Stability results for abstract evolution equations with intermittent time-delay feedback
aa r X i v : . [ m a t h . A P ] J a n Stability results for abstract evolution equations withintermittent time-delay feedback
Cristina Pignotti
Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversit`a di L’AquilaVia Vetoio, Loc. Coppito, 67010 L’Aquila Italy
Abstract
We consider abstract evolution equations with on–off time delay feedback. Without thetime delay term, the model is described by an exponentially stable semigroup. We show that,under appropriate conditions involving the delay term, the system remains asymptoticallystable. Under additional assumptions exponential stability results are also obtained. Concreteexamples illustrating the abstract results are finally given.
In this paper we study the stability properties of abstract evolution equations in presenceof a time delay term.In particular, we include into the model an on–off time delay feedback, i.e. the time delayis intermittently present.Let H be a Hilbert space, with norm k · k , and let A : H → H be a dissipative operator gen-erating a C − semigroup ( S ( t )) t ≥ exponentially stable, namely there are two positive constants M and µ such that k S ( t ) k L ( H ) ≤ M e − µt , ∀ t ≥ , (1.1)where L ( H ) denotes the space of bounded linear operators from H into itself.We consider the following problem (cid:26) U t ( t ) = A U ( t ) + B ( t ) U ( t − τ ) t > ,U (0) = U , (1.2)where τ , the time delay, is a fixed positive constant, the initial datum U belongs to H and, for t > , B ( t ) is a bounded operator from H to H . In particular, we assume that there exists an increasing sequence of positive real numbers { t n } n , with t = 0 , such that1) B ( t ) = 0 ∀ t ∈ I n = [ t n , t n +1 ) , B ( t ) = B n +1 ∀ t ∈ I n +1 = [ t n +1 , t n +2 ) .
1e denote B n +1 = kB n +1 k L ( H ) , n ∈ IN . Moreover, denoted by T n the length of the interval I n , that is T n = t n +1 − t n , n ∈ IN , (1.3)we assume T n ≥ τ, ∀ n ∈ IN . (1.4)Time delay effects are frequently present in applications and concrete models and it is nowwell–understood that even an arbitrarily small delay in the feedback may destabilize a systemwhich is uniformly stable in absence of delay (see e.g. [7, 8, 22, 30]).We want to show that, under appropriate assumptions involving the delay feedback coef-ficients, the size of the time intervals where the delay appears and the parameters M and µ in (1.1), the considered model is asymptotically stable or exponentially stable, in spite of thepresence of the time delay term.Stability results for second–order evolution equations with intermittent damping were firststudied by Haraux, Martinez and Vancostenoble [14], without any time delay term. They con-sidered a model with intermittent on–off or with positive–negative damping and gave sufficientconditions ensuring that the behavior of the system in the time intervals with the standarddissipative damping, i.e. with positive coefficient, prevails over the bad behavior in remainingintervals where the damping is no present or it is present with the negative sign, namely asanti–damping. Therefore, asymptotic/exponential stability results were obtained.More recently Nicaise and Pignotti [23, 24] considered second–order evolution equationswith intermittent delay feedback. These results have been improved and extended to some semi-linear equations in [9]. In the studied models, when the delay term (which possess a destabiliz-ing effect) is not present, a not–delayed damping acts. Under appropriate sufficient conditions,stability results are then obtained. Related results for wave equations with intermittent delayfeedback have been obtained, in 1-dimension, in [12], [13] and [3] by using a different approachbased on the D’Alembert formula. However, this last approach furnishes stability results onlyfor particular choices of the time delay.In the recent paper [28], the intermittent delay feedback is compensated by a viscoelasticdamping with exponentially decaying kernel.The asymptotic behavior of wave–type equations with infinite memory and time delay feed-back has been studied by Guesmia in [11] via a Lyapunov approach and by Alabau–Boussouira,Nicaise and Pignotti [2] by combining multiplier identities and perturbative arguments.We refer also to Day and Yang [6] for the same kind of problem in the case of finitememory. In these papers the authors prove exponential stability results if the coefficient of thedelay damping is sufficiently small. These stability results could be easily extended to a variablecoefficient b ( · ) ∈ L ∞ (0 , + ∞ ) under a suitable smallness assumption on the L ∞ − norm of b ( · ) . In [28], instead, asymptotic stability results are obtained without smallness conditions re-lated to the L ∞ − norm of the delay coefficient. On the other hand, the analysis is restrictedto intermittent delay feedback. Asymptotic stability is proved when the coefficient of the de-lay feedback belongs to L (0 , + ∞ ) and the length of the time intervals where the delay is notpresent is sufficiently large. The same paper considers also problems with on–off anti–dampinginstead of a time delay feedback. Stability results are obtained even in this case under analogousassumptions.The idea is here to generalize the results of [28] by considering abstract evolution equationsfor which, without considering the intermittent delay term, the associated operator generatesan exponentially stable C − semigroup. 2or such a class of evolution equations we already know that, under a suitable smalnesscondition on the delay feedback coefficient, an exponential stability result holds true (see [25]).We want to show that stability results are avalaible also under a condition on the L − norm ofthe delay coefficient, without restriction on the pointwise L ∞ − norm.The paper is organized as follows. In section 2 we give a well–posedness result. In sections3 and 4 we prove asymptotic and exponential stability results, respectively, for the abstractmodel under appropriate conditions. Stability results are established also for a problem withintermittent anti–damping instead of delay feedback in section 5. Finally, in section 6, we givesome concrete applications of the abtract results. In this section we illustrate a well-posedness results for problem (1.2).
