μ(s, ε)} exp \{∫_{s}^{t} α_{m}(τ, ε)dτ\} ds, \\ & + & ∫_{t}^{+∞} \frac{1}{μ(s, ε)} exp \{∫_{t}^{s} α_{m}(τ, ε)τ\} ds \end{aligned}

satisfies

∥U_{m}(t, ε)∥ ≤ M, ∀(t, ε) ∈ [0, ∞) × E.

3 The quasidiagonalization method

We will assume that, for some positive number r, the families of matrices {A(t) : t ∈ J} and {C_{k} : k ∈ ℕ} are contained in the set

M(r) = {F ∈ V^{n×n} : ∥F ∥ ≤ r}.

For each matrix F ∈ M(r) and σ > 0, by Theorem 1.6 in [1], we may choose a nonsingular matrix S such that

S^{−1}F S = Λ(F) + R(F, σ), ∥R(F, σ)∥ ≤ σ/2,  (4)

where Λ(F) denotes the diagonal matrix of eigenvalues of matrix F, ordered by real parts. Let us consider the ball B[F, ρ] = {G ∈ V^{n×n} : ∥f − G∥ ≤ ρ}. For any G ∈ B[F, ρ] we have

S^{−1}GS = Re Λ(F) + i Im Λ(F) + S^{−1}(G − F)S + R(F, σ), i^{2} = −1,

where

Λ(F) = diag{λ_{1}, λ_{2}, …, λ_{n}}, Re Λ(F) = diag{Re λ_{1}, Re λ_{2}, …, Re λ_{n}}.

¿From this decomposition we obtain

S^{−1}GS = Re Λ(G) + i Im Λ(F) + T (F, ρ) + R(F, σ),

where

T (F, ρ) = (Λ(F) − Λ(G)) + S^{−1}(G − F)S.

¿From Hurwitz’s theorem (see [5], page 148), the function L : V^{n×n} → V^{n×n} defined by L(F) = Re Λ(F) is continuous. This assertion implies, for a fixed number σ > 0 and a matrix F ∈ M(r) the existence of a nonsingular matrix S and a ρ > 0, such that if G ∈ B[F, ρ], then

S^{−1}GS = Re Λ(G) + i Im Λ(F) + Γ(F, σ), Γ(F, σ) := T (F, ρ) + R(F, σ),

and ∥Γ(F, σ)∥ ≤ σ. Since M(r) is compact, then given a σ > 0, there exist a covering F = {B[F_{j}, ρ_{j}]}_{j=1}^{m} of M(r), and nonsingular matrices S = {S_{1}, S_{2}, …, S_{m}} having the following property: For a fixed G ∈ M(r) there exists an index j ∈ {1, 2, …, m}, such that G ∈ B[F_{j}, ρ_{j}] and

S_{j}^{−1}GS_{j} = Re Λ(G) + i Im Λ(F_{j}) + Γ_{j}(σ), ∥Γ_{j}(σ)∥ ≤ σ.  (5)

Let ρ > 0 be a Lebesgue number of the covering F. According to H1, there exists a δ > 0, non depending on k, such that for t, s ∈ J_{k}, |t − s| ≤ δ we have ∥A(t) − A(s)∥ < ρ. Let us define

n(k, δ) = inf\{j ∈ ℕ : \frac{t_{k} − t_{k−1}}{j} ≤ δ\},

and the partition of the interval J_{k}:

P_{k} = {t_{0}^{k}, t_{1}^{k}, …, t_{n(k)}^{k}}, t_{0}^{k} = t_{k−1}, t_{n(k)}^{k} = t_{k},

defined by

|t_{i−1}^{k} − t_{i}^{k}| = δ_{k}, i ∈ \overline{1, n(k)}, δ_{k} := \frac{t_{k} − t_{k−1}}{n(k, δ)} .

We emphasize that n(k, δ) = 1 iff t_{k} − t_{k−1} ≤ δ. This and H2 yield

n(k, δ) ≤ L(p, δ)(t_{k} − t_{k−1}), L(p, δ) := max\{\frac{p}{q − 1}, \frac{2}{δ}\}.  (6)

According to the decomposition (5), we may assign to the interval (t_{i−1}^{k}, t_{i}^{k}] a nonsingular matrix S_{k,i} ∈ S and F_{k,i} ∈ {F_{j}}_{j=1}^{m}, such that

S_{k,i}^{−1}A(t)S_{k,i} = ReΛ(t) + iImΛ(F_{k,i}) + Γ_{k,i}(σ), t ∈ (t_{i−1}^{k}, t_{i}^{k}],  (7)

where we have abbreviated Λ(t) = Λ(A(t)) and

∥Γ_{k,i}(σ)∥ ≤ σ.  (8)

Regarding the sequence {C_{k}}_{k=1}^{∞}, we will accomplish a similar procedure. Let us consider a matrix D ∈ M(r) and σ > 0. For some nonsingular matrix T we will have, instead of (4), the decomposition

T^{−1}DT = N(D) + R(D, σ), ∥R(D, σ)∥ < σ,  (9)

where the matrix N(D) is defined by means of the eigenvalues D:

N(D) = diag{μ_{1}, μ_{2}, …, μ_{n}}, |μ_{1}| ≤ |μ_{2}| ≤ … ≤ |μ_{n}|.

