MARTINEZ‡, AND J

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ASYMPTOTIC STABILITY FOR INTERMITTENTLY CONTROLLED SECOND-ORDER EVOLUTION EQUATIONS^∗

A. HARAUX^†, P. MARTINEZ^‡, AND J. VANCOSTENOBLE^‡

Vol. 43, No. 6, pp. 2089–2108

Abstract. Motivated by several works on ordinary differential equations, we are interested in the asymptotic stability ofintermittently controlledpartial differential equations. We give a condition of asymptotic stability for second-order evolution equations uniformly damped by an on/off feedback.

This result extends to the case of partial diﬀerential equations a previous result of R. A. Smith concerning ordinary diﬀerential equations.

Key words. damped wave equation, second-order evolution equations, asymptotic behavior, on-oﬀ damping

AMS subject classiﬁcations. 35L05, 35L10, 35B35, 35B40 DOI.10.1137/S0363012903436569

1. Introduction. Motivated by several works on ordinary differential equations, we are interested in the asymptotic stability of intermittently controlledpartial differential equations. This question has been widely studied in the case of ordinary differential equations (see, for example, [1, 9, 10, 25, 27, 28]). The typical problem is the oscillator damped by anon/off damping:

u+u+a(t)u= 0, t >0, (1.1)

wherea:R+→R+is continuous nonnegative. For each solutionuof (1.1), we deﬁne its energy by

∀t≥0, Eu(t) =1

2u(t)²+1 2u(t)². The derivative of the energy is

E(t) =u(t)u(t) +u(t)u(t) =−a(t)u(t)²,

hence the energy is always nonincreasing, but remains constant on the time intervals for which a = 0, and the decay is “very small” if a is “very small.” Denote :=

limt→∞E(t). Many authors (see, in particular, [1, 9, 10, 25, 27, 28]) investigated the links between the distribution of sets whereais positive and the property= 0.

Assume that there exists a sequence (In)n≥0of disjoint open intervals in (0,+∞), denoted byI_n= (a_n, b_n), whereb_n ≤a_n+1 for alln∈N, and such that

∀t∈I_n, 0< m_n≤a(t)≤M_n <∞.

Roughly speaking, the energy is strictly decreasing on the time intervalsI_n and just nonincreasing elsewhere. It is natural to wonder whether the decay on the time

∗Received by the editors October 16, 2003; accepted for publication (in revised form) July 4, 2004;

published electronically April 14, 2005.

http://www.siam.org/journals/sicon/43-6/43656.html

†Laboratoire Jacques-Louis Lions, U.M.R C.N.R.S. 7598, Universit´e Pierre et Marie Curie, Boite courrier 187, 75252 Paris Cedex 05, France ([email protected]).

‡Laboratoire M.I.P., U.M.R. C.N.R.S. 5640, Universit´e Paul Sabatier Toulouse III, 118 route de Narbonne, 31 062 Toulouse Cedex 4, France ([email protected], [email protected]).

2089

Downloaded 02/27/17 to 134.157.2.3. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

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intervalsIn is suﬃcient to drive the energy to zero. Obviously some condition on the length of the intervalsIn has to be imposed to ensure = 0. Smith [27] proved the followingsuﬃcient condition of asymptotic stability:

Theorem 1.1. [27]. Assume that ∞ n=0

mnTnδ_n²= +∞, (1.2)

wheremn andMn are the minimum and the maximum values ofa(t)inIn,Tn is the length of In andδn = min(Tn,(1 +Mn)⁻¹). Then(1.1)is asymptotically stable; i.e., every solutionuof (1.1)satisﬁesEu(t)→0 ast→ ∞.

For example, in the case of a damping such that 0< m≤a(t)≤M for allt∈In

for alln∈N, the condition (1.2) reduces to ∞ n=0

T_n³= +∞. (1.3)

It is noteworthy that (1.3) is also necessary in the following sense: given ε >0 as small as we want, Pucci and Serrin [25] constructed an example for which the sequence (Tn)n satisﬁes

∞ n=0

T_n³⁻^ε= +∞, while ∞ n=0

T_n³<+∞,

and suitable initial conditions such that the energy decays to some >0.

Note also that, under condition (1.2),the distribution of the intervals In has no importance. Only their size is important.

Condition (1.2) also requires that the damping coeﬃcientais not “too small” or

“too large,” in order to prevent “underdamping” or “overdamping.” These phenomena are also a source of lack of strong stability (see [20, 22, 26], where the stability is studied for the wave equation, but always under the condition that the function a remains positive).

