On conditions for asymptotic stability of dissipative infinite-dimensional systems with intermittent damping

ON CONDITIONS F OR ASYMPTOTIC ST ABILITY OF DISSIP A TIVE INFINITE-DIMENSIONAL SYSTEMS WITH INTERMITTENT D AMPING F ALK M. HANTE, MARIO SIGALOTTI AND MARIUS TUCSNA K Abstract. W e study the asymptotic stability of a dissipative evolution in a Hilb ert space sub ject to i n termitten t damping. W e observ e that, ev en i f the int ermittence satisfies a pers i sten t excitation condition, i f the Hilb ert space is i nfinite-dimensional then the system needs not being asymptotically sta- ble (not even in the weak sense). Exponential stabilit y is reco v ered under a generalized observ abilit y inequality , all o wing for time-domains that are not int erv als. W eak asymptotic stability is obtained under a similarly generalized unique con tin uation pr inciple. Finally , strong asymptotic stability is prov ed for intermittenc es that do not necessarily satisfy some pers i sten t excitation condition, ev aluating their total cont ribution to the deca y of the tr a jectories of the damped system. Our results are discussed using the example of the w a v e equation, Sc hr¨ odinger’s equat ion and, for s trong sta bility , also the s pecial case of finite-dimensional systems. Keyw ords: Int ermittent damping; asymptotic b e havior; persistent excitation; maximal dissipative op era tor. 1. Introduction Consider a sys tem of the form ˙ z = Az + B u with z in some (finite- or infinite- dimensional) Hilb ert space H , B b ounded, and a ssume that there exists a stabilizing feedback law u = u ∗ = K z . Now consider the system ˙ z = Az + α ( t ) B u, (1) where the signal α ta kes v alues in [0 , 1] and α ( t ) = 0 for certain times t (i.e., the control may b e switched off ov er po ssibly non-negligible subsets of time). Under which c o nditions imp osed on α is the clos e d- lo op system (1) with the same control u ∗ asymptotically sta ble? It must b e stressed that a complete knowledge of α (and, in particula r, the precise informa tion on the set of times wher e it v anishe s ) would be a to o r estrictive condition to imp ose on α . W e rather loo k for conditions v alid for a whole c la ss G of functions α and, therefore, we exp ect the closed-lo op systems (1) with u ∗ to b e asymptotica lly stable for every α ∈ G (and, p o s sibly , uniformly with r esp ect to a ll such α ). If α ta kes the v alues 0 and 1 o nly , then the sy stem (1) actually switc hes betw een the uncontrolled sys tem ˙ z = Az and the controlled one ˙ z = Az + B u . If the uncon trolled dynamics are unstable then we should impose on α conditions guaranteeing a sufficient amount of action on the system. Actually , even if they F. M. Hante is with Mathematics Center of Heidelb erg (MA TCH), Int erdisciplinary Cen ter for Scientific Computing (IWR), Im Neuenheimer F eld 368, 69120 Heidelberg, Germany . E-M ail: falk.han te@iwr.un i- heidelberg.de . M. S igalotti is with INRIA Saclay– ˆ Ile-de-F rance, T eam GECO, and CMAP , U M R 7641, ´ Ecole Polytec hnique, R oute de Saclay , 91128 Palaiseau Cedex, F rance. E-Mail: mario.si galotti@i nria.fr . M. T ucsnak is with Institut ´ Elie Cartan (IECN), UM R 7502, BP 239, V an dœuvre-l` es-N ancy 54506, F rance and C O RID A, INRIA Nancy–Grand Es t. E-Mail: tucsnak@ie cn.u- nancy.fr . 1 2 F ALK M . HANTE, MARIO SIGALOTTI AND M ARIUS TUCSNAK are as ymptotically stable, the stability of the ov erall system is not gua ranteed in general (see [12]). The question issued above ma y b e motiv ated by some failur e in the trans mis sion from the co nt roller to the plant, leading to instan ts of time at which the control is s witch ed o ff, o r to some time-v ary ing phenomenon affecting the efficiency of the co ntrol ac tio n. It is also related to problems stemming fro m identification and adaptive control (see, e.g ., [2]). In such type of pro blems, one is lead to consider the stability of linear systems of the kind ˙ z = − P ( t ) z , z ∈ R N , where the ma trix P ( · ) is symmetric non-negative definite. Under which conditions on P is the non- autonomous s ystem stable? An answer for this par ticular case ca n b e found in the seminal pap er [2 1] w hich asserts that, if P ≥ 0 is bo unded and has bo unded deriv ativ e, it is ne c essary and sufficient , for the global exp onent ial stability of ˙ z = − P ( t ) z , that P is also p ersistently exciting , i.e., that there exist µ, T > 0 such that Z t + T t ξ T P ( s ) ξ ds ≥ µ, for all unitary v ectors ξ ∈ R N and all t ≥ 0 . The notion of pers istent excitation, ther efore, appea rs na turally as a reasona ble additional assumption o n α while s tudying the stabiliza tion of (1). The pap ers [7, 8], whose r esults are detailed