We may write (9) in the form

T^{−1}DT = |N(D)|ei Arg(D) + R(D, σ),

where

Arg(D) = diag{arg(μ_{1}), arg(μ_{2}), …, arg(μ_{n})}

and

|N(D)| = diag{|μ_{1}|, |μ_{2}|, …, |μ_{n}}|.

For a matrix C ∈ B[D, ρ], ρ > 0, we write

\begin{aligned}T^{−1}CT & = & |N(C)|ei Arg(D) + (|N(C)| − |N(D)|)ei Arg(D) \\ & + & T^{−1}(C − D)T + R(D, σ), ∥R(D, ρ)∥ ≤ σ. \end{aligned}

The Hurwitz’s theorem implies that the function N : V^{n×n} → V^{n×n} defined by N (C) = |C| is continuous. Since M(r) is compact, then for a given σ > 0, there exists a covering D = {B[D_{i}, ρ_{i}]}m̃ ; i=1 of M(r), and a set of nonsingular matrices T = {T_{1}, T_{2}, …, Tm̃} , such that for each C_{k} there exists a T_{k} ∈ T and

D_{k} ∈ {D_{i}}m̃ ; i=1 such that

T_{k}^{−1}C_{k}T_{k} = |N(C_{k})|ei Arg(D^{k}) + Γ̃_{k}(σ), ∥Γ̃_{k}(σ)∥ ≤ σ.  (10)

4 A change of variables

Let g : [0, 1] → [0, 1] be a strictly increasing function, g ∈ C^{∞}, such that g(0) = g′(0) = g′(1) = 0, g(1) = 1. For an ordered pair (Q, R) of invertible matrices we define

θ : [a, b] → V^{n×n}, θ(t) = Q exp \{g (\frac{t − a}{b − a}) Ln(Q^{−1}R)\} .

The path θ is of class C^{∞}. Moreover θ(t) is a nonsingular matrix for each t, and θ(a) = Q, θ(b) = R, θ′(a) = 0, θ′(b) = 0. In the sequel, we shall say that the path θ splices the ordered pair of matrices (Q, R) on the interval [a, b]. In order to perform a change of variable of system (1), we splice matrices (S_{k,i}, S_{k,i+1}), i ∈ \overline{1, n(k) − 1} on an interval [t_{i}^{k} − ν_{k}(ε)δ_{k,i}/2, t_{i}^{k} + ν_{k}(ε)δ_{k,i}/2], where ν_{k}(ε) = inf{ μ(t, ε) : t ∈ J_{k}}, and δ_{k,i} are small numbers satisfying ν_{k}(ε)δ_{k,i} < δ_{k} and another condition we will specify in the forthcoming definition of number ν (see (13). Let us define the path

θ_{k,i} : [t_{i}^{k} − ν_{k}(ε)δ_{k,i}/2, t_{i}^{k} + ν_{k}(ε)δ_{k,i}/2] → V^{n×n}

splicing the matrices (S_{k,i}, S_{k,i+1}) in the following way

θ_{k,i}(t) = S_{k,i} exp \{g (\frac{t − t_{i}^{k} + ν_{k}(ε)δ_{k,i}}{μ_{k}(ε)δ_{k,i}} ) Ln(S_{k,i}^{−1}S_{k,i+1})\} .

For the constant

K_{1}(σ) = max {(∥S_{k}∥ + ∥Ln(S_{k}^{−1}S_{i})∥) exp {∥Ln(S_{k}^{−1}S_{i})∥} : 1 ≤ k, i ≤ m}

we have the estimates

∥θ_{k,i}(t)∥_{∞} ≤ K_{1}(σ), ∥θ′_{k,i}(t)∥_{∞} ≤ \frac{K_{1}(σ)}{ν_{k}(ε)δ_{k,i}} .  (11)

The matrix T_{k+1} assigned to the impulsive time t_{0}^{k+1} = t_{k+1} = t_{n(k)}^{k} and the matrix S_{k+1,1} are spliced on the interval [t_{0}^{k+1}, t_{0}^{k+1} + μ_{k+1}(ε)δ_{k+1,0}/2] by a path we denote by θ_{k+1,0}. The matrices (S_{k,n(k)}, T_{k+1}) are spliced on the interval [t_{n(k)}^{k} −ν_{k}(ε)δ_{k,n(k)}/2, t_{n(k)}^{k}] by a path we denote by θ_{k,n(k)}. We emphasize that θ_{k+1,0}(t_{k}) = T_{k_{1}} = θ_{k,n(k)}(t_{k}). A special mention deserves the time t = 0 which is not considered as an impulsive time. We will attach to the time t = 0 the matrix S_{1,1}. For these splicing paths are valid similar estimates to (11), with a modified constant for which we maintain the notation K_{1}(σ).