To our knowledge, stability properties for such “intermittently controlled” systems have not yet been studied in the case ofpartialdiﬀerential equations.

In [21], we studied the effect of an on/off feedback on the wave equation. We considered the simplified case of a damping coefficientathat is 2T-periodic and such thata(t) =a0>0 on (0, T) anda(t) = 0 on (T,2T). In particular, the condition (1.2) was always satisfied. And we studied the wave equation damped by aboundaryon/off feedback or by alocally distributedon/off feedback. In both cases, we proved that the situation isradically differentfrom the case of ordinary differential equations. Indeed, we proved that, except for a countable number of exceptional values ofT, asymptotic stability occurs (and more precisely, exponential stability). But, for the exceptional values ofT, asymptotic stability does not occur. This means thatthe distribution of the intervalsInis very important in the case of the locally damped wave equation. This phenomenon is related to the optics rays propagation and the geometric condition of Bardos, Lebeau, and Rauch [2, 3]. See further comments in section 3.3.

In [21], the only case for which the situation was not different from the situation of the ordinary differential equations was the wave equation damped by anuniformly distributedon/off feedback. In that case, asymptotic stability occurs for any value of T. Thus the distribution of the intervals damping has no importance.

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In the present work, we now study the wave equation uniformly damped by a general on/oﬀ feedback (in particular, not necessarily periodic). We prove that“the uniformly damped wave equation behaves exactly like the oscillator”in the sense that Theorem 1.1 is still true.

More generally, we prove this result in an abstract setting that includes both the oscillator and wavelike or platelike equations and that also includes bounded or unboundedandlinear and nonlineardamping operators.

In particular, this gives for the result of Smith a new proof quite diﬀerent from the original one, which was relying on monotonicity properties of the solutions of (1.1). Our method is based on a preliminary result which is interesting in itself: we provide an estimate of the energy decay on ashorttime interval (see Theorem 3.1).

This estimate is true for both ordinary and partial diﬀerential equations.

The paper is organized as follows.

• In section 2, we introduce our abstract setting and we give the result of well- posedness (Theorem 2.1).

• In section 3, we provide an estimate of the energy decay on a short time interval (Theorem 3.1) and we deduce the asymptotic stability result (Theo- rem 3.2) extending the previous result of Smith. Then we make some further comments concerning the case of locally distributed dampings to explain the necessity of considering only uniformly distributed dampings.

• In section 4, we give some examples.

• In section 5, we present another application of the method to the case of a positive-negative damping (Theorem 5.1).

2. Abstract setting and well-posedness. LetH be a real Hilbert space endowed with the scalar product (·,·)H and the norm| · |H.

Assume that A: D(A)⊂ H →H is a linear self-adjoint and coercive operator onH with dense domain. We deﬁneV =D(A^1/2) endowed with the scalar product ((·,·))_V and the norm · V deﬁned by

∀v∈V, v²V =|A^1/2v|²H = Av, v˜ V,V, where ˜A∈ L(V, V) represents the extension ofA.

Also letW be a Hilbert space endowed with the norm · W and such that V →W →H ≡H→W →V

with dense imbeddings. We also assume thatAsatisﬁes the following property:

∃λ0, C0>0, such that,∀λ∈[0, λ0], (2.1)

(I+λA)⁻¹∈ L(W) and (I+λA)⁻¹_L(W)≤C₀. Next we consider atime-dependentoperatorB such that

B∈L^∞(J,Lip (W, W)), B(t)0 = 0, (2.2)

∀t∈J, ∀w, z∈W, B(t)w−B(t)z, w−zW,W ≥ 0, (2.3)

∀t∈J, ∀w∈W, B(t)w, wW,W ≥ b²(t)w²W, (2.4)

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∀t∈J, ∀w, z∈W, B(t)w−B(t)zW ≤ Cb(t)²w−zW, (2.5)

where J = [0, T] withT >0 and whereb(t)≥0 withb ∈L²(J). Note thatB(t) is a priori unbounded and nonlinear. (The choice W =H corresponds to the particular case of a bounded operator.)

Now we consider the following second-order evolution equation u+Au+B(t)u= 0, t >0,

(2.6)

with the initial conditions

u(0) =u0∈V, u(0) =u1∈H (2.7)

and we prove that this problem is well-posed.

