Tempered exponential behavior for a dynamics in upper triangular form

Tempered exponential behavior for a dynamics in upper triangular form

Luís Barreira

and Claudia Valls

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal

Received 18 June 2018, appeared 16 September 2018 Communicated by Christian Pötzsche

Abstract. We consider the problem of whether the existence of a tempered exponential dichotomy for a linear dynamics can be deduced from the same property for the dy- namics restricted to each diagonal entry. More generally, we consider this problem for a dynamics in block upper triangular form. We also obtain corresponding results for a strong tempered exponential dichotomy and for a discrete time dynamics.

Keywords: block triangular equations, tempered exponential dichotomies.

2010 Mathematics Subject Classification: 37D25.

1 Introduction

Any linear dynamics, either autonomous or nonautonomous, can be transformed via a (possi- bly nonautonomous) coordinate change into one in upper triangular form. This is often quite convenient simply because it is easier to deal with a dynamics in upper triangular form. For example, in the case of continuous time this allows one to solve a linear equation by proceed- ing successively from the last component to the first one. More precisely, consider a sequence of n×nmatrices(A_m)_m∈Nand the associated dynamics

x_m+1= A_mx_m, m∈_N.

Then there exists a sequence of n×n orthogonal matrices (U_m)_m∈N such that the matrices Bm = U_m⁻₊¹₁AmUm are upper triangular. In other words, the coordinate change ym = U_m⁻¹xm

given by the matricesU_m leads to a dynamics

y_m+1 =B_my_m, m∈_N,

where all matrices Bm are upper triangular. Similarly, given n×n matrices A(t) varying continuously witht ≥0, consider the linear equation

x⁰ = A(t)x. (1.1)

BCorresponding author. Email: barreira@math.tecnico.ulisboa.pt

Then there exist matrices U(t) varying differentiably with t ≥ 0 such that the coordinate changey(t) =U(t)⁻¹x(t)leads to the equationy⁰ =B(t)y, where the matrices

B(t) =U(t)⁻¹A(t)U(t)−U(t)⁻¹U⁰(t)

are upper triangular for eacht≥0. Both results are well known and follow from a simple ap- plication of the Gram–Schmidt process (see for example [1] for these and other constructions).

When the dynamics is autonomous, it suffices to use the reduction to the Jordan canonical form, both for discrete and continuous time.

As a consequence, there is no loss of generality in considering only linear dynamics that are already in upper triangular form. Incidentally, one can ask whether it is possible to apply further coordinate changes in order to get rid of some elements above the diagonal, if possible bringing the dynamics to a diagonal form. This would certainly make many problems much simpler. Not surprisingly, this is not always possible (see for example [1]). On the other hand, one can still ask whether it is possiblefor some specific propertyto deduce that property solely from the information on the diagonal.

Here we consider the problem of whether the hyperbolicity of a dynamics in upper trian- gular form, and more generally in block upper triangular form, can be deduced from the hy- perbolicity of the dynamics restricted to each diagonal entry. More precisely, we consider the notion of hyperbolicity corresponding to the existence of a tempered exponential dichotomy.

The latter is the natural notion in the context of ergodic theory. In the particular case of a dynamics exhibiting only contraction, equation (1.1) is said to have a tempered exponential contractionif there existλ>0 and a function D:R⁺₀ →_R⁺ satisfying

lim sup

t→+_∞

1

t logD(t)≤0, (1.2)

such that

kx(t)k ≤D(s)e^−λ(t−s)kx(s)k, fort≥s,

where x = x(t) is any solution of the equation (see Section 2 for the notion of a tempered exponential dichotomy). We recall that equation (1.1) is said to have anexponential contraction if there existλ,D>0 such that

kx(t)k ≤De^−λ(t−s)kx(t)k, fort≥ s (1.3) and any solution x = x(t) of the equation. For example, consider an autonomous equation y⁰ = f(y)whose flow ϕ_t preserves a finite measure (such as any Hamiltonian flow restricted to a compact hypersurface, with respect to the Liouville measure). Then, for almost all initial conditionsy, if the linear variational equation

x⁰ = Ay(t)x, where Ay(t) =d_ϕt(y)f,

has only negative Lyapunov exponents, then it has a tempered exponential contraction.

More generally, we consider the problem of whether the hyperbolicity of a dynamics in block upper triangular formcan be deduced from the hyperbolicity of the dynamics restricted to each diagonal block. This includes the upper triangular case as a special case. For example, for continuous time this corresponds to consider the equation

x⁰ = A(t)x+C(t)y,

y⁰ =B(t)y. (1.4)

In Theorem2.1we show that the existence of a tempered exponential dichotomy for equation (1.4) yields the existence of tempered exponential dichotomies for the equations x⁰ = A(t)x and y⁰ = B(t)y, which are associated with the blocks on the diagonal. On the other hand, Theorem2.3shows that under the condition

lim sup

t→+_∞

1

t log D(_t)kC(_t)k≤0,

with D = D(t) as in (1.2) or with a corresponding function in the case of a tempered expo- nential dichotomy, the converse of the statement in Theorem2.1holds. Corresponding results for the notion of an exponential dichotomy (which includes that of an exponential contraction in (1.3) as a particular case) were established by Battelli and Palmer in [2] (see also [4]). To the possible extent we follow their approach in the proofs of Theorems2.1 and2.3. However, we note that none of the mentioned results in the two papers follows from results in the other.