Let us define the intervals

\begin{matrix}I_{k} = [t_{0}^{k+1} − ν_{k}(ε)δ_{k,0}/2, t_{0}^{k+1} + ν_{k+1}(ε)δ_{k+1,0}/2], k = 1, 2, …, \\ I_{k,i} = (t_{i}^{k} − ν_{k}(ε)δ_{k,i}/2, t_{i}^{k} + ν_{k}(ε)δ_{k,i}/2), i ∈ \overline{1, n(k) − 1}, \end{matrix}  (12)

and the number

ν = ∑_{k=1}^{∞} ∑_{i=1}^{n(k)} δ_{k,i}.  (13)

The choice of the numbers δ_{k,i} is at our disposal. Therefore, ν can be made as small as necessary. Let us consider the C^{∞} function

S(t) = \{ \begin{matrix}θ_{k+1,0}(t), t ∈ [t_{0}^{k+1}, t_{0}^{k+1} + ν_{k+1}(ε)δ_{k,0}/2], & k = 0, 1, … \\ S_{k,i}, t ∈ [t_{i}^{k} + ν_{k}(ε)δ_{k,i}/2, t_{i+1}^{k} − ν_{k}(ε)δ_{k,i+1}/2], & i ∈ \overline{0, n(k) − 1}, \\ θ_{k,i}(t), t ∈ [t_{i}^{k} − ν_{k}(ε)δ_{k,i}/2, t_{i}^{k} + ν_{k}(ε)δ_{k,i}/2], & i ∈ \overline{1, n(k) − 1}, \\ θ_{k,n(k)}(t), t ∈ [t_{n(k)−1}^{k} − ν_{k}(ε)δ_{k,n(k)}/2, t_{n(k)}^{k}], & k = 1, 2, … \end{matrix}

From this definition S′(t) = 0 except on the intervals I_{k} and I_{k,i}. Since S(t_{k}) = T_{k}, the change of variable x = S(t)y reduces System (1) to the form

μ(t, ε)y′(t) = (S^{−1}(t)A(t)S(t) − μ(t, ε)S^{−1}(t)S′(t)) y(t), t ≠ t_{k}, (14)
y(t_{k}^{+}) = (|N(C_{k})|ei Arg(D^{k}) + Γ̃_{k}(σ)) y(t_{k}), k ∈ ℕ,

where ∥Γ̃_{k}(σ)∥ ≤ σ. Thus, this change of variable yields a notable simplification of the discrete component of (1). Let us define the left continuous function L : J → V^{n×n} by

L(0) = S_{1,1}, L(t) = S_{k,i}, t ∈ (t_{i−1}^{k}, t_{i}^{k}], i ∈ \overline{1, n(k)}.

From S^{−1}(t)A(t)S(t) = L^{−1}(t)A(t)L(t) + F (t, σ), where

F (t, σ) = S^{−1}(t)A(t)S(t) − L^{−1}(t)A(t)L(t),  (15)

we may write System (14) in the form

rcl μ(t, ε)y′(t) = (L^{−1}(t)A(t)L(t) + F (t, σ) − μ(t, ε)S^{−1}(t)S′(t)) y(t), t ≠ t_{k},
y(t_{k}^{+}) = (N_{k}ei Arg(D^{k}) + Γ̃_{k}(σ)) y(t_{k}), k ∈ ℕ .

From (7) and the definition of the piecewise constant functions

G(t) = ImΛ(F_{k,i}), t ∈ (t_{i−1}^{k}, t_{i}^{k}], Γ(t, σ) = Γ_{k,i}(σ), t ∈ (t_{i−1}^{k}, t_{i}^{k}],  (16)

we can write the last system in the form

μ(t, ε)y′(t) = ( Re Λ(t) + iG(t) + Γ(t, σ) + F (t, σ)
−μ(t, ε)S^{−1}(t)S′(t))y(t), t ≠ t_{k},  (17)
y(t_{k}^{+}) = (N_{k}ei Arg(D^{k}) + Γ̃_{k}(σ)) y(t_{k}), k ∈ ℕ .

Lemma 1

∥Γ(t, σ)∥_{∞} ≤ σ, ∥{Γ̃_{k}(σ)}∥_{∞} ≤ σ,  (18)
∥μ(., ε)^{−1}F (., σ)∥_{1} ≤ K_{2}(σ)ν,  (19)
∫_{s}^{t} ∥S^{−1}(τ)S′(τ)∥dτ ≤ K_{3}(σ)L(δ, p)(t − s), t ≥ s.  (20)

Proof. The first estimate of (18) follows from the definition of function Γ(t, σ) given by (16) and (8), and the second follows from (10). From definition (15), there exists a constant K_{2}(σ) depending only on σ such that

∥F (·, σ)∥_{∞} ≤ K_{2}(σ).