Theorem 2.1. Under the previous assumptions, for any (u0, u1) ∈ V ×H, there exists a unique solution u∈ L²(0, T;V)∩W^1,2(0, T;H)∩W^2,2(0, T;V) with bu∈L²(0, T;W)andB(t)u=b(t)h(t), h(t)∈L²(0, T;W) of

u+Au+B(t)u= 0 inL²(0, T;V) such that

u(0) =u0∈V, u(0) =u1∈H.

In additionu∈C([0, T];V)∩C¹([0, T];H)and the energy of the solutionudeﬁned by

∀t≥0, Eu(t) := 1

2u(t)²V +1

2|u(t)|²H, is absolutely continuous on[0, T] with

E_u(t) =− B(t)u(t), u(t)W,W a.e on {t, b(t)>0}, and

E_u(t) = 0 a.e on{t, b(t) = 0}.

For the proof of Theorem 2.1, we ﬁrst need the following lemma.

Lemma 2.1. Let b=b(t)≥0and consider

u∈L²(0, T;V)∩W^1,2(0, T;H)∩W^2,2(0, T;V) with

bu∈L²(0, T;W),

whereV ⊂W ⊂H with continuous and dense imbeddings. Let f =bg, withg∈L²(0, T;W), and assume

u+Au=f inL²(0, T;V).

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Then, in fact,u∈C([0, T];V)∩C¹([0, T];H) and the energyEu(t)is absolutely continuous on [0, T]with

E_u(t) = g(t), b(t)u(t)W,W.

Proof of Lemma 2.1. Let Jλ = (I +λA)⁻¹ : H → H for λ > 0. We have Jλ∈ L(H),Jλ|V ∈ L(V), andJλ|W ∈ L(W) forλ≤λ0. And by (2.1),{Jλ}0<λ≤λ₀ is uniformly equicontinuous onW →W.

We claim that

∀ϕ∈W, J_λϕ→ϕ in W asλ→0.

(2.8)

Indeed (2.8) is well-known ifϕ∈V, and sinceV is dense inW, the result follows by density.

Then we introduce

uλ:=Jλu and fλ:=Jλ(bg) =bJλg.

We clearly haveu_λ ∈ L²(0, T;D(A)), u_λ ∈L²(0, T;D(A)), and u_λ ∈L²(0, T;V)⊂ L²(0, T;H).

Setting

Eλ(t) :=1

2uλ(t)²V +1

2|u_λ(t)|²H, we have, a.e. on (0, T),

E_λ = ((uλ, u_λ))V + (u_λ, u_λ)H = (Auλ+u_λ, u_λ)H =b(Jλg, u_λ)H, withJ_λg∈L²(0, T;V)⊂L²(0, T;H).

Letα, β be two points of [0, T] such that α < β. Then we have Eλ(β)−Eλ(α) =

β α

(bJλg, u_λ)ds= β

α

(Jλg(s), b(s)u_λ(s))ds.

(2.9)

On the other hand, we can prove

J_λg→g in L²(0, T;W) asλ→0 (2.10)

and

bu_λ=Jλbu→bu in L²(0, T;W) asλ→0.

(2.11)

Indeed, for the ﬁrst property, we notice that Jλ ∈ L(W) for 0 < λ ≤ λ0 with a uniformly bounded norm (by duality, from (2.1)). ThenJ_λg(t)→g(t) asλ→0, in W a.e. on (0, T), and

Jλg(t)−g(t)W ≤Cg(t)W. From Lebesgue’s theorem, it follows that

Jλg−g²W →0 in L¹(0, T) asλ→0, which gives (2.10).

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Next, from (2.8), we also have Jλb(t)u → b(t)u as λ→0 inW a.e. on (0, T), and by (2.1), we have

J_λb(t)u(t)−b(t)u(t)W ≤Cb(t)u(t)W. From Lebesgue’s theorem, it follows that

J_λbu−bu²W →0 in L¹(0, T) as λ→0, which gives (2.11).

Now assume for a moment thatαandβ arebothsuch that [u(α), u(α)]∈V ×H and [u(β), u(β)]∈V ×H.

Then asλ→0, we can pass to the limit in (2.9) to obtain Eu(β)−Eu(α) =

β α

(g(s), b(s)u(s))W,Wds.