We also obtain corresponding results for a strong tempered exponential dichotomy (see Theorems 3.1 and3.3). This correspond to consider both lower and upper bounds along the stable and unstable directions of a tempered exponential dichotomy. Finally, we establish versions of these results for discrete time (see Section4). The arguments follow a similar path to those for continuous time although they require several nontrivial modifications.

2 Continuous time dynamics

Consider the linear equation (1.1) on Rⁿ, where A: I → _R^n×n is a piecewise continuous function on some interval I ⊂ _R(we shall consider the cases I = _R⁺_{0 and}I =_R⁻₀). We write the solutions in the form x(t) = T(t,s)x(s), fort,s ∈ I, where T(t,s) is the evolution family associated with (1.1).

We say that equation (1.1) has atempered exponential dichotomy on I if there exist projections Ptfort∈ I satisfying

P_tT(t,s) =T(t,s)P_s, fort,s ∈ I, (2.1) and there exist λ>0 and a functionD: I →_R⁺_satisfying

lim sup

|t|→+_∞

1

|t|^logD(t)≤0 (2.2)

such that

kT(t,s)P_sk ≤D(s)e^−λ(t−s), fort≥s, (2.3) and

kT(t,s)Qsk ≤D(s)e^−λ(s−t), fort≤s, (2.4) where Q_t = Id_n−P_t for each t (here Id_n is the identity onRⁿ). The sets P_s(_Rⁿ) andQ_s(_Rⁿ) are called, respectively,stableandunstable spacesat times. We note that

Pt =T(t,s)PsT(s,t)

and so in particular P_t = T(t, 0)PT(0,t), where P = P₀. This shows that all the projections P_t are determined by the projection at time 0.

Now we consider a block upper triangular equation (1.4), wherex ∈_R^k andy∈_R^n−k for some integerk∈ (0,n). We write the corresponding evolution family in the form

T(t,s) =

U(t,s) W(t,s) 0 V(t,s)

,

whereU(t,s)andV(t,s)are the evolution families associated, respectively, with the equations x⁰ = A(t)x and y⁰ = B(t)y. (2.5) It follows readily from the variation of constants formula that

W(t,s) =

Z _t

s U(t,τ)C(τ)V(τ,s)dτ. (2.6) We first show that the existence of a tempered exponential dichotomy for equation (1.4) yields the existence of tempered exponential dichotomies for the equations associated with the blocks on the diagonal.

Theorem 2.1. Assume that equation (1.4) has a tempered exponential dichotomy on I = _R0⁺ or I = _R⁻₀ with constantλ. Then the equations (2.5) have tempered exponential dichotomies on I with the same constantλ. Moreover, the projection P₀ associated with the tempered exponential dichotomy for equation(1.4)can be written in the form

P^A LP^B 0 P^B

if I =_R⁺₀ (2.7)

and

P^A L(Id_n−k−P^B)

0 P^B

if I =_R⁻₀, (2.8)

where P^A: R^k →_R^kand P^B: R^n−k →_R^n−kare, respectively, the projections at time0associated with the tempered exponential dichotomies for the equations in(2.5), and where L: P^B(_R^n−k)→P^A(_R^k)^⊥ is the linear map given by

Lv=−

Z _+∞

0

(Id_k−P^A)U(0,s)C(s)V(s, 0)v ds if I =_R⁺₀ (2.9) and L: kerP^B →(kerP^A)^⊥is the linear map given by

Lv=−

Z ₀

−_∞P^AU(_0,s)C(s)V(s, 0)v ds if I =_R⁻_{0. (2.10)} Proof. We start with an auxiliary result forI =_R⁺₀. LetU(t,s)be the evolution family associ- ated with the equationx⁰ = A(t)x.