Moreover, from (15) we observe that F (·, σ) vanishes outside of the intervals I_{k,i} and I_{k}. Therefore, from the definitions (12)-(13) we obtain

∫_{0}^{∞} |\frac{F (t, σ)}{μ(t, ε)}|dt = K_{2}(σ)( ∑_{i,k} ∫_{I_{k,i}} \frac{1}{μ_{k}(ε)}dt + ∑_{k} ∫_{I_{k}} \frac{1}{μ_{k}(ε)}dt) ≤ K_{2}(σ)ν .

In order to obtain (20) we observe that S^{−1}(t)S′(t) vanishes outside of the intervals I_{k,i} and I_{k}. Moreover, there exists a constant K_{3}(σ) depending only on σ, such that on each interval [t_{i−1}^{k}, t_{i}^{k}] we have

∫_{t_{i−1}^{k}}^{t_{i}^{k}} ∥S^{−1}(τ)S′(τ)∥dτ ≤ K_{3}(σ).

From this estimate and (6), it follows

∫_{s}^{t} ∥S^{−1}(τ)S′(τ)∥dτ ≤ K_{3}(σ)L(p, δ)(t − s).

In what follows we unify the notations of the constants K_{i}(σ), i = 1, 2, 3 in a simple constant K(σ).

5 Splitting and dichotomies

We are interested in the proof of existence of a dichotomy for the System (17). In this task we will follow the way indicated by Coppel in [6]: First we split System (17) in two systems of lower dimensions and after this, the Gronwall inequality for piecewise continuous functions [3] will give the required result. Following the ideas of paper [11], we write System (17) in the form:

μ(t, ε)y′(t) = ( Re Λ(t) + iG(t) + Γ(t, σ) + F (t, σ)
−μ(t, ε)S^{−1}(t)S′(t))y(t), t ≠ t_{k},  (21)
Δy(t_{k}) = (B_{k} + Γ̂_{k}(σ)) y(t_{k}^{+}), k ∈ ℕ ,

where Δy(t_{k}) = y(t_{k}^{+}) − y(t_{k}^{−}), B_{k} = I − N_{k}^{−1}e−i Arg(D^{k}), and

Γ̂_{k}(σ) = N_{k}^{−1}e−i Arg(D^{k})Γ_{k}(σ) (N_{k}ei Arg(D^{k}) + Γ_{k}(σ))^{−1} .

From hypotheses H3-H4 and (18) we obtain, for a small σ, the estimate

|Γ̂_{k}(σ)| ≤ \frac{σγ^{2}}{1 − γσ} .  (22)

On the other hand, the fundamental matrix of system

μ(t, ε)w′(t) = (Re Λ(t) + iG(t)) w(t), t ≠ t_{k},
Δw(t_{k}) = B_{k}w(t_{k}^{+}), k ∈ ℕ ,

coincides with the fundamental matrix Z(t, ε) = Z(t) of the diagonal system

μ(t, ε)z′(t) = (Re Λ(t) + iG(t)) z(t), t ≠ t_{k},  (23)
z(t_{k}^{+}) = |N(C_{k})|ei Arg(D^{k})z(t_{k}), k ∈ ℕ ,

which is equal to Z(t) := Φ(t)Ψ(t), where

Ψ(t) = exp \{∫_{0}^{t} \frac{ReΛ(τ) + iG(τ)}{μ(τ, ε)}dτ\} , Φ(t) = ∏_{[0,t)} |N(C_{k})|ei Arg(D^{k}).

For the projection matrix P = diag\{\overbrace{1, 1, …, 1}^{m}, 0, …, 0\}, the function Φ satisfies the following estimates:

∥Φ(t)P ∥ ≤ exp { ∑_{[0,t)} ln |μ_{m}(k)|}.

From definition (3), we may write

∥Φ(t)P ∥ ≤ L exp { ∫_{0}^{t} u_{m}(τ)dτ},
∥Φ^{−1}(t)(I − P)∥ ≤ L exp { ∫_{t}^{0} u_{m+1}(τ)dτ},

where L is a constant depending on the condition H3 only. Since Φ(t) and Ψ(t) commute with P, then for t ≥ s we obtain the following estimates

∥Z(t)P Z^{−1}(s)∥ ≤ L_{1} exp \{ ∫_{s}^{t} (\frac{Reλ_{m}}{μ(·, ε)} + u_{m})(τ)dτ\},  (24)

∥Z(s)(I − P)Z^{−1}(t)∥ ≤ L_{1} exp \{ ∫_{t}^{s} (\frac{Reλ_{m+1}}{μ(·, ε)} + u_{m+1})(τ)dτ\},

where L_{1} is a constant independent of σ and ϵ. In the sequel W (t, s) will denote the matrix: W (t, s) = Z(t)Z^{−1}(s). ¿From (24), for t ≥ s, we have

∥W (t, s)P ∥∥W (s, t)(I − P)∥ ≤ L_{1}^{2} exp { ∫_{s}^{t} α_{m}(τ, ε)dτ}.  (25)

For a given matrix C, we write {C}_{1} = P CP + (I − P)C(I − P).