(2.12)

Now let α be ﬁxed for a while and apply (2.12) with β = βn → t ∈ [0, T] as n → +∞. We obtain that E(βn) is bounded and therefore (replacing if necessary (βn)n by a subsequence) we have

(u(βn), u(βn)) (ϕ, ψ) weakly inV ×H as n→+∞. On the other hand, by the regularity assumptions onu, we have

(u(βn), u(βn))→(u(t), u(t)) strongly inH×V as n→+∞.

It follows that (u(t), u(t)) = (ϕ, ψ) and thereforeu(t)∈V andu(t)∈H. Since this is valid foranyt, (2.12) becomes true for any (α, β).

Now the vectorY(t) = (u(t), u(t)) is weakly continuous on [0, T] and its norm is continuous by (2.12). The remainder of the proof is obvious from (2.12).

Proof of Theorem 2.1. (i) Uniqueness. Let uand ˜u be two solutions with the same initial data. We have

u+Au+B(t)u= 0 and u˜+A˜u+B(t)˜u= 0.

Thenz:= ˜u−usatisﬁes

z+Az=B(t)u−B(t)˜u, with

bz ∈L²(0, T;W) and

B(t)u−B(t)˜u =bg, g∈L²(0, T;W).

From Lemma 2.1, we deduce E_z(t) = g(t), b(t)zW,W =

− B(t)˜u−B(t)u,u˜−uW,W ifb(t)>0,

0 ifb(t) = 0.

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ThusEz(t) = ¹₂[z²V +z²H] is nonincreasing by (2.3). SinceEz(0) = 0, we obtain z≡0, i.e., ˜u≡u.

(ii) Existence. We introduce, forψ∈W, C(t)ψ:=

1 b(t)B(t)

ψ b(t)

ifb(t)>0,

0 ifb(t) = 0.

It is clear thatC(t)∈Lip (W, W) for allt∈J andC(t)ψ₁−C(t)ψ₂W ≤Cψ₁− ψ₂W for allt∈J and ψ₁,ψ₂∈W.

Next, for (u⁰, u¹) given inV ×H and for 0< λ≤λ0, we solve u_λ+Au_λ+J_λB(t)J_λu_λ= 0, t∈J,

uλ(0) =u0, u_λ(0) =u1. (2.13)

We have

uλ∈ C⁰([0, T];V)∩ C¹([0, T];H)∩ C²([0, T];V), and

T 0

BJλu_λ, Jλu_λW,Wds+Eλ(T) =Eλ(0) (2.14)

=1

2[u0²V +u1²H] = E(0), which is ﬁxed. Equation (2.13) can also be written as

u_λ+Au_λ+b(t)J_λC(t)(b(t)J_λu_λ) = 0.

(2.15)

From (2.14) we deduce thatbJ_λu_λ is bounded inW:=L²(J;W). Thus B(t)Jλu_λ=b(t)C(t)(b(t)Jλu_λ) =bhλ,

wherehλ is bounded inW:=L²(J;W).

Finally,uλis bounded inL^∞(J;V)∩W^1,^∞(J;H) and we may assume that there exists a subsequence such that

uλ_n u weakly inL²(J;V)∩H¹(J;H) asn→+∞, and

bJ_λ_nu_λ

n z weakly inW asn→+∞. Since u_λ

n u weakly in H := L²(J;H) as n → +∞, we have (taking the inner product with some test functionϕ∈H¹(J;D(A)), for instance)

bJ_λ_nu_λ

n bu weakly inH asn→+∞. Therefore,bu=z∈ W and

bJλ_nu_λ_n bu weakly inW as n→+∞. On the other hand, we have (taking if necessary a subsequence)

C(t)(bJ_λ_nu_λ

n) ψ weakly inW as n→+∞.

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It is not diﬃcult, using a suitable test function, to check that J_λ_nC(t)(bJ_λ_nu_λ

n) ψ weakly inW asn→+∞, so that, passing to the limit in (2.15),

u+Au+bψ= 0 in L²(J;V).

(2.16)

To obtainbψ=B(t)u, it remains to show that ψ=C(t)bu. (2.17)

We introduceC:W → W deﬁned by

∀ϕ∈ W, (Cϕ)(t) :=C(t)ϕ(t) a.e. onJ.

We remark that

C(bJ_λu_λ), bJ_λu_λ_W,W = T

0

bC(bJ_λu_λ), J_λu_λW,Wdt

= T

0

B(t)Jλu_λ, Jλu_λW,Wdt=E(0)−Eλ(T).

Whereas, due to Lemma 2.1, we have

E(T) + Ψ, buW,W =E(0).