be low, study the case of finite-dimensional systems of the form (1) under the assumption that ther e exist tw o po sitive constants µ, T such that, for every t ≥ 0, Z t + T t α ( s ) ds ≥ µ. (2) Given tw o pos itive real n um ber s µ ≤ T , we say that α is a T - µ PE -signal (sta nding for p ersistently exciting signal ) if it satisfies (2). Note that we do no t c o nsider here any extra a ssumption o n the reg ularity o f the PE - signal α (e.g., having a b ounded deriv ativ e or b e ing piecewise constant). In [7] it is prov ed that if A is neutrally stable (and ( A, B ) is stabilizable) then u ∗ = − B T x stabilizes (1 ) ex po nent ially , uniformly with resp ect to the clas s of T - µ PE-sig nals (see also [2]). The r esults in [7 ] cov er also the fir st non trivial case where A is not sta ble, namely the double in tegrator ˙ z = J 2 z + αb 0 u , wher e J 2 denotes the 2 × 2 Jordan blo ck corresp onding to the eig env alue zero, the co n trol is scala r and b 0 = (0 , 1) T . It is s hown that, for every pair ( T , µ ), there exists a feedback u ∗ = K x suc h that the corresp o nding closed-lo o p system is ex po nent ially stable, uniformly with resp ect to the class of T - µ PE -signals. In [8 ] this last result is extended b y proving that for the single- input ca se ˙ z = Az + α ( t ) bu , u ∈ R , z ∈ R N , there exists a stabilizer unifor m feedback u ∗ = K x for the cla s s of T - µ PE-s ig nals whenever ( A, b ) is controllable and the eigenv alues of A hav e non-p ositive real part. It is shown, moreover, that there exist controllable pairs ( A, b ) for which no s uch stabilizing feedback exists. The scop e of the present pap er is to extend the analysis desc rib ed ab ov e to infinite-dimensional systems. W e fo cus on the case where A g enerates a strong ly contin uous contraction s emigroup and K = − B ∗ , where B ∗ denotes the adjoint of B . This situation co rresp onds to the neutrally stable case studied, in the finite- dimensional setting, in [7 ]. Recall that the linear feedback control ter m B u = − B B ∗ z is a common choice to stabilize a dissipative linear system (see [29 ] and also [13]). 3 The motiv ating exa mple, illustrating the new pheno mena asso cia ted with the new s etting, is the o ne of a string , fixed a t b oth ends, and da mped—when α ( t ) = 1— on a pro p e r sub domain. It is not har d to constr uct (see Exa mple 2.1 for details) an example of p er io dic trav eling w av e on which the damping induced by a certain per io dic nonz e r o s ig nal α (hence, satisfying a p ersistent excitation condition) is ineffective. There fore, the counterpart o f the finite-dimensional stabilizability re s ult do es not hold a nd additiona l assumptions hav e to be made in o rder to gua r antee the stability of the closed- lo op sy stem. The fir st t ype o f results in this dir ection (Section 3) concerns exp onential stabil- it y . W e pr ov e that, if there exist ϑ, c > 0 suc h that Z ϑ 0 α ( t ) k B ∗ e tA z 0 k 2 H dt ≥ c k z 0 k 2 H , for all T - µ PE - signal α ( · ) , (3) then there exist M ≥ 1 and γ > 0 such that the solution z ( t ) of ˙ z = Az − α ( t ) B B ∗ z , (4) satisfies k z ( t ) k H ≤ M e − γ t k z 0 k H uniformly with resp ect to z 0 and α . (See Theore m 3.2.) The co unt erpart of (3) in the unswitched ca s e (i.e., when α ≡ 1) is an observ abilit y inequa lit y for the pair ( A, B ∗ ). Condition (3) ca n a c tua lly be seen as a generalized o bserv abilit y inequality . The pro o f o f Theo rem 3.2 is ba s ed on deducing from (3) a uniform decay for the solutions of (4) o f the squar ed norm, chosen as Lyapuno v function, on time-interv als of length T . The conclusion follows fr om standa rd considera tions on the scala r-v alued Lyapunov function (see, for ins ta nce, [1]). As an applica tio n of the general stability result w e consider the example of the wa ve equation on a N -dimensional do ma in, da mped ev erywhere. It should be stressed that genera lized observ abilit y inequa lities o f the t yp e discussed her e ha ve alr eady b een co ns idered in the literature for the heat eq ua tion with b oundary or lo cally distributed cont rol ([9, 20, 24, 34]). The second t ype of results presented in this pap er (Section 4 ) deals with weak stability . W e prov e that, if there exists ϑ > 0 such that Z ϑ 0 α ( s ) k B ∗ e sA z 0 k 2 H ds 6 = 0 for all z 0 6 = 0 and all T - µ sig nal α, then the solution t 7→ z ( t ) of system (4) converges weakly to 0 in H as t → ∞ for any initial da ta z 0 ∈ H and any T - µ PE-signa l α . (See Theorem 4 .2.) The counterpart of such condition in the unswitched ca s e is the unique contin uation prop erty , ensuring appr oximate controllabilit y (see, e.g., [33]). The pro of of Theo- rem 4.2 is