Theorem 2.1
For any initial datum U ∈ H there exists a unique (mild) solution U ∈ C ([0 , ∞ ); H ) of problem (1 . . Moreover, U ( t ) = S ( t ) U + Z t S ( t − s ) B ( s ) U ( s − τ ) ds . (2.1) Proof.
We prove the existence and uniqueness result on the interval [0 , t ]; then the global resultfollows by translation (cfr. [23]). In the time interval [0 , t ] , since B ( t ) = 0 ∀ t ∈ [0 , t ) , thenthere exists a unique solution U ∈ C ([0 , τ ] , H ) satisfying (2 . . The situation is different in thetime interval [ t , t ] where the delay feedback is present. In this case, we decompose the interval[ t , t ] into the successive intervals [ t + jτ, t + ( j + 1) τ ) , for j = 0 , . . . , N − , where N is suchthat t + ( N + 1) τ ≥ t . The last interval is then [ t + N τ, t ] . Now, first we look at the problemon the interval [ t , t + τ ] . Here U ( t − τ ) can be considered as a known function. Indeed, for t ∈ [ t , t + τ ] , then t − τ ∈ [0 , t ] , and we know the solution U on [0 , t ] by the first step. Thus,problem (1 .
2) may be reformulated on [ t , t + τ ] as (cid:26) U t ( t ) = A U ( t ) + g ( t ) t ∈ ( τ, τ ) ,U ( τ ) = U ( τ − ) , (2.2)where g ( t ) = B ( t ) U ( t − τ ) . This problem has a unique solution U ∈ C ([ τ, τ ] , H ) (see e.g. Th.1.2, Ch. 6 of [27]) given by U ( t ) = S ( t − τ ) U ( τ − ) + Z tτ S ( t − s ) g ( s ) ds, ∀ t ∈ [ τ, τ ] . Proceedings analogously in the successive time intervals [ t + jτ, t + ( j + 1) τ ) , we obtain asolution on [0 , t ] . Let T ∗ be defined as 3 ∗ := 1 µ ln M , (3.1)where M and µ are the constants in (1 . , that is T ∗ is the time for which M e − µT ∗ = 1 . We can state a first estimate on the intervals I n where the delay feedback is not present. Proposition 3.1
Assume T n > T ∗ . Then, there exists a constant c n ∈ (0 , such that k U ( t n +1 ) k ≤ c n k U ( t n ) k , (3.2) for every solution of problem (1 . . Proof.
Observe that in the time interval I n = [ t n , t n +1 ] the delay feedback is not presentsince B ( t ) ≡ . Thus, (3.2) easily follows from (1.1) with √ c n = M e − µT n < M e − µT ∗ = 1 . Let us now introduce the Lyapunov functional F ( t ) = F ( U, t ) := 12 k U ( t ) k + 12 Z tt − τ kB ( s + τ ) k L ( H ) k U ( s ) k ds . (3.3) Proposition 3.2
Assume , . Moreover, assume T n ≥ τ, ∀ n ∈ IN . Then, F ′ ( t ) ≤ B n +1 k U ( t ) k , t ∈ I n +1 = [ t n +1 , t n +2 ] , ∀ n ∈ IN . (3.4) for any solution of problem (1 . . Proof.
By differentiating the energy F ( · ) , we have F ′ ( t ) = h U ( t ) , A U ( t ) i + h U ( t ) , B ( t ) U ( t − τ ) i + 12 kB ( t + τ ) k L ( H ) k U ( t ) k − kB ( t ) k L ( H ) k U ( t − τ ) k . Then, since the operator A is dissipative, one can estimate F ′ ( t ) ≤ kB ( t ) k L ( H ) k U ( t ) kk U ( t − τ ) k + 12 kB ( t + τ ) k L ( H ) k U ( t ) k − kB ( t ) k L ( H ) k U ( t − τ ) k . (3.5)Therefore, from Cauchy–Schwarz inequality, F ′ ( s ) ≤ kB ( t ) k L ( H ) k U ( t ) k + 12 kB ( t + τ ) k L ( H ) k U ( t ) k . Now, observe that, since T n ≥ τ, for every n ∈ IN , if t belongs to I n +1 then t + τ belongs to I n +1 or to I n +2 . In the first case kB ( t ) k L ( H ) = B n +1 while, in the second case kB ( t ) k L ( H ) = 0 . Thus (3.4) is proved. 4 heorem 3.3
Assume and T n ≥ τ for all n ∈ IN . Moreover assume T n > T ∗ , for all n ∈ IN , where T ∗ is the time defined in (3 . . Then, if ∞ X n =0 ln (cid:2) e B n +1 T n +1 ( c n + T n +1 B n +1 ) (cid:3) = −∞ , (3.6) the equation (1 . is asymptotically stable, namely any solution U of (1 . satisfies k U ( t ) k → for t → + ∞ . Proof.