Since

E(T)≤lim inf

n→+∞ Eλ_n(T), we obtain

E(0)− Ψ, bu_W,W ≤lim inf

n→+∞Eλ_n(T)

= lim inf

n→+∞(E(0)− C(bJλ_nu_λ_n), bJλ_nu_λ_nW,W)

=E(0)−lim sup

n→+∞ C(bJ_λ_nu_λ

n), bJ_λ_nu_λ

n_W,W. Hence

lim sup

n→+∞ C(bJλ_nu_λ_n), bJλ_nu_λ_n_W,W≤ Ψ, bu_W,W. Then we can apply the following lemma.

Lemma 2.2. Let W be a Hilbert space and let C : W → W be monotone and Lipschitz continuous. Assume that(zn)n is a sequence ofW such thatzn z weakly inW andCzn Ψweakly inW asn→+∞.

If

lim sup

n→+∞ Czn, znW,W ≤ Ψ, zW,W, thenΨ =Cz

For the proof of this lemma, let K : W → W be the duality map. Then C :=

KC:W → W satisﬁes the assumptions of Proposition 2 of [8, p. 41]. (See also Brezis [5].)

This provides (2.17) and the proof of Theorem 2.1 is ﬁnished.

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3. Asymptotic stability.

3.1. An energy decay estimate on a short time interval. Assume that (2.1)–(2.5) hold. In order to study the asymptotic behavior of the energy, we ﬁrst prove the following result, interesting in itself, concerning theestimate of energy decay on a short interval of time.

Theorem 3.1. Let T >0 be ﬁxed and assume that there exist M, m > 0 such that

∀t∈(0, T), ∀v∈W, B(t)v, vW,W ≥mv²W, (3.1)

and

∀t∈(0, T), ∀v∈W, B(t)v²W ≤M B(t)v, vW,W. (3.2)

Then there exists c > 0 (independent of T) such that, for all (u0, u1)∈V ×H, the solution uof (2.6)–(2.7)satisﬁes

E(T) ≤ 1

1 +c_T₋₃_+T₋₁^m_{+M mT}₋₁ E(0).

(3.3)

Theorem 3.1 is interesting in itself because it provides an estimate of the decay of the energy that is valid fort small. In particular,E(t)< E(0). It has to be noted that, in general, estimates of the decay of the energy are provided fortlarge enough, even in the case of uniformly distributed damping terms. Of course if the damping is locally distributed in the domain, it is impossible to expect that E(t)< E(0) for t >0 small. (See, for example, [11, 12, 15, 16, 17, 18, 19, 23, 24] for classical estimates of the energy decay whentis large enough.)

Proof of Theorem 3.1. Following [7], we set θ(t) =t²(T−t)². Note that

∀t∈[0, T], |θ(t)|=|2t(T−t)(T−2t)| ≤2T θ^1/2(t), (3.4)

max

t∈[0,T]θ(t) = T⁴ 16, (3.5)

and

T 0

θ(t)dt=T⁵ 30. (3.6)

We also introduceKW, K_W >0 such that

∀v∈W, K_W |v|H ≤ vW ≤KWvV =KW|A^1/2v|H. (3.7)

First note that the energy ofuis nonincreasing and satisﬁes E(0)−E(T) =

T 0

Bu, uW,W ≥ 0.

(3.8)

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Multiplying (2.6) byθu, we obtain T

0

θ|A^1/2u|²H =− T

0

θ u+Bu, uV,V

= T

0

((θu), u))H− T

0

θ Bu, uW,W

= T

0

θ(u, u)H+ T

0

θ|u|²H− T

0

θ Bu, uW,W

≤ε T

0

θ²(t)|u|²H+ 1 4ε

T 0

|u|²H+ T

0

θ(t)|u|²H

+η T

0

θu²W + 1 4η

T 0

θBu²W, for allε, η >0. Using (3.7), (3.4), (3.5) and (3.2) we deduce

T 0

θ|A^1/2u|²H≤4K_W² K_W ²T²ε

T 0

θ|A^1/2u|²H+ 1 4ε

T 0

|u|²H

+T⁴ 16

T 0

|u|²H+K_W² η T

0

θ|A^1/2u|²H+T⁴ 16

M 4η

T 0

Bu, uW,W.