based on a co mpactness a rgument. The theo rem is a pplied to the cas e of a Schr¨ odinger equation with in ternal control lo calized on a subdo ma in. The gener - alized unique contin uation prop er ty is then recovered by a n analyticity argument (Priv alo v’s theore m) a nd s tandard unique contin uation (Holmgr en’s theo r em). Finally , a third type o f results (Section 5) conce rns s tr ong (but no t nec e ssarily exp onential) stability . In the spirit of [14], instea d o f imp os ing conditions on α which a re satisfied on every time-window of prescr ib e d length, we admit the “exci- tations” to be r arefied in time and o f v ariable duration. Stability is guar anteed b y asking that the tota l con tribution of the excitations, suitably summed up, is “large enough”. Mo re precisely , it is pr ov ed that if there exist ρ > 0 and a contin uous function c : (0 , ∞ ) → (0 , ∞ ) such that for all T > 0, (3) holds true with µ = ρT and c = c ( T ), and if ther e exists a sequenc e of disjoint interv als ( a n , b n ) in [0 , ∞ ) with R b n a n α ( t ) dt ≥ ρ ( b n − a n ) and P ∞ n =1 c ( b n − a n ) = ∞ , then the solution z ( · ) of (4) satisfies k z ( t ) k H → 0 a s t → ∞ . A function c ( · ) as a b ov e is ex plic itely found 4 F ALK M . HANTE, MARIO SIGALOTTI AND M ARIUS TUCSNAK in the case of the uniformly da mped wa v e equation (it is o f orde r T 3 for T small) and als o in the finite-dimensio na l case, where it is o f the s ame o r der as the one computed b y Seidman in the unswitched c a se ([2 8 ]). A large literature is devoted to conditio ns ensuring s tability of second o rder systems with time-v arying param- eters, mostly but not ex clusively in the finite-dimensional setting. L e t us mention, for instance, [15, 16, 25, 3 0] a nd the alr eady cited pap er [14]. In tegral conditions in space, instead of in time, guara nteeing stabilizability of sys tems whos e uncon- trolled dy na mics a r e given by a contraction semigr oup hav e also been studied. Let us mention, for instance, [18, 2 2, 31] for the wav e equation and [6] for the plate equation. In analogy with the function c ( · ) in tro duced ab ov e, in the mentioned pap ers the correct weigh t has to b e used in or der to s um up the con tributions of the damping co efficients at different p o in ts. An interesting q uestion, with p ossible applications to bang- bang control, would b e to combine these tw o type of results, i.e., to co nsider controls supp or ted in “sufficiently la rge” measurable subsets of the time-space domain. 2. Preliminaries Let H be a Hilb ert space with scalar pro duct h· , ·i H . In the following, we study systems mo deled as      ˙ z ( t ) = Az ( t ) + α ( t ) Bu ( t ) u ( t ) = − B ∗ z ( t ) z (0) = z 0 (5) with A : H ⊃ D ( A ) → H being a (p ossibly un bounded) linear op erator gener - ating a strongly contin uous cont raction semigroup { e tA } t ≥ 0 , B : U → H being a bo unded linear oper ator o n some Hilb ert space U , B ∗ : H → U being its adjoint and α : [0 , ∞ ) → [0 , 1] b eing some signal poss ibly tuning the feedbac k co ntrol ter m B u ( t ) = − B B ∗ z ( t ). Note that under the ab ove as sumptions, the op erator A is maximal dissipative (see [33, Pro p o sition 3 .1.13]). W e are interested in co nditions on A , B and o n a cla ss of signals G ensuring the asymptotic decay of s olutions z 7→ z ( t ) of (5 ) to the origin in a suitable sense as time t tends to infinity—independently of the initial data z 0 ∈ H a nd of the sp ecific α ( · ) chosen in G . The in teresting case is when the uncontrolled evolution doe s not generate a strict contraction, i. e., when k e tA k = 1 for t ≥ 0, so that the energy of the system may stay cons ta nt in the absence of damping. The clas ses o f s ig nals w e mostly deal with are those defined by p ersistent exci- tation conditions. The latter are defined as follows: given tw o p ositive c o nstants T and µ satisfying µ ≤ T , w e sa y that a measur able s ig nal α ( · ) is a T - µ PE-signal if it satisfies Z t + T t α ( s ) ds ≥ µ, for all t ∈ [0 , ∞ ) . (6) W e note a t this p oint tha t we will not make any further smoo thness assumption on α ( · ) for our s tability results in this pap er. Thus our analysis takes into a ccount the mo deling o f a brupt actuator failures up to the extremal case when the system switches betw een an uncontrolled evolution when α ( t ) = 0 and a fully co ntrolled evolution when α ( t ) = 1. C o nditions o f the t ype (6) how ev er mea n that to some extent the feedback co ntrol is active. Solutions of (5) have to b e interpreted in the mild sense, i.e., for a ny t ≥ 0 and z 0 ∈ H , the solution z ( · ) of (5), ev a luated at time t , is given by z ( t ) = e tA z 0 − Z t 0 e ( t − s ) A α ( s ) B