Lemma 2.2. Assume that the equation x⁰ = A(t)x onR^k has a tempered exponential dichotomy on R⁺₀ with constantλand projections P_t. Moreover, let P_tbe another family of projections such that

PtU(t,s) =U(t,s)Ps, for t,s≥0. (2.11) Then the equation x⁰ = A(t)x has a tempered exponential dichotomy with projections Pt if and only if P₀(_R^k) = P₀(_R^k), in which case the equation has a tempered exponential dichotomy onR⁺₀ with projections P_t, constantλand function D given by

D(s) =D(s) +D(₀)D(s)kP₀−P₀k_{. (2.12)}

Proof of the lemma. To the possible extent we follow similar arguments in [3] for uniform ex- ponential dichotomies. One can easily verify that if the equation has a tempered exponential dichotomy onR⁺₀ with projectionsP_t, then

P₀(_R^k) =

v∈_R^k : sup

t≥0

kT(t, 0)vk< +_∞

=P₀(_R^k)

(in other words, the stable space is uniquely determined and coincides with the set of all initial conditions leading to bounded solutions). Now assume thatP0(_R^k) =P0(_R^k). Then

P₀P₀ =P₀ and P₀P₀= P₀, which implies that

P₀−P₀ =P₀(P₀−P₀) = (P₀−P₀)Q₀. (2.13) It follows from the existence of a tempered exponential dichotomy for the equationx⁰ = A(t)x that

kU(t, 0)(P0−P0)vk=kU(t, 0)P0(P0−P0)vk

≤D(0)e^−λtk(P₀−P₀)vk

=D(0)e^−λtk(P₀−P₀)Q₀vk

≤D(₀)e^−λtkP₀−P₀k · kQ₀vk

=D(0)e^−λtkP₀−P₀k · kU(0,s)U(s, 0)Q₀vk

=D(0)e^−λtkP₀−P₀k · kU(0,s)QsU(s, 0)vk

≤D(0)D(s)e^−λ(s+t)kP₀−P₀k · kU(s, 0)vk

(2.14)

fort,s≥0 andv∈_R^k. Therefore,

kU(t,s)Psvk ≤ kU(t,s)Psvk+kU(t,s)(Ps−Ps)vk

=kU(t,s)Psvk+kU(t, 0)(P0−P0)U(0,s)vk

≤ D(s)e^−λ(t−s)kvk+D(0)D(s)e^−λ(t−s)kP₀−P₀k · kvk

= D(s)e^−λ(t−s)kvk

whenevert ≥s≥0, with Das in (2.12). Similarly, lettingQ_t =Id_k−P_t we obtain kU(t,s)Q_tvk ≤ kU(t,s)Q_svk+kU(t,s)(P_s−P_s)vk

=kU(t,s)Q_svk+kU(t, 0)(P₀−P₀)U(0,s)vk

≤ D(s)e^−λ(t−s)kvk+D(0)D(s)e^−λ(t−s)kP0−P0k · kvk

= D(s)e^−λ(t−s)kvk

(2.15)

whenever s ≥ t ≥ 0. This shows that the equation x⁰ = A(t)x has a tempered exponential dichotomy with projectionsP_t.

One can readily obtain a corresponding version of Lemma2.2forI =_R⁻₀.

We proceed with the proof of the theorem. We first show that the equation x⁰ = A(t)x has a tempered exponential dichotomy on the interval I. Let E₁ ⊂ _R^k be the vector space of all initial conditions at time 0 for which the solutions of x⁰ = A(t)x are bounded on I and let

E₂ be any complement ofE₁ in R^k. Moreover, let F₁ ⊂ _R^n−k be the vector space of all initial conditionsvat time 0 for whichV(t, 0)vis bounded on I and the equation

x⁰ = A(t)x+C(t)V(t, 0)v (2.16) has a bounded solution on the intervalI. Finally, let F₂be any complement ofF₁inR^n−k.

We first show that given v ∈ F₁, there exists a unique bounded solution x_v of equation (2.16) on I with x_v(₀)∈ E₂. Let xbe a bounded solution of equation (2.16). We note that ¯xis another bounded solution of (2.16) (for the samev) if and only if x−x¯ is a bounded solution of x⁰ = A(t)x, that is, if and only if x(0)−x¯(0) ∈ E₁. This aside remark can be used to establish the existence and uniqueness ofx_v, as follows.

Given bounded solutions x and ¯x of equation (2.16) with x(0), ¯x(0) ∈ E₂, it follows from the remark that x−x¯ is a bounded solution of x⁰ = A(t)x with x(0)−x¯(0) ∈ E₂. By the choice of E₁ and E₂, this yields that x(₀) = x¯(₀)_{and so} x = x. For the existence we take a¯ bounded solution x of equation (2.16) with x(0) = u₁+u₂, where u₁ ∈ E₁ and u₂ ∈ E₂ (it exists since v ∈ F₁). Then for the solution ¯x of equation (2.16) with ¯x(0) = u₂ ∈ E₂ we have x(₀)−x¯(₀) =u₁ ∈E₁ and so, by the remark, we conclude that ¯x is bounded.

Using the solutionxv we define a linear operator L: F₁ →E₂ byLv= xv(0). We note that (u,v)is the initial condition of a bounded solution of equation (1.4) on I if and only if

u−Lv∈ E₁ and v∈ F₁. (2.17)

Moreover, for I = _R⁺₀, let P^A: R^k → _R^k and Q: R^n−k → _R^n−k be, respectively, the projec- tions onto the first components of the splittingsE₁⊕E₂ and F₁⊕F₂. Finally, for I = _R⁻₀, let P^A: R^k →_R^k and Q: R^n−k →_R^n−k be, respectively, the projections onto the second compo- nents of the splittingsE₁⊕E₂ andF₁⊕F₂.