We chooseεandη such that 4 K_W²

K_W ²T²ε=K_W² η= 1/4, hence

1

4ε = 4K_W²

K_W ²T²; 1

4η =K_W² . Thus we obtain

1 2

T 0

θ|A^1/2u|²H≤ 1 4ε

T 0

|u|²H+T⁴ 16

T 0

|u|²H+T⁴ 16

M 4η

T 0

Bu, uW,W

=

4 K_W²

K_W ²T²+T⁴ 16

T 0

|u|²H+K_W² T⁴M 16

T 0

Bu, uW,W. Hence,

T 0

θ|A^1/2u|²H ≤F(T) T

0

|u|²H+K_W² M T⁴ 8

T 0

Bu, uW,W,

where

F(T) := 8 K_W²

K_W ²T²+T⁴ 8 . Using

∀t∈[0, T], 2E(t) =|A^1/2u(t)|²H+|u(t)|²H ≥2E(T),

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we deduce T

0

θ(2E(T)− |u|²H)≤F(T) T

0

|u|²H+K_W² M T⁴ 8

T 0

Bu, uW,W.

Hence, using (3.5) and (3.7), E(T)

T 0

θ≤ 1

2 F(T) +T⁴ 16

T 0

|u|²H+K_W² M T⁴ 16

T 0

Bu, uW,W

≤ 1

K²_W 4 K_W²

K_W ²T²+3T⁴ 32

T

0

u²W +K_W² M T⁴ 16

T 0

Bu, uW,W.

Thus, using (3.6), there exists a constantc >0 (independent ofT) such that E(T)≤1

c(T⁻³+T⁻¹) T

0

u²W +1 cM T⁻¹

T 0

Bu, uW,W.

Using (3.1) and (3.8), we ﬁnd E(T)≤ 1

cm(T⁻³+T⁻¹) T

0

Bu, uW,W+1 cM T⁻¹

T 0

Bu, uW,W

= 1

cm(T⁻³+T⁻¹+M mT⁻¹)(E(0)−E(T)).

Hence

E(T) ≤ 1

1 +c_T₋₃_+T₋₁^m_{+M mT}₋₁ E(0).

3.2. A condition for asymptotic stability. Assume that (2.1)–(2.5) hold for any T >0. Then (2.6)–(2.7) is well-posed and it follows from Theorem 3.1 that the result of Smith [27] may be extended to the case of problem (2.6)–(2.7).

Theorem 3.2. Consider a sequence(In)n≥0of disjoint open intervals in(0,+∞), denoted by In = (an, bn), where bn ≤an+1 for all n ∈ N, and assume that, for all n≥0, there existMn, mn>0 such that

∀t∈I_n, ∀v∈W, B(t)v, vW,W ≥m_nv²W, (3.9)

and

∀t∈In, ∀v∈W, B(t)v²W≤Mn B(t)v, vW,W. (3.10)

Assume that the following condition holds:

∞ n=0

m_nT_nmin T_n², 1 1 +mnMn

= +∞, (3.11)

whereTn denotes the length of In. Then (2.6)–(2.7)is asymptotically stable; i.e., for all(u0, u1)∈V ×H, the solution uof(2.6)–(2.7) satisﬁesEu(t)→0 ast→ ∞.

Remark 1. Moreover, the proof of Theorem 3.2 also provides an estimate of the decay of the energy: if there existsC >0 such that

∀n∈N, u_n:=m_nT_nmin T_n², 1 1 +mnMn

≤C,

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then there existsω >0 such that

∀n∈N,∀t≥bn, E(t)≤E(0) exp

−ω n p=0

up

.

Remark 2. Note that condition (1.2) implies condition (3.11). Note also that, in the case of ordinary diﬀerential equations, Pucci and Serrin [25] improved the condition of [27] and proved asymptotic stability under the following condition:

∞ n=0

mnTnmin T_n², 1 1 +^m_Tⁿ

n

Ina

= +∞.

We do not know if this weaker condition is also suﬃcient in the case of the partial diﬀerential equations (2.6).

Proof of Theorem 3.2. For all n ≥ 0, we denote In = (an, bn) and we apply Theorem 3.1 to the time intervalIn instead of (0, T), which implies

E(bn)≤ 1 1 +ckn

E(an), where, for alln≥0,

kn := mn

Tn⁻³+Tn⁻¹+M_nm_nTn⁻¹

>0.

Using that the energy is nonincreasing, we deduce, for alln≥0, E(an+1)≤E(bn)≤ 1

1 +ckn

E(an)

≤ _n

p=0

1 1 +ck_p

E(a0)≤ _n

p=0

1 1 +ck_p

E(0).