B ∗ z ( s ) ds. 5 F or any measur able signal α and for a ny finite time-hor izon ϑ ≥ 0 , there exists a unique mild so lution z ( · ) ∈ C ([0 , ϑ ]; H ) (see, e. g., [4]). Occa s ionally , we write z ( t ; z 0 ) to indicate the dependency of the mild solution on the initial data z 0 . As recalled in the intro ductio n, it is shown in [7 ] that for H = R N and ( A, B ) controllable (with A a dissipative N × N -matrix) the solutions of (5) sa tisfy k z ( t ) k ≤ M e − γ t k z 0 k , t ≥ 0 uniformly in α s a tisfying (6), in the sense that the c onstants M and γ dep end only o n A, B , µ and T . Such result do e s not ex tend in full generality to infinite- dimensional spaces. W e see this from the following example with ( A, B ) b eing a controllable pair, made o f a skew-adjoin t (and th us dissipative) op er a tor A and a bo unded op erator B . Example 2.1 . (String equation) Let us cons ide r a damped string of length nor mal- ized to one with fixed endpoints. Its dynamics can b e describ ed by v tt ( t, x ) = v xx ( t, x ) − α ( t ) d ( x ) 2 v t ( t, x ) , ( t, x ) ∈ (0 , ∞ ) × (0 , 1) , (7) v (0 , x ) = y 0 ( x ) , x ∈ (0 , 1) , (8) v t (0 , x ) = y 1 ( x ) , x ∈ (0 , 1) , (9) v ( t, 0) = v ( t, 1 ) = 0 , t ∈ (0 , ∞ ) , (10) where d ∈ L ∞ (0 , 1) and α ∈ L ∞ ([0 , ∞ ) , [0 , 1]). W e can express such dynamics as a system of t ype (5 ) with H = U = H 1 0 (0 , 1) × L 2 (0 , 1), z ( t ) = ( v ( t, · ) , v t ( t, · )), A ( z 1 ( t ) , z 2 ( t )) = ( z 2 ( t ) , ∂ xx z 1 ( t )), B ( z 1 ( t ) , z 2 ( t )) = (0 , dz 2 ( t )). The op erato r A is dissipative taking, as no rm in H , k ( z 1 , z 2 ) k 2 = k ∂ x z 1 k 2 L 2 (0 , 1) + k z 2 k 2 L 2 (0 , 1) . Assume that d = χ ω (11) for so me pr op er subinterv a l ω of (0 , 1). Then there exist T ≥ µ > 0 , a T - µ PE -signal α , and a corresp onding nonzer o per io dic solution. This follows fro m the results in [19] (see also [14]) and can be illustra ted by an explicit counterexample expresse d in terms of d’Alem ber t solutions. Let ω = ( a, b ) and assume, without loss of g enerality , that b < 1. Set b ′ = 1+ b 2 . T a ke T = 2 and µ = 1 − b ′ . Then α = ∞ X k =0 χ [2 k − µ, 2 k + µ ) is a T - µ signal and v ( t, x ) = ∞ X k =0 ( χ [ b ′ +2 k, 1+2 k ] ( x + t ) − χ [ − 1 − 2 k, − b ′ − 2 k ] ( x − t )) is a pe r io dic, nonzero, mild solution of (7), (10) corr esp onding to α . Notice, in particular, tha t even we a k a symptotic s ta bilit y fails to hold in this case. ⋄ The sco p e of the r eminder of the pa pe r is to understand to which exten t the finite-dimensional results obtained in [7] may b e extended to the case where H is infinite-dimensional. A crucial remar k in this p ersp ective is the following ener gy decay estimate. Let V ( z ) = 1 2 k z k 2 H (12) 6 F ALK M . HANTE, MARIO SIGALOTTI AND M ARIUS TUCSNAK denote the “energy” in H and obser ve that we have V ( z ( t + s )) − V ( z ( t )) ≤ − Z t + s t α ( s ) k B ∗ z ( s ) k 2 U ds for all s ≥ 0 , (13) so that V ( · ) is non- incr easing along tra jectories for all signals α ( · ). This can be shown b y a s tandard approximation argument [23, Theorem 2.7] and using that A is maximal dissipative. The estimate provided by the following lemma w ill b e a key to ol in the pro o f of some of the results in this paper . Lemma 2.1. L et 0 ≤ a ≤ b < ∞ . Then, for any me asura ble funct ion α : [0 , ∞ ) → [0 , 1] , the solution z ( · ) of system (5) satisfies V ( z ( b )) − V ( z ( a )) ≤ − (2 + 2( b − a ) 2 k B k 4 ) − 1 Z b − a 0 α ( t + a ) k B ∗ e tA z ( a ) k 2 U dt. Pr o of. Let φ a ( t ) = e ( t − a ) A z ( a ) , t ≥ a, and ψ a ( · ) be the mild so lution of ( ˙ ψ a ( t ) = Aψ a ( t ) − α ( t ) B B ∗ z ( t ) , t ≥ a, ψ a ( a ) = 0 . Observe that φ a ( t ) + ψ a ( t ) = e ( t − a ) A z ( a ) − Z t a e ( t − τ ) A α ( τ ) B B ∗ z ( τ ) dτ = z ( t ) , t ≥ a, (14 ) and that sup ξ ∈ [ a,b ] k ψ a ( ξ ) k 2 H ≤ ( b − a ) k B k 2 Z b a k α ( t ) B ∗ z ( t ) k 2 U dt, (15) bec ause, for ξ ∈ [ a, b ], k ψ a ( ξ ) k 2 H ≤ Z ξ a k e ( ξ − t ) A k H k B kk α ( t ) B ∗ z ( t ) k U dt ! 2 ≤ k B k 2 Z ξ a k α ( t ) B ∗ z ( t ) k U dt ! 2 ≤ ( ξ − a ) k B k 2 Z ξ a k α ( t ) B ∗ z ( t ) k 2 U dt, where we used that k e tA k ≤ 1 . F ro m inequalit y (15) we get Z b a α ( t ) k B ∗ ψ a ( t ) k 2 U dt ≤ ( b − a ) k B ∗ k 2 sup t ∈ [ a,b ] k ψ a ( t ) k 2 H ≤ ( b − a ) 2 k B k 2 k B ∗ k 2 Z b a k α ( t ) B ∗ z ( t ) k 2 U dt. (16) Moreov er, using (14), we o bta in Z b a α ( t ) k B ∗ φ a ( t ) k 2 U dt = Z b a α ( t ) k B ∗ ( z ( t ) − ψ a ( t )) k 2 U dt ≤ 2 Z b a α ( t ) k B ∗ z ( t ) k 2 U dt + Z b a α ( t ) k B ∗ ψ a ( t ) k 2 U dt ! . (17) 7 Plugging (16) in (17), we get Z b a α ( t ) k B ∗ φ a ( t ) k 2 U dt ≤ 2(1 + ( b − a ) 2 k B k 4 ) Z b a k α ( t ) B ∗ z ( t ) k 2 U dt. (18) W e ge t fro m the energy inequality (13) combined with (18) that V ( z ( b )) − V ( z ( a )) ≤ − Z b a α ( t ) k B ∗ z ( t ) k 2 U dt ≤ − 1 2(1 + ( b − a ) 2 k B k 4 ) Z b a α ( t ) k B ∗ φ a ( t ) k 2 U dt = − 1 2(1 + ( b − a ) 2 k B k 4 ) Z b − a 0 α ( a + t ) k B ∗ e tA z ( a ) k 2 U dt, concluding the pro of.  