Note that from (3.4) we obtain F ′ ( t ) ≤ B n +1 F ( t ) , t ∈ I n +1 = [ t n +1 , t n +2 ) , n ∈ IN . Then, by integrating on the time interval I n +1 ,F ( t n +2 ) ≤ e B n +1 T n +1 F ( t n +1 ) , ∀ n ∈ IN . (3.7)From the definition of the Lyapunov functional F , F ( t n +1 ) = 12 k U ( t n +1 ) k + 12 Z t n +1 t n +1 − τ kB ( s + τ ) k L ( H ) k U ( s ) k ds . (3.8)Note that, for t ∈ [ t n +1 − τ, t n +1 ) , then t + τ ∈ [ t n +1 , t n +1 + τ ) and therefore, since | I n +2 | ≥ τ it results t + τ ∈ I n +1 ∪ I n +2 . Now, if t + τ ∈ I n +2 , then B ( t + τ ) = 0 . Otherwise, if t + τ ∈ I n +1 , then kB ( t + τ ) k = B n +1 . Then, from (3.8) we deduce F ( t n +1 ) = 12 k U ( t n +1 ) k + 12 B n +1 Z min( t n +2 − τ,t n +1 ) t n +1 − τ k U ( s ) k ds , (3.9)since if t n +1 > t n +2 − τ, then B ( t ) = 0 for all t ∈ [ t n +2 , t n +1 + τ ) ⊂ [ t n +2 , t n +3 ) . Then, since k U ( · ) k is decreasing in the intervals I n (the operator A is dissipative and B ( t ) ≡ F ( t n +1 ) ≤ k U ( t n +1 ) k + 12 T n +1 B n +1 k U ( t n +1 − τ ) k ≤ k U ( t n +1 ) k + 12 T n +1 B n +1 k U ( t n ) k . (3.10)Using this last estimate in (3.7), we obtain k U ( t n +2 ) k ≤ F ( t n +2 ) ≤ e B n +1 T n +1 ( c n + T n +1 B n +1 ) k U ( t n ) k , ∀ n ∈ IN , (3.11)where we have used also the estimate (3.2). By iterating this argument we arrive at k U ( t n +2 ) k ≤ Π nk =0 e B k +1 T k +1 ( c k + T k +1 B k +1 ) k U k , ∀ n ∈ IN . (3.12)Now observe that k U ( t ) k is not decreasing in the whole (0 , + ∞ ) . However, it is decreasingfor t ∈ [ t n , t n +1 ), n ∈ IN , where the destabilizing delay feedback does not act and so k U ( t ) k ≤ k U ( t n ) k , ∀ t ∈ [ t n , t n +1 ) . (3.13)5oreover, from (3.10), for t ∈ [ t n +1 , t n +2 ) we have k U ( t ) k ≤ F ( t ) ≤ e B n +1 T n +1 ( c n + B n +1 T n +1 ) k U ( t n ) k , (3.14)where in the second inequality we have used (3.2).Then, we have asymptotic stability ifΠ nk =0 e B k +1 T k +1 ( c k + T k +1 B k +1 ) −→ , for n → ∞ , or equivalently ln h Π nk =0 e B k +1 T k +1 ( c k + T k +1 B k +1 ) i −→ −∞ , for n → ∞ , namely under the assumption (3.6). This concludes the proof. Remark 3.4
In particular, (3.6) is verified if the following conditions are satisfied: ∞ X n =0 B n +1 T n +1 < + ∞ and ∞ X n =0 ln c n = −∞ . (3.15)Indeed, it is easy to see that (3.15) is equivalent to ∞ X n =0 B n +1 T n +1 < + ∞ and ∞ X n =0 ln( c n + B n +1 T n +1 ) = −∞ (3.16)and that (3.16) implies (3.6).Therefore, from (3 . , we have stability if kB ( t ) k ∈ L (0 , + ∞ ) and, for instance, the lengthof the good intervals I n is greater than a fixed time ¯ T , ¯ T > T ∗ and ¯ T ≥ τ, namely T n ≥ ¯ T , ∀ n ∈ IN . Indeed, in this case there exists ¯ c ∈ (0 ,
1) such that 0 < c n < ¯ c . If we assume that the length of the delay intervals, namely the time intervals where thedelay feedback is present, is lower than the time delay τ, that is T n +1 ≤ τ, ∀ n ∈ IN . (3.17)we can prove another asymptotic stability result which is, in some sense, complementary to theprevious one.In this case we can directly work with k U ( t ) k instead of passing trough the function F ( · ) . We can give the following preliminary estimates on the time intervals I n +1 , n ∈ IN . Proposition 3.5
Assume . Moreover assume T n +1 ≤ τ and T n ≥ τ , ∀ n ∈ IN . Then,for t ∈ I n +1 , ddt k U ( t ) k ≤ B n +1 k U ( t ) k + B n +1 k U ( t n ) k . (3.18)6 roof: By differentiating k U ( t ) k we get ddt k U ( t ) k = 2 h U ( t ) , A U ( t ) i + 2 h U ( t ) , B ( t ) U ( t − τ ) i . Then, by using the dissipativness of the operator A ,ddt k U ( t ) k ≤ h U ( t ) , B ( t ) U ( t − τ ) i . Hence, from 2) , ddt k U ( t ) k ≤ B n +1 k U ( t ) k + B n +1 k U ( t − τ ) k . We can now conclude observing that since T n +1 ≤ τ and T n ≥ τ , then for t ∈ I n +1 it is t − τ ∈ I n . Then, since k U ( t ) k is decreasing in I n , the estimate in the statement is proved.The stability result follows. Theorem 3.6
Assume , T n +1 ≤ τ and T n ≥ τ , ∀ n ∈ IN . Moreover assume T n > T ∗ , for all n ∈ IN , where T ∗ is the time defined in (3 . . If ∞ X n =0 ln (cid:2) e B n +1 T n +1 ( c n + 1 − e − B n +1 T n +1 ) (cid:3) = −∞ , (3.19) then every solution U of (1 . satisfies k U ( t ) k → for t → + ∞ . Proof.