Now we consider the projectionPgiven by P=

P^A LQ

0 Q

if I =_R⁺₀ and

P=

P^A L(Id_n−k−Q)

0 Q

if I =_R⁻₀.

It follows readily from the characterization of the initial conditions of the bounded solutions of equation (1.4) in (2.17) that P(_Rⁿ)(both when I = _R⁺₀ and I = _R⁻₀) is the vector space of all initial conditions at time 0 leading to bounded solutions. In short,P(_Rⁿ) =P₀(_Rⁿ), where P_tare the original projections with respect to which equation (1.4) has a tempered exponential dichotomy. It follows from Lemma 2.2 (and its corresponding version when I = _R⁻₀) that equation (1.4) has a tempered exponential dichotomy on I with respect to the projections

P_t =T(t, 0)PT(0,t), (2.18) whereT(t,s)is the evolution family associated with equation (1.4), with constantλand func- tionD.

LetQ_s =Id_n−P_s. For eachu∈ _R^k ands∈ I we have

P_s(u, 0) = (P_s^Au, 0) and Q_s(u, 0) = (Q_s^Au, 0), (2.19) where

P_s^A=U(s, 0)P^AU(_0,s) _and Q_s^A=_Idk−P_s^A.

Therefore, for each u∈_R^k andt≥ switht,s∈ I we have

kU(t,s)P_s^Auk=kT(t,s)(P_s^Au, 0)k=kT(t,s)P_s(u, 0)k

≤ D(s)e^−λ(t−s)k(u, 0)k= D(s)e^−λ(t−s)kuk, (2.20) using the normk(u,v)k=_max{kuk_,kvk}_foru∈_R^k_andv ∈_R^n−k. Similarly, for eachu∈_R^k andt≤ switht,s∈ I we have

kU(t,s)Q_s^Auk= kT(t,s)Q_s^A(u, 0)k=kT(t,s)Q_s(u, 0)k

≤ D(s)e^−λ(s−t)k(u, 0)k=D(s)e^−λ(s−t)kuk_. ^(2.21) This shows that the equation x⁰ = A(t)x has a tempered exponential dichotomy on I with projectionsP_t^A.

Before showing that the equation y⁰ = B(t)y has a tempered exponential dichotomy we obtain identity (2.9) for v∈ F₁. By the variation of constants formula, fort≥ 0 andv ∈ F₁ we have

P_t^Axv(t) =U(t, 0)P₀^Axv(0) +

Z _t

0 U(t,τ)P_τ^AC(τ)V(τ, 0)v dτ and

Q_t^Ax_v(t) =U(t, 0)Q₀^Ax_v(0) +

Z _t

0 U(t,τ)Q^A_τC(τ)V(τ, 0)v dτ.

The last identity is equivalent to

Q₀^Ax_v(0) =U(0,t)Q^A_t x_v(t)−

Z _t

0 U(0,τ)Q^A_τC(τ)V(τ, 0)v dτ. (2.22) Since the functionxv is bounded, we haveC=sup_t≥0kxv(t)k<+_∞and

kU(0,t)Q^A_t x_v(t)k ≤CDe^−λt. Hence, taking limits in (2.22) whent→+_∞we obtain

Q₀^Axv(0) =−

Z _+∞

0 U(0,τ)Q_τ^AC(τ)V(τ, 0)v dτ.

Recall that by construction we have x_v(₀)∈ E₂ and soQ₀^Ax_v(₀) =x_v(₀). Therefore, Lv=x_v(0) =−

Z +_∞

0 U(0,τ)Q_τ^AC(τ)V(τ, 0)v dτ, (2.23) which establishes identity (2.9). Identity (2.10) can be obtained in a similar manner.

Finally, we show that the equationy⁰ =B(t)yhas a tempered exponential dichotomy on I. Consider the adjoint equation

x⁰ =−A(t)^∗x

y⁰ =−C(t)^∗x−B(t)^∗y (2.24) and write it in the form

z⁰ =−B(t)^∗z−C(t)^∗w,

w⁰ =−A(t)^∗w, (2.25)

taking (z,w) = (y,x). One knows from the theory that the adjoint equation (2.24) also has a tempered exponential dichotomy on I, with the same constant λand functionD, and with projections Id_n−P^∗_t (see (2.18)). This readily implies that equation (2.25) has a tempered exponential dichotomy on I with projection at time 0 given by

Id_n−k−Q^∗ −Q^∗L^∗ 0 Id_k−(_P^A)^∗

if I =_R⁺₀

and

Id_n−k−Q^∗ (Id_n−k−Q^∗)L^∗ 0 Id_k−(P^A)^∗

if I =_R⁻₀,

with the same constantλand function D. Thus, one can proceed as in (2.19), (2.20) and (2.21) to conclude that the equationy⁰ = −B(t)^∗yhas a tempered exponential dichotomy on I with projections at time 0 equal to Id_n−k−Q^∗, with the same data. This implies that the equation y⁰ = B(t)y has also a tempered exponential dichotomy, with projection at time 0 equal to P^B = Q, thus leading to the projections in (2.7) and (2.8). In particular, identity (2.23) holds for

v∈ F₁=Q(_R^n−k) =P^B(_R^n−k)_, with a corresponding remark for I =_R⁻₀.