Other useful facts which are used r ep eatedly b elow ar e the following remark s on the class of T - µ PE -signals. W e note that if α ( · ) is a T - µ PE-signa l, then for every t 0 ≥ 0, the same is true for α ( t 0 + · ). Moreover, the s et of all T - µ PE-s ignals is weakly- ∗ compact, i. e., for any sequence ( α n ( · )) n ∈ N in this set, there exists a subsequence ( α n ( ν ) ( · )) ν ∈ N such that for some T - µ P E-signa l α ∞ ( · ) Z ∞ 0 α ∞ ( s ) g ( s ) ds = lim ν →∞ Z ∞ 0 α n ( ν ) ( s ) g ( s ) ds for all g ∈ L 1 ([0 , ∞ )) . (19) The existence of a function α ∞ ∈ L ∞ ([0 , ∞ ) , [0 , 1]) satisfying (19) follows from the weak- ∗ compactness of L ∞ ([0 , ∞ ) , [0 , 1]) a nd one recov ers (6) for α ∞ by choosing as g in (19) the indicator function of the int erv al [ t, t + T ]. 3. Exponential st ability under persistent excit a tion W e next show that, under the following conditio n, asymptotic exp onential sta- bilit y holds. Hyp othesi s 3. 1. Ther e exist two c onstants c, ϑ > 0 such that Z ϑ 0 α ( t ) k B ∗ e tA z 0 k 2 U dt ≥ c k z 0 k 2 H , for al l z 0 ∈ H and al l T - µ PE-signals α ( · ) . (20) Theorem 3 . 2. Under Hyp othesis 3.1, ther e exist two c o nstants M ≥ 1 and γ > 0 such that the mild solut ion z ( · ) of s yst em (5) satisfies k z ( t ) k H ≤ M e − γ t k z 0 k H , t ≥ 0 , (21) for any initial data z 0 ∈ H and any T - µ PE-signal α ( · ) . Pr o of. Fix so me T - µ PE-sig nal α ( · ) and so me s ≥ 0, and define V by (12). Lemma 2.1 with a = s a nd b = s + ϑ , where ϑ is a s in Hyp othesis 3.1, then yields V ( z ( s + ϑ )) − V ( z ( s )) ≤ − 1 2(1 + ϑ 2 k B k 4 ) Z ϑ 0 α ( t + s ) k B ∗ e tA z ( s ) k 2 U dt. Again using that α ( · + s ) is a T - µ PE -signal, Hypothes is 3.1 then implies V ( z ( s + ϑ )) − V ( z ( s )) ≤ − c (1 + ϑ 2 k B k 4 ) V ( z ( s )) . The desired estimate (21) then follows from standard arg umen ts.  8 F ALK M . HANTE, MARIO SIGALOTTI AND M ARIUS TUCSNAK Example 3.1 b elow illustr ates an applica tion of Theorem 3.2. W e consider aga in the model of a damp ed string intro duced in Example 2.1, replacing the lo calized damping g iven in (11) (whic h, a s we prov ed, gives r ise to non-stabiliza bility) b y a damping acting almo st e verywhere. The a rgument is presented for the g e ne r al ca se of the damped wav e equation (the string cor r esp onding to the ca se N = 1). Example 3.1 . (W av e equation) Let N ≥ 1 and consider a N -dimensional version of system (7)–(10) in tro duced in Example 2.1: v tt ( t, x ) = ∆ v ( t, x ) − α ( t ) d ( x ) 2 v t ( t, x ) , ( t, x ) ∈ (0 , ∞ ) × Ω , (22) v (0 , x ) = y 0 ( x ) , x ∈ Ω , (23) v t (0 , x ) = y 1 ( x ) , x ∈ Ω , (24) v ( t, x ) = 0 , ( t, x ) ∈ (0 , ∞ ) × ∂ Ω , (25) where Ω is a bounded doma in in R N and d ∈ L ∞ (Ω) satisfies | d ( x ) | ≥ d 0 > 0 for almost all x ∈ Ω . W e claim that in this c a se Hyp othesis 3.1 is satisfied with ϑ = T , taking H = H 1 0 (Ω) × L 2 (Ω) with norm k ( z 1 , z 2 ) k 2 = k∇ z 1 k 2 L 2 (Ω) + k z 2 k 2 L 2 (Ω) . Denote by ( φ n ) n ∈ N an orthono r mal basis o f L 2 (Ω) made of eigenfunctions of the Laplace–Dirichlet oper a tor on Ω. F or ea ch n ∈ N , let λ n > 0 be the e ig env alue corres p o nding to φ n . Recall that λ n go es to infinit y as n → ∞ . Let t 7→ z ( t ) = ( v ( t, · ) , v t ( t, · )) b e a solutio n o f (22)–(2 5) with initial condi- tion ( y 0 ( · ) , y 1 ( · )) = ( P n ∈ N a n φ n ( · ) , P ∞ n =1 √ λ n b n φ n ( · )), where ( √ λ n a n ) n ∈ N and ( √ λ n b n ) n ∈ N belo ng to ℓ 2 . By definition, k z (0) k 2 H = P n ∈ N λ n ( a 2 n + b 2 n ) and v ( t, x ) = X n ∈ N a n φ n ( x ) cos( p λ n t ) + ∞ X n =1 b n φ n ( x ) sin( p λ n t ) . Then Z T 0 α ( t ) k B ∗ z ( t ) k 2 U dt ≥ d 2 0 Z T 0 Z Ω α ( t ) | v t ( x, t ) | 2 dx dt = d 2 0 Z T 0 Z Ω α ( t ) X n ∈ N λ n ( − a n sin( p λ n t ) + b n cos( p λ n t )) φ n ( x ) ! 