For t ∈ I n +1 = [ t n +1 , t n +2 ) , from estimate (3.18) we have k U ( t ) k ≤ e B n +1 ( t − t n +1 ) n k U ( t n +1 ) k + B n +1 Z tt n +1 k U ( t n ) k e − B n +1 ( s − t n +1 ) ds o . Then we deduce k U ( t ) k ≤ e B n +1 T n +1 k U ( t n +1 ) k + e B n +1 ( t − t n +1 ) k U ( t n ) k h − e − B n +1 ( t − t n +1 ) i , and therefore k U ( t ) k ≤ e B n +1 T n +1 k U ( t n +1 ) k + e B n +1 T n +1 k U ( t n ) k − k U ( t n ) k , for t ∈ I n +1 = [ t n +1 , t n +2 ) , n ∈ IN . Now we use the estimate (3.2) obtaining k U ( t n +2 ) k ≤ e B n +1 T n +1 (cid:0) c n + 1 − e − B n +1 T n +1 (cid:1) k U ( t n ) k , n ∈ IN . Thus, k U ( t n +2 ) k ≤ h Π nk =0 e B k +1 T k +1 ( c k + 1 − e − B k +1 T k +1 ) i k U k . (3.20)Then the asymptotic stability result follows ifΠ nk =0 e B k +1 T k +1 (cid:0) c k + 1 − e − B k +1 T k +1 (cid:1) → , for n → ∞ , namely if ∞ X n =0 ln h e B n +1 T n +1 ( c n + 1 − e − B n +1 T n +1 ) i → −∞ , for n → ∞ . emark 3.7 Observe that, when the odd intervals I n +1 have length lower or equal than thetime delay τ , the assumption (3.19) is a bit less restrictive than (3.6) . Indeed, e B n +1 T n +1 ( c n + 1 − e − B n +1 T n +1 ) < e b n +1 T n +1 ( c n + B n +1 T n +1 ) , ∀ n ∈ IN . Remark 3.8
Arguing as in Remark 3.4 we can show that condition (3.19) is verified, in par-ticular, if (3.15) holds true.
Under additional assumptions on the coefficients T n , B n +1 , c n , exponential stability resultshold true. Theorem 4.1
Assume . Moreover, assume T n = T ∀ n ∈ IN , (4.1) with T ≥ τ and T > T ∗ , where T ∗ is the constant defined in (3 . ,T n +1 = ˜ T ∀ n ∈ IN (4.2) and sup n ∈ IN e B n +1 ˜ T ( c + B n +1 ˜ T ) = d < , (4.3) where c = M e − µT . Then, there exist two positive constants
C, α such that k U ( t ) k ≤ Ce − αt k U k , t > , (4.4) for any solution of problem (1 . . Proof.
Note that, from the definition of the constant c, estimate (3.2) holds with c n = c, ∀ n ∈ IN . Thus, from (4.3) and (3.11) we obtain k U ( T + ˜ T ) k ≤ d k U k , and then, k U ( n ( T + ˜ T )) k ≤ d n k U k , ∀ n ∈ IN . Therefore, k U ( t ) k satisfies an exponential estimate like (4.4) (see Lemma 1 of [12]).Concerning the case of small delay intervals, namely | I n +1 | ≤ τ, ∀ n ∈ IN , one can statethe following asymptotic stability result. Theorem 4.2
Assume . Moreover assume T n = T ∀ n ∈ IN , with T ≥ τ and T > T ∗ , where the time T ∗ is defined in (3 . ,T n +1 = ˜ T , with ˜ T ≤ τ ∀ n ∈ IN (4.5)8 nd sup n ∈ IN e B n +1 ˜ T ( c + 1 − e − B n +1 ˜ T ) = d < , (4.6) where c = M e − µT . Then, there exist two positive constants
C, α such that k U ( t ) k ≤ Ce − αt k U k , t > , (4.7) for any solution of (1 . . Proof.