Definition 2 By a splitting of System (21), we mean the existence of a function T : J → V^{n×n} with the following properties:

T1: T is continuously differentiable on each interval J_{k},

T2: For each impulsive time t_{k}, there exists the right hand side limit T (t_{k}^{+}),

T3: T (t) is invertible for each t ∈ J_{k}. T (t_{k}^{+}) are invertible for all k, T4: The functions T and T^{−1} are bounded,

T5: The change of variables y(t) = T (t)z(t) reduces System (21) to

μ(t, ε)z′(t) = ( Re Λ(t) + iG(t) + {(Γ(t, σ) + F (t, σ)) T (t)}_{1}
−μ(t, ε){S^{−1}(t)S′(t)T (t)}_{1})z(t), t ≠ t_{k},  (26)
Δz(t_{k}) = (B_{k} + {Γ̂_{k}(σ)}_{1}) z(t_{k}^{+}), k ∈ ℕ .

For ordinary differential equations, problem T1-T5 was solved in [6]. For difference equations, it was solved in [14]. The problem of splitting for impulsive equations is treated in [11]. None of the cited works study the splitting of system (21), where the unbounded coefficient {S^{−1}(t)S′(t)}_{1} appears.

Following the general setting of [6, 14, 11], we will seek a function T in the form T (t) = I + H(t), where H ∈ BC({t_{k}}), ∥H∥_{∞} ≤ 1/2, such that T satisfies conditions T1-T5. In the following we use the notations

H_{k} = H(t_{k}), H_{k}^{+} = H(t_{k}^{+}).

Let us consider the following operators: The operator of continuous splitting

O(H)(t) = ∫_{t_{0}}^{t} \frac{1}{μ(s, ε)}W (t, s)P (I − H(s))(Γ(s, σ)

+F (s, σ))(I + H(s))(I − P)W (s, t)ds

− ∫_{t}^{∞} \frac{1}{μ(s, ε)}W (t, s)(I − P)(I − H(s))(Γ(s, σ)

+F (s, σ))(I + H(s))P W (s, t) ds;

the operator of discrete splitting

D(H)(t) = ∑_{[t_{0},t)} W (t, t_{k})P (I − H_{k})Γ̃_{k}(σ)(I + H_{k}^{+})(I − P)W (t_{k}^{+}, t)

− ∑_{[t,∞)} W (t, t_{k})(I − P)(I − H_{k})Γ̃_{k}(σ)(I + H_{k}^{+})P W (t_{k}^{+}, t);

and the operator of impulsive splitting

S(H)(t)

= − ∫_{t_{0}}^{t} W (t, s)P (I − H(s))(S^{−1}(s)S′(s)(I + H(s))(I − P)W (s, t)ds

+ ∫_{t}^{∞} W (t, s)(I − P)(I − H(s))S^{−1}(s)S(s)(I + H(s))P W (s, t)ds .

Raúl Naulin 235 Lemma 2 Uniformly with respect to t_{0} ∈ J, for some constant L_{2} non depending on σ nor on ε, we have the following estimates

∥O(H)∥_{∞} ≤ L_{2}(σ + K(σ)ν),  (27)

and

∥D(H)(t)∥_{∞} ≤ L_{2}σ.  (28)

Proof. From condition H5 and (25) we have the estimate

∥O(H)(t)∥ = ∫_{t_{0}}^{t} \frac{9L_{1}^{2}}{4μ(s, ε)} exp \{ ∫_{s}^{t} α_{m}(τ, ε)dτ\} (∥Γ(s, σ)∥ + ∥F (s, σ)∥) ds

+ ∫_{t}^{∞} \frac{9L_{1}^{2}}{4μ(s, ε)} exp \{ ∫_{t}^{s} α_{m}(τ, ε)dτ\} (∥Γ(s, σ)∥ + ∥F (s, σ)∥) ds

≤ \frac{9L_{1}^{2}}{4} (σ∥U_{m}(·, ε)∥_{∞} + ∫_{t_{0}}^{∞} \frac{∥F (s, σ)∥}{μ(s, ε)}ds) .

Now the estimate (26) follows from (18) and H5, for some constant L_{2}.

For a fixed t > 0, let us consider the impulsive times divided as follows:

t_{1} < t_{2} < … < t_{k} < t ≤ t_{k+1} < t_{k+2} < …

From (17) and (24) we can write the estimate

∥D(H)(t)∥

≤ \frac{9L_{1}^{2}σ}{4} ∑_{i=1}^{k} exp \{ ∫_{t_{i}}^{t} α_{m}(τ, ε)dτ\} + \frac{9L_{1}^{2}σ}{4} ∑_{i=k+1}^{∞} exp \{ ∫_{t}^{t_{i}} α_{m}(τ, ε)dτ\}
≤ \frac{9L_{1}^{2}σ}{4} (2 + ∑_{i=1}^{k−1} exp \{ ∫_{t_{i}}^{t} α_{m}(τ, ε)dτ\} + ∑_{i=k+2}^{∞} exp \{ ∫_{t}^{t_{i}} α_{m}(τ, ε)dτ\})
≤ \frac{9L_{1}^{2}σ}{4} (2 + ∑_{i=1}^{k−1} \frac{1}{t_{i} − t_{i−1}} ∫_{t_{i−1}}^{t_{i}} exp \{ ∫_{s}^{t} α_{m}(τ, ε)dτ\}ds