Since the energy is nonincreasing, in order to prove Theorem 3.2, it is suﬃcient to prove thatE(a_n+1)→0 asn→ ∞. Thus it is suﬃcient to prove that

+∞ p=0

1 1 +ckp

= 0, or

+∞

p=0

ln 1

1 +ckp

=−∞.

Ifkp →0 asp→ ∞, then the result follows, and ifkp→0 asp→ ∞, then it reduces to prove that+∞

p=0kp= +∞. This condition is equivalent to (3.11) since 1

2mnTnmin T_n², 1 1 +m_nM_n

≤kn= mnTn 1

T_n² + 1 +mnMn

≤mnTnmin T_n², 1 1 +mnMn

,

which ends the proof of Theorem 3.2. Note also that condition (1.2) implies condition (3.11) since

m_nT_nδ_n² ≤m_nT_nmin T_n², 1 1 +mnMn

,

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which proves Remark 2. It remains to prove Remark 1. Usingkp≥up/2, we have E(b_n)≤

n p=0

1

1 +cup/2E(0) = exp

− n p=0

ln (1 +cu_p/2)

E(0).

Since ln (1 +cx/2)≥ln (1 +cC/2)x/C for allx∈(0, C), we obtain E(bn)≤exp

−ln (1 +cC/2) C

n p=0

up

E(0).

3.3. Further comments. The main restrictive assumption of our general setting is that the damping termB(t)uis assumed to beuniformlydistributed in space.

However, this restriction is crucial if we want to consider an on/off damping. Our result does not apply to an on/off damping that is only locallydistributed in space, even if the geometric condition of Bardos, Lebeau, and Rauch [2, 3] is satisfied.

Let us explain why the case of a locally distributed on/oﬀ damping, for example, B(t)u=a(t, x)u, is out of reach, at least under such a general form. Indeed, even for a very simple example, the situation is complicated and the statement of the results depends on a lot of parameters.

Let us consider the one-dimensional wave equation in (0,1):

⎧⎨

⎩

u−u_xx=−b(t)c(x)u, x∈(0,1), t >0,

u(t,0) =u(t,1) = 0, t >0,

u(0,·) =u0∈H₀¹(0,1), u(0,·) =u1∈L²₀(0,1).

(3.12)

Here we considera(t, x) =b(t)c(x) and we can distinguish three cases.

1. The locally distributed (non on/oﬀ ) case. Our result does not apply to this case. Actually, it was not the purpose of the present paper, since this case has been widely studied in the literature. Let us recall some well-known results.

First, we consider the time-independent case:

a(t, x) =c(x), i.e.,b(t)≡1, with

c(x)≥c0>0 for allx∈ω,

whereωis an open subset of (0,1). Ifωis nonempty, then asymptotic stability holds.

More generally, this result is well known in higher dimension spaces, provided thatω satisﬁes the geometric condition of Bardos, Lebeau, and Rauch [2, 3]. On the other hand, this kind of result may be extended to the time-dependent case

a(t, x) =b(t)c(x), provided that

0< σ(t)≤b(t)≤1/σ(t), with the condition

+∞ 0

σ(τ)dτ = +∞.

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See, for example, [20, 22, 26]. Notice that this allows us to consider a time-dependent damping, but not an on/oﬀ damping, since the assumptionb(t)>0 is needed.

2. The uniformly distributed on/oﬀ case. In the present paper, we consider a damping that is uniformly distributed in space, but is allowed to be on/oﬀ in time.

For example, we assume

a(t, x) =b(t)c(x)≥c0b(t),

withc0>0 and whereb(t) = 0 on an inﬁnite union of intervals. In this case, Theorem 3.2 gives a sharp condition of asymptotic stability. (See section 4 for several examples of application of Theorem 3.2.)

3. The locally distributed on/oﬀ case. Now let us turn to the more general case of a locally distributed on/oﬀ damping, and let us see why its study is out of reach, at least in a general setting.

We consider the following “simple” example:

c(x) =χ_ω(x), where ω= (1/2−λ,1/2 +λ), with 0< λ≤1/2, andb:R+→R+ isT-periodic such that

b(t) = 1 on [0, T) and b(t) = 0 on [T,2T).

Notice that, for all 0< λ≤1/2,ωsatisﬁes the geometric condition of Bardos, Lebeau, and Rauch [2, 3]. However, this is not suﬃcient to insure asymptotic stability in this case. Indeed the following result holds.