Our second result gives a suficient condition for the converse of the statement in Theo- rem2.1.

Theorem 2.3. Assume that the equations x⁰ = A(t)x and y⁰ = B(t)y have tempered exponential dichotomies on I= _R⁺₀ or I =_R⁻₀ with constantλ, function D and, respectively, projections P^Aand P^Bat time0. If

lim sup

|t|→+_∞

1

t log D(t)kC(t)k ≤0, (2.26) then equation(1.4)has a tempered exponential dichotomy on I with any constant less thanλ.

Proof. Consider the projections

P_t =T(t, 0)PT(_0,t)_,

withPas in (2.7) or (2.8), respectively, when I =_R⁺₀ or I =_R⁻₀. We claim that P_t =

P_t^A R(t) 0 P_t^B

, where

P_t^A=U(t, 0)P^AU(0,t) and P_t^B =V(t, 0)P^BV(0,t), with

R(t) = −

Z _t

0 U(t,τ)P_τ^AC(τ)(Id_n−k−P_τ^B)V(τ,t)dτ

−

Z _+∞

t U(t,τ)(Id_k−P_τ^A)C(τ)P_τ^BV(τ,t)dτ

(2.27)

whenI =_R⁺_{0 and}

R(t) = −

Z _t

−_∞U(t,τ)P_τ^AC(τ)(Id_n−k−P_τ^B)V(τ,t)dτ

−

Z ₀

t U(t,τ)(Id_k−P_τ^A)C(τ)P_τ^BV(τ,t)dτ

(2.28)

when I =_R⁻₀. Clearly,

R(0) =

(LP^B if I =_R⁺₀,

L(Id_n−k−P^B) if I =_R⁻₀. (2.29) Identities (2.27) and (2.28) can be established as follows. For I = _R0⁺ it follows readily from (2.7) that

P_t= T(t, 0)

P^A LP^B 0 P^B

T(0,t)

=

U(t, 0) W(t, 0) 0 V(t, 0)

P^A LP^B 0 P^B

U(0,t) W(0,t) 0 V(0,t)

=

P_t^A P_t^AU(t, 0)W(0,t) + [U(t, 0)L+W(t, 0)]V(0,t)P_t^B

0 P_t^B

. Using (2.6) we find that

R(t) =P_t^AU(t, 0)W(0,t) + [U(t, 0)L+W(t, 0)]V(0,t)P_t^B

=U(t, 0)R(0)V(0,t) +

Z _t

0 U(t,τ)C(τ)P_τ^B−P_τ^AC(τ)]V(τ,t)dτ.

Using (2.8) one can readily obtain a corresponding property when I = _R⁻₀. Identities (2.27) and (2.28) follow now in a straightforward manner from (2.9) and (2.10) together with (2.29).

We use the former identities to show that the equation x⁰ = A(t)x has a tempered ex- ponential dichotomy on I with projections P_t. First we show that (2.3) holds for t ≥ s with t,s ∈ I, for some constantλand some function Dsatisfying (2.2). Note that

T(t,s)P_s=

U(t,s)P_s^A W(t,s)P_s^B+U(t,s)R(s) 0 V(t,s)P_s^B

. (2.30)

We have

W(t,s)P_s^B =

Z _t

s U(t,τ)P_τ^AC(τ)P_τ^BV(τ,s)dτ +

Z _t

s

U(t,τ)(_Idk−P_τ^A)C(τ)P_τ^BV(_τ,s)dτ.

When I =_R⁺₀ we obtain

U(t,s)R(s) =−

Z _s

0 U(t,τ)P_τ^AC(τ)(Id_n−k−P_τ^B)V(τ,s)dτ

−

Z _+∞

s U(t,τ)(Id_k−P_τ^A)C(τ)P_τ^BV(τ,s)dτ and hence,

W(t,s)P_s^B+U(t,s)R(s) =

Z _t

s U(t,τ)P_τ^AC(τ)P_τ^BV(τ,s)dτ

−

Z +_∞

t U(t,τ)(Id_k−P_τ^A)C(τ)P_τ^BV(τ,s)dτ

−

Z _s

0 U(t,τ)P_τ^AC(τ)(Id_n−k−P_τ^B)V(τ,s)dτ.

(2.31)

Similarly, when I =_R⁻₀ we obtain W(t,s)P_s^B+U(t,s)R(s) =

Z _t

s U(t,τ)P_τ^AC(τ)P_τ^BV(τ,s)dτ

−

Z ₀

t U(t,τ)(Id_k−P_τ^A)C(τ)P_τ^BV(τ,s)dτ

−

Z _s

−_∞U(t,τ)P_τ^AC(τ)(Id_n−k−P_τ^B)V(τ,s)dτ.