+ ∑_{i=k+2}^{∞} \frac{1}{t_{i+1} − t_{i}} ∫_{t_{i}}^{t_{i+1}} exp \{ ∫_{t}^{s} α_{m}(τ, ε)dτ\}ds)

From (2) and H2 we obtain

∥D(H)(t)∥ ≤ \frac{9L_{1}^{2}σp}{4(K − 1)}(2 + \frac{p}{4(q − 1)} ∫_{0}^{t} \frac{1}{μ(s, ε)} exp \{ ∫_{s}^{t} α_{m}(τ, ε)dτ\}ds

+\frac{p}{4(q − 1)} ∫_{t}^{∞} \frac{1}{μ(s, ε)} exp \{ ∫_{t}^{s} α_{m}(τ, ε)dτ\}).

From this estimate it follows (28) for some constant L_{2}. ♢ 236 Exponential dichotomies for linear systems

The estimate of operator S is more complicated. From (25) we obtain

∥S(H)(t)∥ ≤ I_{1}(t) + I_{2}(t),

where

I_{1}(t) = ∫_{t_{0}}^{t} exp { ∫_{s}^{t} α_{m}(τ, ε)dτ}∥S^{−1}(s)S′(s)∥ds ,
I_{2}(t) = ∫_{t}^{∞} exp { ∫_{t}^{s} α_{m}(τ, ε)dτ}∥S^{−1}(s)S′(s)∥ds.

We can write I_{1} in the form

I_{1}(t) = ∫_{t_{0}}^{t} exp \{ ∫_{s}^{t} α_{m}(τ, ε)dτ\}\frac{d}{ds} ∫_{t}^{s} ∥S^{−1}(ξ)S′(ξ)∥dξds .

Integration by parts gives

I_{1}(t) = exp { ∫_{t_{0}}^{t} α_{m}(τ, ε)dτ} ∫_{t_{0}}^{t} ∥S^{−1}S′∥(u)du
− ∫_{t_{0}}^{t} α_{m}(s, ε) exp { ∫_{s}^{t} α_{m}(τ, ε)dτ} ∫_{s}^{t} ∥S^{−1}S′∥(u)du .

Taking into account the estimate (20) we obtain

I_{1}(t) ≤ K(σ)L(δ, p) exp { ∫_{t_{0}}^{t} α_{m}(τ, ε)dτ}(t − t_{0})
−K(σ)L(δ, p) ∫_{t_{0}}^{t} α_{m}(s, ε) exp { ∫_{s}^{t} α_{m}(τ, ε)dτ}(t − s)ds .

Once again, integrating by parts the last integral, from the right hand side of this inequality we obtain

I_{1}(t) ≤ K(σ)L(δ, p) ∫_{t_{0}}^{t} exp { ∫_{s}^{t} α_{m}(τ, ε)dτ}ds .  (29)

By similar tokens

I_{2}(t) ≤ K(σ)L(δ, p) ∫_{t}^{∞} exp { ∫_{t}^{s} α_{m}(τ, ε)dτ}ds .  (30)

Using (2) and the hypothesis H5 we obtain the estimate

I_{i}(t) ≤ MK(σ)L(δ, p)∥μ(·, ε)∥_{∞}, i = 1, 2 .

Thus, for a given α > 0, if ∥μ(·, ε)∥_{∞} is small enough, we will have

∥S(H)(t)∥ ≤ α .  (31)

Theorem 1 The conditions H1-H5 imply, for a small values of the norm {μ(·, ε)}, the existence of a function T : [t_{0}, ∞) → V^{n×n} satisfying T1-T5. Moreover ∥T ∥ ≤ \frac{3}{2}, ∥T^{−1}∥ ≤ 2.

Proof. According to Lemma 4 and Lemma 5, the operator T = O + D + S, for small values of σ, ν and α (see (31), satisfies

T : {H ∈ BC({t_{k}}) : ∥H∥_{∞} ≤ 1/2} → {H ∈ BC({t_{k}}) : ∥H∥_{∞} ≤ 1/2}.

Also, for small values of σ, ν and α this operator is a contraction. This and further details of this theory are well known for exponential dichotomies. The corresponding result for the dichotomy (24) are similar [6, 14, 12]. ♢

Once we have split (17), we write System (26) in the form

μ(t, ε)z′(t) = ( Re Λ(t) + iG(t) + {(Γ(t, σ) + F (t, σ)) T (t)}_{1}
−μ(t, ε){S^{−1}(t)S′(t)T (t)}_{1})z(t), t ≠ t_{k},  (32)
z(t_{k}^{+}) = (N_{k}ei Arg(D^{k}) + {G_{k}(σ)}_{1}) z(t_{k}), k ∈ ℕ ,

where

G_{k}(σ) = (I − N_{k}ei Arg(D^{k}){Γ̂_{k}(σ)}_{1})^{−1} N_{k}ei Arg(D^{k}) − N_{k}ei Arg(D^{k}).