2 dx dt = d 2 0 X n ∈ N λ n Z T 0 α ( t )( − a n sin( p λ n t ) + b n cos( p λ n t )) 2 dt, where we used that, for all n, m ∈ N , Z 1 0 φ n ( x ) φ m ( x ) dx = δ nm . W e are le ft to prov e that there e x ist c 0 > 0 independent of n ∈ N , a n , b n ∈ R a nd of the T - µ signal α such that Z T 0 α ( t )( − a n sin( p λ n t ) + b n cos( p λ n t )) 2 dt ≥ c 0 ( a 2 n + b 2 n ) . (26) F or every ǫ ∈ (0 , 1 ), let A ǫ n = { t ∈ [0 , T ] | | − a n sin( p λ n t ) + b n cos( p λ n t ) | > ǫ p a 2 n + b 2 n } . (27) 9 Notice that − a n sin( √ λ n t ) + b n cos( √ λ n t ) = p a 2 n + b 2 n sin( √ λ n t + θ n ) for some θ n ∈ R . Hence, | − a n sin( p λ n t ) + b n cos( p λ n t ) | ≤ λ n p a 2 n + b 2 n | t − t 0 | for every t 0 belo nging to { t 0 | sin( λ n t 0 + θ n ) = 0 } = π λ n Z − θ n λ n . In particula r, [0 , T ] \ A ǫ n is cont ained in the set o f po int s with a dis tance from π λ n Z − θ n λ n smaller than ǫ/λ n , i.e., in the unio n of interv als of length 2 ǫ/λ n centered a t elements o f π λ n Z − θ n λ n . Therefor e, meas( A ǫ n ) ≥ T − 2 ǫ λ n #  [0 , T ] ∩  π λ n Z − θ n λ n  ≥ T − 2 ǫ λ n  T λ n π + 1  ≥ T  1 − 2 ǫ π  − 2 ǫ min n ∈ N λ n . (28) Thu s, the mea s ure of A ǫ n tends to T as ǫ go es to zero , uniformly with resp ect to the triple ( n, a n , b n ). In par ticula r, ther e exis ts ¯ ǫ > 0 suc h that for every n ∈ N , a n , b n ∈ R and every T - µ signal α , Z A ¯ ǫ n α ( t ) dt ≥ µ 2 . Then Z T 0 α ( t )( − a n sin( p λ n t ) + b n cos( p λ n t )) 2 dt ≥ ¯ ǫ 2 ( a 2 n + b 2 n ) Z A ¯ ǫ n α ( t ) dt ≥ µ ¯ ǫ 2 2 ( a 2 n + b 2 n ) , proving (26) with c 0 = µ ¯ ǫ 2 / 2. ⋄ R emark 3.1 . The example presented ab ov e shows that the sufficient condition for asymptotic stabilit y of abs tract second order evolution equations with on/off damp- ing consider e d in [14] is not necess a ry , as detailed here b elow. T he question of its necessity had b een raised in [10, p. 2522]. Extending a result of [30] for ordinary differential equations, it was shown in [1 4] that existence of a sequence of open dis jo int interv als I n of length T n such that ∞ X n =1 m n T n min  T 2 n , (1 + m n M n ) − 1  = ∞ (29) and existence of constants M n ≥ m n > 0 such that m n ≤ α ( t ) ≤ M n , t ∈ I n , (30) implies asymptotic stability of sys tems whose prototype is (2 2)–(25). T a k ing, for exa mple, I n = ( s n , s n + 1 n ) with s n = n − 1 X k =1 2 k and α ( · ) piecewis e constant such that (30) holds with m n = M n = 1 , the sum in (29) conv erges, but for T = 2 , Z t + T t α ( s ) ds ≥ µ, t ≥ 0 , for some µ > 0 , as it eas ily follows b y no ticing that lim t → + ∞ R t + T t α ( s ) ds = 1 / 2 . ⋄ 10 F ALK M . HANTE, MARIO SIGALOTTI AND M ARIUS TUCSNAK Another ex ample that one could consider is the Sc hr¨ odinger equation with in- ternal damping. Beca us e of the infinite sp eed of pr o pagation of the Schr¨ odinger equation, it is a natural question whether, differently form the cas e of the wav e equation, stability ca n b e achiev ed by a lo ca lized damping. W e ar e not able to give an answer to this question (detailed b elow), which w e leave as an op en problem. Example 3.2 . (Sch r¨ oding er eq uation) Consider iy t ( t, x ) + y xx ( t, x ) + i α ( t ) d ( x ) 2 y ( t, x ) = 0 , ( t, x ) ∈ (0 , ∞ ) × (0 , 1) , (3 1) y ( t, 0) = y ( t, 1) = 0 , t ∈ (0 , ∞ ) , (32) y (0 , x ) = y 0 ( x ) , t ∈ (0 , 1) , (33) with d ( · ) ∈ L ∞ (0 , 1) and α ( · ) b eing a T - µ PE -signal. Assume that d = χ ω with ω = ( a, b ) a nonempt y subinterv al of (0 , 1). In o rder to wr ite system (31)–(33) in the for m (5), let H = U = L 2 (0 , 1), define A as Az = iz xx , acting on D ( A ) = H 2 (0 , 1) ∩ H 1 0 (0 , 1), and let B : z 7→ χ ω z be the m ultiplication o per ator by the function χ ω = d . Then, for y 0 ∈ H , the mild solution z ( · ) of (5) with this choice o f A a nd B co rresp onds to the weak s o lution y ( · ) of (31)– (3 3) (see [3]). Since A is skew-adjoint, in order to apply T heo rem 3.2 we should prov e that Hypo thesis 3.1 holds true. More explicitly , w e should prov e that there exist ϑ, c > 0 such that, for each z 0 ∈ L 2 (0 , 1) and each T - µ PE-signal α , Z ϑ 0 Z b a α ( t ) | ( e tA z 0 )( x ) | 2 dx dt ≥ c Z 1 0 | z 0 ( x ) | 2 dx. In o rder to fix the ideas , let us take ϑ = T > µ . The question can be rephra sed by asking whether there exists c > 0 such that for every Ξ ⊂ [0 , T ] of mea s ure eq ual to µ , Z Ξ Z b a | X n ∈ N h φ n , z 0 i L 2 (0 , 1) φ n ( x ) e in 2 π 2 t | 2 dx dt ≥ c Z 1 0 | z 0 ( x ) | 2 dx, (34) with φ n ( x ) = √ 2 sin( nπ x ). T his problem is, up to our knowledge, op en. The question is somehow related with a discussion presented b y Seidman in [27], where it is co njectured that, amo ng a ll s uch sets Ξ, the maximal cons ta nt in (34) (uniform with resp ect to z 0 ) is obtained for interv a ls. Notice that in the case ω = (0 , 1 ) inequality (34) is s atisfied b ecaus e the L 2 norm of z ( t ) is constant with resp ect to t . Because of the full damping in space, the techniques dev elop ed by F attorini in [9] would a lso apply , yielding the required generalized observ abilit y inequality . ⋄ 4. Weak st ability under persistent excit a tio n Our main result is tha t weak asymptotic stability holds when the pair ( A, B ) ha s the following T - µ PE unique contin uation prop er t y , which weak ens Hyp othesis 3 .1. Hyp othesi s 4. 