The proof is analogous to the one of Theorem 4.1 .
With analogous technics we can also deal with an intermittent anti–damping term. Moreprecisely, let us consider the model (cid:26) U t ( t ) = A U ( t ) + B ( t ) U ( t ) t > ,U (0) = U , (5.1)where τ is the time delay, the initial datum U belongs to H and, for t > , B ( t ) is a boundedoperator from H such that hB ( t ) U, U i ≥ , ∀ U ∈ H . Thus B ( t ) U ( t ) is an anti–damping term (cfr. [14]). In particular we consider an intermittentfeedback, that is we assume that there exists an increasing sequence of positive real numbers { t n } n , with t = 0 , such that3) B ( t ) = 0 ∀ t ∈ I n = [ t n , t n +1 ) , B ( t ) = D n +1 ∀ t ∈ I n +1 = [ t n +1 , t n +2 ) . We denote D n +1 = kD n +1 k L ( H ) , n ∈ IN . As before, denote by T n the length of the interval I n , that is T n = t n +1 − t n , n ∈ IN . Note that Proposition 3.1, which gives an observability estimate on the intervals I n wherethe anti-damping is not present, still holds true. Concerning the time intervals I n +1 where theanti-damping acts one can obtain the following estimate. Proposition 5.1
Assume and . For every solution of problem (5 . ,ddt k U ( t ) k ≤ D n +1 k U ( t ) k , t ∈ I n +1 = [ t n +1 , t n +2 ] , ∀ n ∈ IN . Proof.
Being A dissipative, the estimate follows immediately from 3).From Proposition 5.1 we deduce an asymptotic stability result.9 heorem 5.2 Assume . Moreover assume T n > T ∗ , for all n ∈ IN , where T ∗ is the timedefined in (3 . . If ∞ X n =0 ln (cid:0) e D n +1 T n +1 c n (cid:1) = −∞ , (5.2) then the problem (5 . is asymptotically stable, that is any solution U of (5 . satisfies k U ( t ) k → for t → + ∞ . Proof.
From Proposition 5.1 we have the estimate ddt k U ( t ) k ≤ D n +1 k U ( t ) k , t ∈ I n +1 = [ t n +1 , t n +2 ] , ∀ n ∈ IN . This implies k U ( t n +2 ) k ≤ e D n +1 T n +1 k U ( t n +1 ) k , ∀ n ∈ IN . (5.3)Then, from estimate (3.2) which is always valid of course in the time intervals without damping, k U ( t n +2 ) k ≤ e D n +1 T n +1 c n k U ( t n ) k , ∀ n ∈ IN . (5.4)By repeating this argument we obtain k U ( t n +2 ) k ≤ Π nk =0 e D k +1 T k +1 c k k U k , ∀ n ∈ IN . (5.5)Therefore, asymptotic stability is ensured ifΠ nk =0 e D k +1 T k +1 c k −→ , for n → ∞ , or equivalently ln (cid:16) Π nk =0 e D k +1 T k +1 c k (cid:17) −→ −∞ , for n → ∞ . This concludes.
Remark 5.3
In particular (5.2) is verified under the following assumptions: ∞ X n =0 D n +1 T n +1 < + ∞ and ∞ X n =0 ln c n = −∞ . (5.6)Under additional assumptions on the problem coefficients T n , D n +1 , c n , an exponentialstability result holds. Theorem 5.4
Assume and T n = T ∀ n ∈ IN , (5.7) with T > T ∗ , where the time T ∗ is defined in (3 . . Assume also that T n +1 = ˜ T ∀ n ∈ IN (5.8) and sup n ∈ IN e D n +1 ˜ T c = d < , (5.9) where, c = M e − µT . Then, there exist two positive constants
C, α such that k U ( t ) k ≤ Ce − αt k U k , t > , (5.10) for any solution of problem (5 . . Concrete examples
In this section we illustrate some eaxamples falling into the previous abstract setting.