Theorem 3.3. [21, Theorem 2.3, p. 340].

(i)If

1

T ∈2N and 2λ < T

,

then there exists initial condition (u₀, u₁)∈H₀¹(0,1)×L²(0,1) such that the energy of the solution of(3.12) remains constant with time: E(t) =E(0)>0for all t >0.

(ii)If

1

T ∈2N and 2λ > T

, or 1

T ∈2N ,

then the energy of the solutions of (3.12)decays uniformly exponentially to 0.

This result shows that the situation is much more complicated because both diﬃculties (coming from the fact the damping is locally distributed in space and is on/oﬀin time) are considered. Notice that Theorem 3.3 is proved in [21] without using the notion of optic rays. However, the results can be explained with the following comments related to optic rays propagation.

First consider the case of a uniformly distributed damping, i.e.,λ= 1/2. In this case, Theorem 3.3 insures that asymptotic stability holds for allT >0. Notice that, in this case, it is clear that each optic ray crosses the damping space region during a period when the damping is eﬀective.

Next consider the more interesting case of a locally distributed damping: 0 <

λ < 1/2. Now there are some values of T and some values of λ for which some optic rays cross the space damping region only when the feedback is not active. For example, take T = 1/2,λ < T /2 = 1/4, and consider the optic ray that leaves the

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pointx=T /2 = 1/4 and that goes to the left (toward the pointx= 0) at timet= 0.

This ray describes the segment [1/4,3/4] (that contains the damping region) during the time intervals [T,2T], [3T,4T], . . . ,thus during periods whenb(t) = 0. The same situation occurs as soon as 1/T ∈ 2N with 2λ < T. In these cases, Theorem 3.3 provides negative results of stability, while it provides positive results in the other cases.

We see that all these results are coherent with the optic ray condition known for time-independent feedbacks [2, 3]. But now, the fact that the feedback depends on time has to be taken into account. And it seems to be crucial that each optic ray crosses the damping space region during a period when the damping is eﬀective.

4. Some examples. We ﬁrst consider the case ofordinary diﬀerential equations that has been widely studied (see, for example, [1, 9, 10, 25, 27, 28]).

Example 1 (the oscillator equation). Assume a ∈ L^∞_loc(R+) is a nonnegative function whose minimum and maximum values inI_n are denoted byα_n andA_n.

WithH =V =D(A) = R, Au=u, andB(t) = a(t)Id, Theorem 3.2 applies to the oscillator equation (1.1) (withm_n:=α_nandM_n :=A_n). Since (1.2) implies condition (3.11), this again gives Theorem 1.1 with a small improvement of the suﬃcient condition. (In particular, this gives a new proof of the result of Smith, very diﬀerent from the original proof based on monotonicity properties.)

TakingB(t) =a(t)f, we may also consider the nonlinear oscillator u+u+a(t)f(u) = 0, t >0,

where we assume that f ∈ C¹(R) is such that f(0) = 0 and 0 < β ≤ f ≤ B.

Hence B deﬁned byB(t)v:=a(t)f(v) satisﬁes (3.9) and (3.10) with m_n:=βα_n and M_n :=BA_n.

Next we assume that Ω is a bounded open set ofR^N with regular boundary and we turn to the case ofuniformly damped partial diﬀerential equationswith abounded damping operator.

Example 2 (a wave equation). Leta1, a2 ∈L^∞_loc(R+) be two nonnegative func- tions such thateither

a2(t) = 0 a.e. onR+

or

∃C >0, a₁(t)≤Ca₂(t) for a.e. t∈R+

and

∀n∈N, αn≤a1(t) +a2(t)≤An for a.e. t∈In. Consider alsof ∈C¹(R) such thatf(0) = 0 and 0< β≤f ≤B.

Then we study the following damped wave equation:

⎧⎨

⎩

u−∆u+a₁(t)f(u)−a₂(t)∆u= 0, x∈Ω, t >0,

u= 0, x∈∂Ω, t >0,

u(t= 0)∈H₀¹(Ω), u(t= 0)∈L²(Ω).

(4.1)

We choose H = L²(Ω), A = −∆ (with Dirichlet boundary conditions), D(A) = H²∩H₀¹(Ω), andB(t)v:=a₁(t)f(v)−a₂(t)∆v. The choice ofW depends ona₂: in the casea₂≡0,W =H; otherwiseW =V.