By (2.26), givenε>0, there existsd>0 such that

D(t)kC(t)k ≤de^ε|t|, fort ∈ I. (2.32) Hence, whenevert≥s≥0 we have

kW(t,s)P_s^B+U(t,s)R(s)k ≤

Z _t

s D(τ)e^{−λ(t−τ)}kC(τ)kD(s)e^{−λ(τ−s)}dτ +

Z +_∞

t D(τ)e^{−λ(τ−t)}kC(τ)kD(s)e^{−λ(τ−s)}dτ +

Z _s

0 D(τ)e^{−λ(t−τ)}kC(τ)kD(s)e^{−λ(s−τ)}dτ

≤dD(s)

ce^−µ(t−s)+ ^e

−(λ−ε)(t−s)+εs

2λ−ε + ^e

−(λ−ε)(t−s)+εs

2λ−ε

≤dD(s)e^εse^−µ(t−s)

c+ ² 2λ−ε

,

(2.33)

for some constantsc>0 andµ<λ(independent ofε) provided thatεis sufficiently small so thatµ< λ−ε. The first term follows from noting that

Z _t

s e^ετe^{−λ(t−τ)}e^{−λ(τ−s)}dτ=

Z _t

s e^ετe^−λ(t−s)dτ

≤ (t−s)e^εte^−λ(t−s)≤ce^{−µ(t−s)+εs}, for some constants as above. It follows from (2.2) and (2.33) that

lim sup

s→+_∞

1

s log kW(t,s)P_s^B+U(t,s)R(s)ke^µ(t−s)

≤ε. (2.34)

In view of identity (2.30), it follows from (2.34) and the arbitrariness of ε that (2.3) holds whenevert≥s ≥0 withλreplaced byµandDreplaced by the function

s7→sup

t≥s

kW(t,s)P_s^B+U(t,s)R(s)ke^µ(t−s) .

The case whenI =_R⁻₀ can be treated similarly (it only requires replacing in the integrals the lower limit 0 by−_∞and the upper limit+_∞by 0).

Now we show that (2.4) holds for t ≤ s with t,s ∈ I, for some constant λ and some functionDsatisfying (2.2). First note that

T(t,s)Q_s=

(Id_k−P_t^A)U(t,s) (Id_k−P_t^A)W(t,s)−R(t)V(t,s) 0 (Id_n−k−P_t^B)V(t,s)

.

Again, for simplicity of the exposition, we consider only I =_R⁺₀. We have R(t)V(t,s) = −

Z _t

0

U(t,τ)P_τ^AC(τ)(_Idn−k−P_τ^B)V(_τ,s)dτ

−

Z _+∞

t

(Id_k−P_τ^A)C(τ)P_τ^BV(τ,s)dτ and

(Id_k−P_t^A)W(t,s) =

Z _t

s U(t,τ)(Id_k−P_τ^A)C(τ)V(τ,s)dτ.

Hence,

(Id_k−P_t^A)W(t,s)−R(t)V(t,s) =

Z +_∞

s U(t,τ)(Id_k−P_τ^A)C(τ)P_τ^BV(τ,s)dτ

−

Z _t

s U(t,τ)(Id_k−P_τ^A)C(τ)(Id_n−k−P_τ^B)V(τ,s)dτ +

Z _t

0 U(t,τ)P_τ^AC(τ)(Id_n−k−P_τ^B)V(τ,s)dτ, which implies that whenever 0≤t≤ swe have

k(_Idk−P_t^A)W(t,s)−R(t)V(t,s)k

≤dD(s)e^εs

e^{(λ−ε)(t−s)+εs}

2λ−ε +ce^µ(t−s)+ ^e

(λ−ε)(t−s)+εs

2λ−ε

≤dD(s)e^εs 2

2λ−ε +c

e^λ(t−s)

(2.35)

for some constants c > 0 and µ < λ (independent ofε) provided that ε is sufficiently small so that µ < λ−ε. Proceeding as in (2.34), it follows from (2.35) that (2.4) holds whenever 0 ≤ t ≤ s, withλ replaced byµand D replaced by some other function. This completes the proof of the theorem.

Now we give a counterexample to the converse of Theorem 2.3 when condition (2.26) is not satisfied.