Let H be a real Hilbert space and let A : D ( A ) → H be a positive self–adjoint operatorwith a compact inverse in H. Denote by V := D ( A ) the domain of A . Let us consider the problem u tt ( x, t ) + Au ( x, t ) − Z ∞ µ ( s ) Au ( x, t − s ) ds + b ( t ) u t ( x, t − τ ) = 0 t > , (6.1) u ( x, t ) = 0 on ∂ Ω × (0 , + ∞ ) , (6.2) u ( x, t ) = u ( x, t ) in Ω × ( −∞ , u belongs to a suitable space, the constant τ > µ : [0 , + ∞ ) → [0 , + ∞ ) satisfiesi) µ ∈ C (IR + ) ∩ L (IR + );ii) µ (0) = µ > R + ∞ µ ( t ) dt = ˜ µ < µ ′ ( t ) ≤ − δµ ( t ) , for some δ > . Moreover, the function b ( · ) ∈ L ∞ loc (0 , + ∞ ) is a function which is zero intermittently. That is, weassume that for all n ∈ IN there exists t n > , with t = 0 and t n < t n +1 , such that1 w ) b ( t ) = 0 ∀ t ∈ I n = [ t n , t n +1 ) , w ) | b ( t ) | ≤ b n +1 = 0 ∀ t ∈ I n +1 = [ t n +1 , t n +2 ) . Stability result for the above problem were firstly obtained in [28]. We want to show thatthey can also be obtained as particular case of previous abstract setting.To this aim, following Dafermos [5], we can introduce the new variable η t ( x, s ) := u ( x, t ) − u ( x, t − s ) . (6.4)Then, problem (6.1)–(6.3) may be rewritten as u tt ( x, t ) = − (1 − ˜ µ ) Au ( x, t ) − Z ∞ µ ( s ) Aη t ( x, s ) ds − b ( t ) u t ( x, t − τ ) in Ω × (0 , + ∞ ) , (6.5) η tt ( x, s ) = − η ts ( x, s ) + u t ( x, t ) in Ω × (0 , + ∞ ) × (0 , + ∞ ) , (6.6) u ( x, t ) = 0 on ∂ Ω × (0 , + ∞ ) , (6.7) η t ( x, s ) = 0 in ∂ Ω × (0 , + ∞ ) , t ≥ , (6.8) u ( x,
0) = u ( x ) and u t ( x,
0) = u ( x ) in Ω , (6.9) η ( x, s ) = η ( x, s ) in Ω × (0 , + ∞ ) , (6.10)11here u ( x ) = u ( x, , x ∈ Ω ,u ( x ) = ∂u ∂t ( x, t ) | t =0 , x ∈ Ω ,η ( x, s ) = u ( x, − u ( x, − s ) , x ∈ Ω , s ∈ (0 , + ∞ ) . (6.11)Set L µ ((0 , ∞ ); V ) the Hilbert space of V − valued functions on (0 , + ∞ ) , endowed with the innerproduct h ϕ, ψ i L µ ((0 , ∞ ); V ) = Z ∞ µ ( s ) h A / ϕ ( s ) , A / ψ ( s ) i H ds . Let H be the Hilbert space H = V × H × L µ ((0 , ∞ ); V ) , equipped with the inner product * uvw , ˜ u ˜ v ˜ w + H := (1 − ˜ µ ) h A / u, A / ˜ u i H + h v, ˜ v i H + Z ∞ µ ( s ) h A / w, A / ˜ w i H ds . (6.12)Denoting by U the vector U = ( u, u t , η ) , the above problem can be rewritten in the form (1.2),where B U = B ( u, v, η ) = (0 , bv,
0) and A is defined by A uvw := v (1 − ˜ µ ) Au + R ∞ µ ( s ) Aw ( s ) ds − w s + v , (6.13)with domain (cfr. [26]) D ( A ) := (cid:8) ( u, v, η ) T ∈ H (Ω) × H (Ω) × L µ ((0 , + ∞ ); H (Ω)) :(1 − ˜ µ ) u + R ∞ µ ( s ) η ( s ) ds ∈ H (Ω) ∩ H (Ω) , η s ∈ L µ ((0 , + ∞ ); H (Ω)) (cid:9) . (6.14)It has been proved in [10] that the above system is exponentially stable, namely that the operator A generates a strongly continuos semigroup satisfying the estimate (1 . , for suitable constants.Moreover, it is well-known that, the operator A is dissipative. Therefore, our previous resultsapply to this model.As a concrete example we can consider the wave equation with memory. More precisely, letΩ ⊂ IR n be an open bounded domain with a smooth boundary ∂ Ω . Let us consider the initialboundary value problem u tt ( x, t ) − ∆ u ( x, t ) + Z ∞ µ ( s )∆ u ( x, t − s ) ds + b ( t ) u t ( x, t − τ ) = 0 in Ω × (0 , + ∞ ) , (6.15) u ( x, t ) = 0 on ∂ Ω × (0 , + ∞ ) , (6.16) u ( x, t ) = u ( x, t ) in Ω × ( −∞ , . (6.17)This problem enters in previous form (6 . − (6 . , if we take H = L (Ω) and the operator A defined by A : D ( A ) → H : u → − ∆ u, D ( A ) = H (Ω) ∩ H (Ω) . The operator A is a self–adjoint and positive operator with a compact inverse in H and issuch that V = D ( A / ) = H (Ω) . Under the same conditions that before on the memory kernel µ ( · ) and on the function b ( · ) , previous asymptotic/exponential stability results are valid. The case b constant has been studiedin [2]. In particular, we have proved that the exponential stability is preserved, in presence ofthe delay feedback, if the coefficient b of this one is sufficiently small. The choice b constantwas made only for the sake of clearness. The result in [2] remains true if instead of b constantwe consider b = b ( t ) , under a suitable smallness condition on the