1. Ther e exists ϑ > 0 such that for al l T - µ PE-signals α ( · ) Z ϑ 0 α ( t ) k B ∗ e tA z 0 k 2 U dt = 0 ⇒ z 0 = 0 . (35) W e will prove the fo llowing. Theorem 4.2. Under H yp othesis 4.1, t he mild solution t 7→ z ( t ) of system (5) c o nver g es we akly to 0 in H as t → ∞ for any initial data z 0 ∈ H and any T - µ PE-signal α ( · ) . 11 Pr o of. It suffices to prov e that, for each z 0 ∈ H a nd fo r each T - µ PE-signal α ( · ), the we ak ω - limit set ω ( z 0 , α ( · )) = { z ∈ H | there exists a sequence { s n } n ∈ N , s n → ∞ , so that z ( s n ; z 0 ) ⇀ z as n → ∞} is non-empt y and, taking ϑ > 0 a s in Hypo thes is 4.1, z ∞ ∈ ω ( z 0 , α ( · )) ⇒ ∃ α ∞ T - µ P E -signal s . t. Z ϑ 0 α ∞ ( t ) k B ∗ e tA z ∞ k 2 U dt = 0 . ( 36) The assertio n of the theorem then follows from (35). Let z 0 ∈ H and a T - µ -p ersistent excita tion sig nal α ( · ) be given. Let z ( · ; z 0 ) b e the unique mild solution of the system (5) and define V a s in (12). F ro m the energy inequality (13), one obta ins tha t the weak ω -limit set ω ( z 0 , α ( · )) is non-empty . So le t z ∞ ∈ ω ( z 0 , α ( · )) b e an element of the weak ω - limit set and let { s n } n ∈ N , s n → ∞ b e a sequence of times such that z ( s n ; z 0 ) ⇀ z ∞ as n → ∞ . W e co nsider the tr a nslations z n ( s ) = z ( s + s n ; z 0 ) α n ( s ) = α ( s + s n ) and we note tha t z n ( · ) is the mild solution of system (5) for the T - µ PE- signal α n ( · ) and initial condition z n (0) = z ( s n ; z 0 ), i. e., z n ( · ) satisfies z n ( s ) = e sA z ( s n ; z 0 ) − Z s 0 e ( s − t ) A α n ( t ) B B ∗ z n ( t ) dt. (37) Therefore, w e ha ve the energy estimates V ( z n ( s )) − V ( z ( s n ; z 0 )) ≤ − Z s 0 α n ( t ) k B ∗ z n ( t ) k 2 U dt for all s ≥ 0 . (38) F ro m (38) a nd (13) w e get k z n ( s ) k H ≤ k z ( s n ; z 0 ) k H ≤ k z 0 k H , s ≥ 0 , (39) and th us, for a ny ϑ ≥ 0, we hav e that { z n ( · ) } n ∈ N is a b ounded subset of C ([0 , ϑ ]; H ). Cho ose ϑ > 0 as in Hyp othesis 4 .1. W e cla im tha t z n ( s ) ⇀ z ∞ ( s ) as n → ∞ , for a ll s ∈ [0 , ϑ ] , (40) where z ∞ ( · ) is the mild solution of the undamped equation ( ˙ z ( s ) = Az ( s ) z (0) = z ∞ . (41) Indeed, muc h a s in [5], w e can show that { z n } n ∈ N is equicont inuous in C ([0 , ϑ ]; H w ), where H w is H endowed with the weak top olog y . T o verify this, let s r ց s in [0 , ϑ ] and select some ψ ∈ H . F r om (37), we hav e that |h z n ( s r ) − z n ( s ) , ψ i| ≤    [ e s r A − e sA ] z ( s n ; z 0 ) , ψ    + Z s 0    D [ e ( s r − t ) A − e ( s − t ) A ] α n ( t ) B B ∗ z n ( t ) , ψ E    dt + Z s r s    h e ( s r − t ) A α n ( t ) B B ∗ z n ( t ) , ψ i    dt. (42) Moreov er, using (39 ), we hav e that k α n ( t ) B B ∗ z n ( t ) k H ≤ k α n ( t ) k R k B B ∗ k L ( H ) k z n ( t ) k H ≤ cons t . k z 0 k H and, as prov ed in [5, Theorem 2.3], a r = sup k φ k H ≤ 1 , 0 ≤ t ≤ s |h [ e ( s − t ) A − e ( s r − t ) A ] φ, ψ i| → 0 as r → ∞ . 12 F ALK M . HANTE, MARIO SIGALOTTI AND M ARIUS TUCSNAK Thu s, from (42) we get |h z n ( s r ) − z n ( s ) , ψ i| ≤ const .a r k z 0 k H + const . | s r − s | , and hence |h z n ( s r ) − z n ( s ) , ψ i| → 0 uniformly as r → ∞ . (43) Similarly , one shows that (43) holds for s r ր s in [0 , ϑ ]. Th us, { z n } n ∈ N is equico n- tin uous in C ([0 , ϑ ]; H w ). Again using that { z n ( s ) | n ∈ N , s ∈ [0 , ϑ ] } is b ounded in H by (39 ), we ma y view { z n } n ∈ N as an eq uibo unded set o f c urves in H endow ed with the metr ized w eak to po logy . Hence we can apply the Arzela–Ascoli theo rem for metr ic space s to conclude that there ex is ts z ∞ ( · ) ∈ C ([0 , ϑ ]; H w ) a nd a sub- sequence that we r e-lab el by n ∈ N so that z n ( s ) ⇀ z ∞ ( s ) unifor mly o n [0 , ϑ ] as ν → ∞ . Moreov er, for any ψ ∈ H we hav e from (37) b y adding and subtrac ting h e ( s − t ) A α n ( t ) B B ∗ z ∞ ( t ) , ψ i under the integral that h z n ( s ) , ψ i = h e sA z ( s n ; z 0 ) , ψ i − Z s 0 α n ( t ) h e ( s − t ) A B B ∗ z ∞ ( t ) , ψ i dt − Z s 0 α n ( t ) h e ( s − t ) A B B ∗ [ z n ( t ) − z ∞ ( t )] , ψ i dt. (44) Using that α n ( t ) is a bo unded sequence for t ∈ [0 , s ] and that h e ( s − t ) A B B ∗ [ z n ( t ) − z ∞ ( t )] , ψ i → 0 as ν → ∞ for all t ∈ [0 , s ], we can conclude from the dominated conv ergence theo rem that Z s 0 α n ( t ) h e ( s − t ) A B B ∗ [ z n ( t ) − z ∞ ( t )] , ψ i dt → 0 , a s n → ∞ . Hence, by sequential weak ∗ -compactness of L ∞ ([0 , ∞ ); [0 , 1]), we can extract an- other subsequenc e that we again r e-lab el by n ∈ N and pa s s to the limit in (44), obtaining that, for every s ≥ 0, h z ∞ ( s ) , ψ i = h e sA z ∞ , ψ i − Z s 0 h e ( s − t ) A α ∞ ( t ) B B ∗ z ∞ ( t ) , ψ i dt, (45) where α ∞ ( · ) aga in is a T - µ P E -signal. Since (45) holds