Example 2.4. Consider the triangular equation

x⁰ =2x+ye^at, y⁰ =−2y (2.36)

with a>0. Both linear equations

x⁰ =2x, y⁰ =−2y

have a tempered exponential dichotomy on R⁺₀ with constant function D and projections, respectively,P^A =0 andP^B =1. SinceC(t) =e^at, we have

lim sup

t→+_∞

1

t log(D(t)kC(t)k) =_{lim sup}

t→+_∞

1

te^at =a>₀

and so (2.26) is not satisfied. Now we show that the triangular equation (2.36) has no tempered exponential dichotomy onR⁺₀. The solutions are

x(t) =

x(0)− ^y(0) a−3

e^2t+y(0)e^(a−2)ta−3, y(t) =y(0)e^−2t

ifa6=3 and

x(t) =x(0)e^t+y(0)te^t, y(t) =y(0)e^−2t

if a = 3. It follows from the proof of Theorem 2.1 that if equation (2.36) has a tempered exponential dichotomy, then the projection Phas rank 1. In view of the first component this happens if and only if a ≤ 2 and the initial condition is a scalar multiple of (_1,a−3)_{. On} the other hand, by Theorem 2.1 we have P = ⁰_{0 1}^c for some c ∈ _{R. For} a ≤ 2 the matrix U(t) =T(t, 0)associated with (2.36) is

U(t) = ^e2t

e^2t−e^(a−2)t 3−a

0 e^−2t

!

and so

U(t)PU(t)⁻¹= ⁰

[(3−a)c+1]e^4t−e^at 3−a

0 1

! . Clearly,

lim inf

t→+_∞

1 t log

[(₃−a)c+₁]e^4t−e^at >₀

for all a ∈ (0, 2] and so, for a > 0 equation (2.36) does not have a tempered exponential dichotomy onR⁺₀.

3 Strong tempered exponential dichotomies

We say that equation (1.1) has astrong tempered exponential dichotomy on an interval I if there exist projectionsP_t fort∈ I satisfying (2.1) and there exist constantsµ>λ>0 and a function D: I →_R⁺ satisfying (2.2) such that

kT(t,s)P_sk ≤D(s)e^−λ(t−s), kT(s,t)P_tk ≤D(t)e^µ(t−s) (3.1) and

kT(s,t)Qtk ≤D(t)e^−λ(t−s), kT(t,s)Qsk ≤D(s)e^µ(t−s) (3.2) fort ≥s, whereQ_t =Id_n−P_t for eacht.

Theorem 3.1. Assume that equation(1.4)has a strong tempered exponential dichotomy on I =_R0⁺or I =_R⁻₀ with constantsλandµ. Then the equations x⁰ = A(t)x and y⁰ =B(t)y have strong tempered exponential dichotomies on I with the same constantsλandµ.

Proof. For simplicity of the exposition we consider only the case of I = _R⁺₀. LetU(t,s)be the evolution family associated with the equationx⁰ = A(t)x.

Lemma 3.2. Assume that the equation x⁰ = A(t)x onR^khas a strong tempered exponential dichotomy on R⁺₀ with constants λ,µand projections P_t. Moreover, let P_t be another family of projections sat- isfying (2.11). Then the equation x⁰ = A(t)x has a strong tempered exponential dichotomy with projections P_t if and only if P₀(_R^k) = P₀(_R^k), in which case the equation has a strong tempered exponential dichotomy onR⁺₀ with projections P_t, constantsλ,µand function D given by(2.12).

Proof of the lemma. The lemma can be established along the lines of the proof of Lemma 2.2 and so we only outline the differences. Namely, we only need to obtain the second bounds in (3.1) and (3.2). Proceeding as in the proof of Lemma2.2(see (2.14)) we have

kU(_s,t)_Ptvk ≤ kU(_s,t)_Ptvk+kU(_s,t)(_Pt−Pt)_vk

=kU(s,t)P_tvk+kU(s, 0)(P₀−P₀)U(0,t)vk

≤D(t)e^µ(t−s)kvk+D(0)D(t)e^−λ(t+s)kP₀−P₀k · kvk

=D(t)e^µ(t−s)kvk

fort ≥sandv∈_R^k, withDas in (2.12). Similarly, lettingQ_t =Id_k−Ptwe obtain kU(s,t)Q_tvk ≤ kU(s,t)Q_tvk+kU(s,t)(P_t−P_t)vk

=kU(s,t)Q_tvk+kU(s, 0)(P₀−P₀)U(0,t)vk

≤ D(t)e^µ(t−s)kvk+D(0)D(t)e^λ(t+s)kP0−P0k · kvk

= D(t)e^µ(t−s)kvk

for t ≤ s. This shows that the equation x⁰ = A(t)x has a strong tempered exponential di- chotomy with projectionsPt.