From (22) we obtain

∥G_{k}(σ)∥ ≤ L_{3}σ, L_{3} = 2∥{C_{k}}∥_{∞}, if 0 < 2σ < ∥{C_{k}}∥_{∞}^{−1}.  (33)

The right hand side equation of (32) commute with projection P. Therefore,

(32) may be written as two systems of dimensions m and n − m,

μ(t, ε)z′_{j}(t) = ( Re Λ_{j}(t) + iG_{j}(t) + Γ_{j}(t, σ) + F_{j}(t, σ)
+μ(t, ε)V_{j}(t))z_{i}(t), t ≠ t_{k},  (34)
z_{j}(t_{k}^{+}) = (N_{k,j}ei Arg(D^{k,j}) + G_{k,j}(σ)) z_{j}(t_{k}), k ∈ ℕ ,  (35)

where j = 1, 2. The matrices Λ_{1}(t), Λ_{2}(t) are defined by

Λ_{1}(t) = {λ_{1}(t), λ_{2}(t), …, λ_{m}(t)}, Λ_{2}(t) = {λ_{m+1}(t), λ_{m+2}(t), …, λ_{n}(t)},

and similarly the diagonal matrices G_{j}(t), N_{k,j} and D_{k,j} are defined. The matrices G_{k,j}(σ) satisfy estimate (33). Γ_{j}(t, σ) has the estimate (18), where instead of σ it is necessary to write 3σ, F_{j}(t, σ) has the estimate (19) and

∥ ∫_{s}^{t} V_{j}(τ)dτ∥ ≤ 3∥ ∫_{s}^{t} S^{−1}(τ)S′(τ)dτ∥ ≤ 3L(δ, p)K(σ)(t − s), t ≥ s.

The Gronwall inequality for piecewise continuous functions [3] gives the following estimates for Z_{i}(t), the fundamental matrices of systems (34), j = 1, 2:

∥Z_{1}(t)Z_{1}^{−1}(s)∥ ≤ L exp { ∫_{s}^{t} μ_{1}(τ, ε)dτ}, s ≤ t,
∥Z_{2}(t)Z_{2}^{−1}(s)∥ ≤ L exp { ∫_{s}^{t} μ_{2}(τ, ε)dτ}, t ≤ s,

where L is a constant non depending on ε neither on σ, and

μ_{1}(t, ε) = \frac{Re(λ_{m}(t))}{μ(t, ε)} + u_{m}(t) + L_{4}σ + 3L(δ, p)K(σ),

μ_{2}(t, ε) = \frac{Re(λ_{m+1}(t))}{μ(t, ε)} + u_{m+1}(t) + L_{4}σ + 3L(δ, p)K(σ),

with a constant L_{4} = 3 + L_{3}. Since the decoupled system (34) is kinetically similar to System (1), we obtain for this system the following

Theorem 2 If the hypotheses H1-H5 are fulfilled, then for a small value of ∥μ(·, ε)∥ the System (1) has the following (μ_{1}, μ_{2})-dichotomy:

∥X(t, ε)P X^{−1}(s, ε)∥ ≤ L exp { ∫_{s}^{t} μ_{1}(τ, ε)dτ}, s ≤ t,  (36)
∥X(t, ε)P X^{−1}(s, ε)∥ ≤ L exp { ∫_{s}^{t} μ_{2}(τ, ε)dτ}, t ≤ s ,

where L is a constant independent of ε and σ.

6 Dichotomies for linear differential systems

In this section we present some applications of formulas (36).

The case ∥μ(·, ε)∥_{∞} ≤ ε

Theorem 3 Under conditions H1-H5, if ∥μ(·, ε)∥ ≤ ε, ε ∈ (0, ∞), then there exists a positive number ε_{0} such that for each ε ∈ (0, ε_{0}), the impulsive system (1) has the dichotomy (36).

In the particular case μ(t, ε) = ε, we obtain the system

εx′(t) = A(t)x(t), t ≠ t_{k}, J = [0, ∞),  (37)
x(t_{k}^{+}) = C_{k}x(t_{k}^{−}), k ∈ ℕ = {1, 2, 3, …},

and the dichotomy (36) has the form

\begin{aligned}μ_{1}(t, ε) & = & \frac{Re(λ_{m}(t)) + εu_{m}(t) + L_{4}εσ + 3εL(δ, p)K(σ)}{ε} , \\ μ_{2}(t, ε) & = & \frac{Re(λ_{m+1}(t)) + εu_{m+1}(t) + L_{4}εσ + 3εL(δ, p)K(σ)}{ε} . \end{aligned}

Considering in (37) C_{k} = I for k ∈ N, we obtain that the solutions of this sytems coincide with the solutions of the ordinary system with a small and a positive parameter at the derivative

εy′(t) = A(t)y(t).  (38)