L ∞ − norm of b ( · ) . On thecontrary here we give stability results without restrictions on the L ∞ − norm of b ( · ) , even if onlyfor on–off b ( · ) . Our results also apply to Petrovsky system with viscoelastic damping with Dirichlet andNeumann boundary conditions: u tt ( x, t ) + ∆ u ( x, t ) − Z ∞ µ ( s )∆ u ( x, t − s ) ds + b ( t ) u t ( x, t − τ ) = 0 in Ω × (0 , + ∞ ) , (6.18) u ( x, t ) = ∂u∂ν = 0 on ∂ Ω × (0 , + ∞ ) , (6.19) u ( x, t ) = u ( x, t ) in Ω × ( −∞ , . (6.20)This problem enters into the previous abstract framework, if we take H = L (Ω) and theoperator A defined by A : D ( A ) → H : u → ∆ u, where D ( A ) = H (Ω) ∩ H (Ω) , with H (Ω) = n v ∈ H (Ω) : v = ∂v∂ν = 0 on ∂ Ω o . The operator A is a self–adjoint and positive operator with a compact inverse in H and is suchthat V = D ( A / ) = H (Ω) . Therefore, under the same conditions that before on the memory kernel µ ( · ) and on thefunction b ( · ) , previous asymptotic/exponential stability results are valid. Here we consider the wave equation with local internal damping and intermittent delayfeedbck. More precisely, let Ω ⊂ IR n be an open bounded domain with a boundary ∂ Ω of class C . Denoting by m the standard multiplier m ( x ) = x − x , x ∈ IR n , let ω be the intersectionof Ω with an open neighborhood of the subset of ∂ ΩΓ = { x ∈ ∂ Ω : m ( x ) · ν ( x ) > } . (6.21)Fixed any subset ω ⊆ Ω , let us consider the initial boundary value problem13 tt ( x, t ) − ∆ u ( x, t ) + aχ ω u t ( x, t ) + b ( t ) χ ω u t ( x, t − τ ) = 0 in Ω × (0 , + ∞ ) , (6.22) u ( x, t ) = 0 on ∂ Ω × (0 , + ∞ ) , (6.23) u ( x,
0) = u ( x ) and u t ( x,
0) = u ( x ) in Ω , (6.24)where χ ω i denotes the characteristic function of ω i , i = 1 , , a is a positive number and b in L ∞ (0 , + ∞ ) is an on–off function satysfying (1 w ) and (2 w ) of subsection 6.1. The initia datum( u , u ) belongs to H (Ω) × L (Ω) . This problem enters into our previous framework, if we take H = L (Ω) and the operator A defined by A : D ( A ) → H : u → − ∆ u, where D ( A ) = H (Ω) ∩ H (Ω) . Now, denoting U = ( u, u t ) , the problem can be restated in the abstract form (1.2) where B U = B ( u, v ) = (0 , b ( t ) χ ω v ) and A is defined by A (cid:18) uv (cid:19) := (cid:18) v − Au − aχ ω v (cid:19) , (6.25)with domain D ( A ) × L (Ω) in the Hilbert space H = H × H. Concerning the the part without delay feedback, namely the locally damped wave equation u tt ( x, t ) − ∆ u ( x, t ) + aχ ω u t ( x, t ) = 0 in Ω × (0 , + ∞ ) , (6.26) u ( x, t ) = 0 on ∂ Ω × (0 , + ∞ ) , (6.27) u ( x,
0) = u ( x ) and u t ( x,
0) = u ( x ) in Ω , (6.28)it is well–known that, under the previous Lions geometric condition on the set ω (or underthe more general assumption of control geometric property [4]) where the frictional damping islocalized, an exponential stability result holds (see e.g. [4, 16, 17, 18, 19, 20, 21, 31]). This isequivalent to say that the strongly continuous semigroup generated by the operator A associatedto (6 . − (6 . , namely the one defined in (6.25), satisfies (1.1). As well–known, the operator A is dissipative. Thus previous abstract stability results are valid also for this model. We emphasizethe fact that the set ω may be any subset of Ω , not necessarily a subset of ω . On the contrary,in previous stability results for damped wave equation and intermittent delay feedback (see e.g.[24, 9]) the set ω has to be a subset of ω . On the other hand, now the standard (not delayed)frictional damping is always present in time while in the quoted papers it is on–off like the delayfeedback and it acts only on the complementary time intervals with respect to this one.
References [1] F. Alabau-Boussouira, P. Cannarsa and D. Sforza. Decay estimates for second order evo-lution equations with memory.
J. Funct. Anal. , 254:1342–1372, 2008.[2] F. Alabau-Boussouira, S. Nicaise and C. Pignotti. Exponential stability of the wave equationwith memory and time delay.
New Prospects in Direct, Inverse and Control Problems forEvolution Equations, Springer Indam Series , 10:1–22, 2014.143] K. Ammari, S. Nicaise and C. Pignotti. Stabilization by switching time–delay.