We proceed with the proof of the theorem. In view of Theorem2.1(see (2.20) and (2.21)), for the equationx⁰ = A(t)xit remains to obtain the second bounds in (3.1) and (3.2). For each u∈_R^k andt≥s ≥0 we have

kU(s,t)P_t^Auk= kT(s,t)(P_t^Au, 0)k=kT(s,t)P_t(u, 0)k

≤ D(t)e^µ(t−s)k(u, 0)k= D(t)e^µ(t−s)kuk,

using again the norm k(u,v)k= max{kuk,kvk}foru ∈_R^k andv∈ _R^n−k. Similarly, for each u∈_R^k ands≥t ≥0 we have

kU(s,t)Q_t^Auk= kT(s,t)Q_t^A(u, 0)k=kT(s,t)Q_t(u, 0)k

≤ D(t)e^µ(s−t)k(u, 0)k= D(t)e^µ(s−t)kuk.

This shows that the equationx⁰ = A(t)xhas a strong tempered exponential dichotomy onR⁺₀ with projections P_t^A. Proceeding analogously with the adjoint equation (see the proof of Theorem 2.1), we conclude that the equation y⁰ = B(t)y has a strong tempered exponential dichotomy onR⁺₀.

We note that the projectionP₀associated with the strong tempered exponential dichotomy for equation (1.4) can be written in the form (2.7) and (2.8), respectively, where P^A and P^B are, respectively, the projections at time 0 associated with the strong tempered exponential dichotomies for the equations in (2.5), and whereLis the linear map given by (2.9) and (2.10), respectively.

The following result is a partial converse of Theorem3.1.

Theorem 3.3. Assume that the equations x⁰ = A(t)x and y⁰ = B(t)y have strong tempered exponen- tial dichotomies on I = _R⁺₀ or I = _R⁻₀ with constantsλ,µ, function D and, respectively, projections P^Aand P^B at time0. If condition(2.26)holds, then equation (1.4)has a strong tempered exponential dichotomy on I with any constants, respectively, less thanλand greater thanµ.

Proof. The statement can be established as in the proof of Theorem2.3and so we only outline the differences. For simplicity of the exposition we assume thatI =_R⁺₀ (the case whenI =_R⁻₀ can be treated similarly). Using (2.32), for t ≥ s ≥ 0 the quantity W(s,t)P_t^B+U(s,t)R(t) satisfies (see (2.31)).

It follows in a similar manner as in (2.31) that fort ≥swe have W(s,t)P_t^B+U(s,t)R(t) =−

Z _t

s U(s,τ)P_τ^AC(τ)P_τ^BV(τ,t)dτ

−

Z _+∞

s U(s,τ)(Id_k−P_τ^A)C(τ)P_τ^BV(τ,t)dτ

−

Z _t

0 U(s,τ)P_τ^AC(τ)(Id_n−k−P_τ^B)V(τ,t)dτ.

Now using (3.1) and (3.2) we readily get kW(s,t)P_t^B+U(s,t)R(t)k

≤

Z _t

s

D(τ)e^µ(τ−s)kC(τ)kD(t)e^µ(t−τ)dτ +

Z _t

s D(τ)e^{−λ(τ−s)}kC(τ)kD(t)e^µ(t−τ)dτ +

Z +_∞

t D(τ)e^{−λ(τ−s)}kC(τ)kD(t)e^{−λ(τ−t)}dτ +

Z _s

0 D(τ)e^µ(τ−s)kC(τ)kD(t)e^{−λ(τ−t)}dτ +

Z _t

s D(τ)e^{−λ(s−τ)}kC(τ)kD(t)e^{−λ(τ−t)}dτ

≤dD(t)e^µ(t−s) Z _t

s e^ετdτ+dD(t)e^µt+λs Z _t

s e^{−(µ+λ)τ+ετ}dτ +dD(t)e^λ(t+s)

Z +_∞

t e^{−2λτ+ετ}dτ+dD(t)e^λt−µs Z _s

0 e^{(µ−λ)τ+ετ}dτ +dD(t)e^λ(t−s)

Z _t

s e^ετdτ

≤dD(t)

ce^{ν(t−s)+εt}+ ^e

µ(t−s)+εs

µ+λ−ε+ ^e

−λ(t−s)+εt

2λ−ε + ^e

λ(t−s)+εs

µ−λ+ε +ce^{ν(t−s)+εt}

≤dD(t)e^εt

2ce^ν(t−s)+ ^e

µ(t−s)

µ+λ−ε + ^e

−λ(t−s)

2λ−ε + ^e

λ(t−s)

µ−λ+ε

≤dD(t)e^εte^ν(t−s)

2c+ ² 2λ−ε

+ ¹

µ−λ+ε

,

for some constantsc>_{0 and}ν >µ(independent ofε) provided thatεis sufficiently small so that 2λ>ε.

In a similar manner as we did in (2.35) we get, fort ≥s

k(Id_k−P_t^A)W(t,s)−R(t)V(t,s)k ≤cD˜ (t)e^εte^ν(t−s)

for some constants ˜c>_{0 and}ν>µ(independent ofε) provided thatεis sufficiently small.