Denoting by Y (t, ε) the fundamental matrix of System (38), from (36) we obtain the dichotomy

∥Y (t, ε)P Y^{−1}(s, ε)∥ ≤ K exp { ∫_{s}^{t} μ_{1}(τ, ε)dτ}, s ≤ t,
∥Y (t, ε)(I − P)Y^{−1}(s, ε)∥ ≤ K exp { − ∫_{t}^{s} μ_{2}(τ, ε)}, t ≤ s,

where

μ_{1}(t, ε) = \frac{Re(λ_{m}(t)) + L_{4}εσ + εL(δ, 0)K(σ)}{ε},

μ_{2}(t, ε) = \frac{Re(λ_{m+1}(t)) + L_{4}εσ + 3εL(δ, 0)K(σ)}{ε}.

If Re(λ_{m}(t)) ≤ −α < 0 and Re(λ_{m}(t)) ≥ β > 0, for all values of t, for a small ε_{0}, we obtain for (38) the dichotomy

∥Y (t, ε)P Y^{−1}(s, ε)∥ ≤ L exp \{−\frac{α}{2ε}(t − s)\} , s ≤ t,

∥Y (t, ε)(I − P)Y^{−1}(s, ε)∥ ≤ L exp \{\frac{β}{2ε}(t − s)\}, t ≤ s,

for ε ∈ (0, ε_{0}] and L is independent of ε. This dichotomy was obtained by Chang [7] for almost periodic systems and by Mitropolskii-Lykova [9] for a system (38) which function A(t) is uniformly continuous on J.

The case μ(t, ε) = μ(t) → 0, if t → ∞

In this case the condition lim_{t→∞} μ(t) = 0 allows to obtain a small value of |μ(t, ε)| if we consider t ∈ [t_{0}, ∞). All the reasoning leading to Theorem 2 can be acomplished on the interval [t_{0}, ∞) instead of [0, ∞).

Theorem 4 If we assume valid H1-H5, where U(t, ε) is defined with

α_{m}(t, ε) = \frac{λ_{m}(t) − λ_{m+1}(t)}{μ(t)} + u_{m}(t) − u_{m+1}(t),

(therefore U(t, ε) does not depend on ε), then the impulsive system

μ(t)x′(t) = A(t)x(t), t ≠ t_{k}, J = [0, ∞)
x(t_{k}^{+}) = C_{k}x(t_{k}^{−}), k ∈ ℕ = {1, 2, 3, …} ,

has the dichotomy

∥X(t)P X^{−1}(s)∥ ≤ K exp {∫_{s}^{t} μ_{1}(τ)dτ} , s ≤ t,
∥X(t)(I − P)X^{−1}(s)∥ ≤ K exp {∫_{t}^{s} μ_{2}(τ)} , t ≤ s ,

where

μ_{1}(t) = \frac{Re(λ_{m}(t)) + L_{4}μ(t)σ + μ(t)L(δ, 0)K(σ)}{μ(t)},

μ_{2}(t) = \frac{Re(λ_{m+1}(t)) + L_{4}σμ(t) + 3μ(t)L(δ, 0)K(σ)}{μ(t)}.

As an application of the above formula let us consider the ordinary system

μ(t)x′(t) = A(t)x(t), lim_{t→∞} μ(t) = 0.  (39)

Theorem 5 If A(·) satisfies H1 and the function U_{m}(t) defined in H5 with

α_{m}(t, ε) = \frac{λ_{m}(t) − λ_{m+1}(t)}{μ(t)},

is bounded, then system (39) has the dichotomy (36), where

μ_{1}(t) = \frac{Re(λ_{m}(t)) + 3σμ(t) + μ(t)L(δ, 0)K(σ)}{μ(t)},

μ_{2}(t) = \frac{Re(λ_{m+1}(t)) − 3σμ(t) − 3μ(t)L(δ, 0)K(σ)}{μ(t)}.

The above theorem gives conditions of existence of a (μ_{1}, μ_{2})- dichotomy for (39) with an unbounded function μ(t)^{−1}A(t). These systems have been studied in [13].

References

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[11] R. Naulin, R.; Pinto, M.: Splitting of linear differential systems with impulsive effect, to appear in Rocky Mountain J. Math. (1999).

[12] R. Naulin, R.; Pinto, M.: Quasi-diagonalization of linear impulsive systems and applications, J. Math. Anal. Appl., Vol. 208, 281-297 (1997).

[13] R. Naulin, R.; Pinto, M.: Dichotomies for differential systems with unbounded coefficients”, Dyn. Syst. Appl. , 3, 333-348 (1994).

[14] Papashinopoulos, G.: Some roughness results concerning reducibility for linear difference equations, Internat. J. Math. & Math. Sci. , Vol. 11, N^{o} 4, 793-804 (1988).

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Raúl Naulin Departamento de Matemáticas, Universidad de Oriente Apartado 245, Cumaná 6101-A, Venezuela e-mail: rnaulin@cumana.sucre.udo.edu.ve