Asymptot.Anal. , 83:263–283, 2013.[4] C. Bardos, G. Lebeau and J. Rauch. Sharp sufficient conditions for the observation, controland stabilization of waves from the boundary.
SIAM J. Control Optim. , 30:1024–1065,1992.[5] C.M. Dafermos. Asymptotic stability in viscoelasticity.
Arch. Rational Mech. Anal. , 37:297–308, 1970.[6] Q. Dai and Z. Yang. Global existence and exponential deacay of the solution for a vis-coelastic wave equation with a delay.
Z. Angew. Math. Phys. , 65:885–903, 2014.[7] R. Datko. Not all feedback stabilized hyperbolic systems are robust with respect to smalltime delays in their feedbacks.
SIAM J. Control Optim. , 26:697–713, 1988.[8] R. Datko, J. Lagnese and M. P. Polis. An example on the effect of time delays in boundaryfeedback stabilization of wave equations.
SIAM J. Control Optim. , 24:152–156, 1986.[9] G. Fragnelli and C. Pignotti, Stability of solutions to nonlinear wave equations with switch-ing time-delay.
Dyn. Partial Differ. Equ. , 13:31–51, 2016.[10] C. Giorgi, J.E. Mu˜noz Rivera and V. Pata. Global attractors for a semilinear hyperbolicequation in viscoelasticity.
J. Math. Anal. Appl. , 260:83–99, 2001.[11] A. Guesmia. Well–posedness and exponential stability of an abstract evolution equationwith infinite memory and time delay.
IMA J. Math. Control Inform. , 30:507–526, 2013.[12] M. Gugat. Boundary feedback stabilization by time delay for one-dimensional wave equa-tions.
IMA J. Math. Control Inform. , 27:189–203, 2010.[13] M. Gugat and M. Tucsnak. An example for the switching delay feedback stabilization of aninfinite dimensional system: the boundary stabilization of a string.
Systems Control Lett. ,60:226–233, 2011.[14] A. Haraux, P. Martinez and J. Vancostenoble. Asymptotic stability for intermittentlycontrolled second–order evolution equations.
SIAM J. Control Optim. , 43:2089–2108, 2005.[15] M. Kirane and B. Said-Houari. Existence and asymptotic stability of a viscoelastic waveequation with a delay.
Z. Angew. Math. Phys. , 62:10651082, 2011.[16] V. Komornik.
Exact controllability and stabilization, the multiplier method , volume 36 of
RMA . Masson, Paris, 1994.[17] V. Komornik and P. Loreti.
Fourier series in control theory , Springer Monographs inMathematics. Springer-Verlag, New York, 2005.[18] J. Lagnese. Control of wave processes with distributed control supported on a subregion.
SIAM J. Control Optim. , 21:68–85, 1983.[19] I. Lasiecka and R. Triggiani. Uniform exponential decay in a bounded region with L (0 , T ; L (Σ))-feedback control in the Dirichlet boundary conditions. J. Differential Equa-tions , 66:340–390, 1987. 1520] J. L. Lions.
Contrˆolabilit´e exacte, perturbations et stabilisation des syst`emes distribu´es,Tome 1, volume 8 of
RMA . Masson, Paris, 1988.[21] K. Liu. Locally distributed control and damping for the conservative systems.
SIAM J.Control and Optim. , 35:1574–1590, 1997.[22] S. Nicaise and C. Pignotti. Stability and instability results of the wave equation with adelay term in the boundary or internal feedbacks.
SIAM J. Control Optim. , 45:1561–1585,2006.[23] S. Nicaise and C. Pignotti. Asymptotic stability of second–order evolution equations withintermittent delay.
Adv. Differential Equations , 17:879–902, 2012.[24] S. Nicaise and C. Pignotti. Stability results for second–order evolution equations withswitching time-delay.
J. Dynam. Differential Equations , 26:781–803, 2014.[25] S. Nicaise and C. Pignotti. Exponential stability of abstract evolution equations with timedelay.
Journal of Evolution Equations , 15:107–129, 2015.[26] V. Pata. Stability and exponential stability in linear viscoelasticity.
Milan J. Math. , 77:333–360, 2009.[27] A. Pazy.
Semigroups of linear operators and applications to partial differential equations ,Vol. 44 of
Applied Math. Sciences.
Springer-Verlag, New York, 1983.[28] C. Pignotti. Stability results for second-order evolution equations with memory and switch-ing timedelay.
J. Dynam. Differential Equations , doi:10.1007/s10884-016-9545-3, in press,2016.[29] J. Rauch and M. Taylor. Exponential decay of solutions to hyperbolic equations in boundeddomains.
Indiana Univ. Math. J. , 24:79–86, 1974.[30] G. Q. Xu, S. P. Yung and L. K. Li. Stabilization of wave systems with input delay in theboundary control.
ESAIM: Control Optim. Calc. Var. , 12:770–785, 2006.[31] E. Zuazua. Exponential decay for the semi-linear wave equation with locally distributeddamping.
Comm. Partial Differential Equations , 15:205–235, 1990.
E-mail address,
Cristina Pignotti: [email protected]@univaq.it