for all ψ ∈ H , we hav e z ∞ ( s ) = e sA z ∞ − Z s 0 e ( s − t ) A α ∞ ( t ) B B ∗ z ∞ ( t ) dt. Next we show that Z s 0 e ( s − t ) A α ∞ ( t ) B B ∗ z ∞ ( t ) dt = 0 . (46) Since V ( z n (0)) is b ounded and monotone, it ha s a limit V ∗ = lim n →∞ V ( z n ), so that Z ϑ 0 α n ( t ) k B ∗ z n ( t ) k 2 U dt ≤ V ( z n (0)) − V ∗ → 0 as n → ∞ . Hence Z ϑ 0 α n ( t ) k B ∗ z n ( t ) k 2 U dt → 0 as n → ∞ . (47) Morov er, (40) and α n ∗ ⇀ α ∞ imply lim inf n →∞ Z ϑ 0 α n ( t ) k B ∗ z n ( t ) k 2 U dt ≥ Z ϑ 0 α ∞ ( t ) k B ∗ z ∞ ( t ) k 2 U dt. (48) T o see this, obser ve that (40) implies k B ∗ z ∞ ( t ) k U ≤ lim inf n →∞ k B ∗ z n ( t ) k U , for all t ∈ [0 , ϑ ] . (49) 13 Fix any ǫ > 0 and define, for all m ∈ N , S ǫ m = { t ∈ [0 , ϑ ] | k B ∗ z n ( t ) k 2 U ≥ k B ∗ z ∞ ( t ) k 2 U − ǫ for all n ≥ m } . F ro m (49) we hav e [0 , ϑ ] = [ m S ǫ m ( S ǫ m ⊇ S ǫ m − 1 ) , hence there exists m ( ǫ ) such that | S ǫ m ( ǫ ) | > T − ǫ . Then w e hav e, for n ≥ m ( ǫ ), Z T 0 α n ( t ) k B ∗ z n ( t ) k 2 U dt = Z S ǫ m ( ǫ ) α n ( t )  k B ∗ z n ( t ) k 2 U − k B ∗ z ∞ ( t ) k 2 U + ǫ  dt + Z S ǫ m ( ǫ ) α n ( t )  k B ∗ z ∞ ( t ) k 2 U − ǫ  dt + Z [0 ,ϑ ] \ S ǫ m ( ǫ ) α n ( t ) k B ∗ z n ( t ) k 2 U dt. (50) The fir st int egral in the r ight-hand side of (50) is no n- negative b eca use k B ∗ z n ( t ) k 2 U − k B ∗ z ∞ ( t ) k 2 U + ǫ ≥ 0 for all t ∈ S ǫ m ( ǫ ) and α n ( t ) ≥ 0 for all t ∈ [0 , ϑ ]. The second int egral is bo unded from below b y Z S ǫ m ( ǫ ) α n ( t ) k B ∗ z ∞ ( t ) k 2 U dt − ǫϑ ≥ Z ϑ 0 α n ( t ) k B ∗ z ∞ ( t ) k 2 U dt − ǫ (const. + ϑ ) as it fo llows from (3 9). Finally , the third integral is non-neg ative, again beca us e α n ( t ) ≥ 0 for all t ∈ [0 , ϑ ]. Thus, for a ll n ≥ m ( ǫ ), Z ϑ 0 α n ( t ) k B ∗ z n ( t ) k 2 U dt ≥ Z ϑ 0 α n ( t ) k B ∗ z ∞ ( t ) k 2 U dt − ǫ ( C + ϑ ) . Hence, by the conv ergence α n ( · ) ∗ ⇀ α ∞ ( · ), lim inf n →∞ Z ϑ 0 α n ( t ) k B ∗ z n ( t ) k 2 U dt ≥ Z ϑ 0 α ∞ ( t ) k B ∗ z ∞ ( t ) k 2 U dt − ǫ ( C + T ) , proving (48) fro m the fa c t that ǫ is a rbitrar y . F ro m (48) a nd (4 7), we hav e 0 = lim n →∞ Z ϑ 0 α n ( t ) k B ∗ z n ( t ) k 2 U dt = Z ϑ 0 α ∞ ( t ) k B ∗ z ∞ ( t ) k 2 U dt, (51) so either α ∞ ( t ) = 0 or B ∗ z ∞ ( t ) = 0 for almost every t ∈ [0 , ϑ ]. This prov es (46) and hence z ∞ ( · ) solves, as claimed, the undamp ed equation (41). Finally , since z ∞ solves (41), we hav e z ∞ ( s ) = e sA z ∞ and thus (51) implies (36), as required.  In the example below w e go back to the internally damp ed Sch r¨ oding er equation considered in Example 3.2, where we were no t able to co nclude whether such equa- tion is strong ly stable, uniformly with r esp ect to all T - µ sig nals ( T > µ > 0 given). W e prove here, in the general N -dimensional case, that weak stability holds true. Example 4.1 . (Schr¨ odinger equation) Let Ω b e a b ounded domain of R N , N ≥ 1, and consider the int ernally damp ed Schr¨ odinger equa tion iy t ( t, x ) + ∆ y ( t, x ) + iα ( t ) d ( x ) 2 y ( t, x ) = 0 , ( t, x ) ∈ (0 , ∞ ) × Ω , (52) y ( t, x ) = 0 , t ∈ (0 , ∞ ) × ∂ Ω , (53) y (0 , x ) = y 0 ( x ) , t ∈ Ω , (54 ) 14 F ALK M . HANTE, MARIO SIGALOTTI AND M ARIUS TUCSNAK where d ( · ) b elongs to L ∞ (Ω) a nd α ( · ) is a T - µ PE -signal. Assume that there ex ist d 0 > 0 and an op en nonempty ω ⊂ Ω such that | d ( x ) | ≥ d 0 for a. e. x in ω . (55) As in Example 3.2, sys tem (52)–(54) ca n b e written in the form (5) with H = U = L 2 (Ω), Az = i ∆ z , D ( A ) = H 2 (Ω) ∩ H 1 0 (Ω), and B : z 7→ dz . Since A is skew-adjoin t, it genera tes a contraction s e migroup. As to apply Theorem 4.2, it r e ma ins to show that the pair ( A, B ) has the T - µ PE unique co ntin uation pr op erty s tated in Hyp othesis 4.1. T o this end, fix so me z 0 ∈ L 2 (Ω), some T - µ P E -signal α ( · ), and c hoo se any ϑ > T − µ . Observe that, since Z ϑ 0 α ( t ) k B ∗ e tA z 0 k 2 U dt = Z ϑ 0 α ( t ) k de tA z 0 k 2 H dt, then either α ( t ) = 0 or de tA z 0 = 0 fo r almost every t ∈ [0 , ϑ ]. But (6) implies that α ( · ) > 0 on a set Ξ ⊂ (0 , ϑ ) with meas(Ξ) ≥ ϑ − T + µ > 0 a nd (55) yields d ( · ) 6 = 0 a. e. on the op en set ω ⊂ Ω. Hence, ( t, x ) 7→ ( e tA z 0 )( x ) ≡ 0 on Ξ × ω. Let us now adapt the unique co nt inuation a r gument prop osed in [26] in or de r to prov e that ( t, x ) 7→ ( e tA z 0 )( x ) = 0 on the o p en set (0 , ϑ ) × ω . W rite the sp ectr um of A as ( λ k ) k ∈ N (with eigenv a lues rep ea ted according to their mult iplicities). Then the sequence ( iλ k ) k ∈ N is contained in R and is b ounded from b elow. Denote by ( φ k ) k ∈ N an orthonormal basis of H such that Aφ k = λ k φ k . Fix any ϕ ∈ L 2 ( ω ) and conside r the function F : t 7→ P k ∈ N e λ k t h φ k , z 0 ih φ k , ϕ i . Notice that F ( t ) = h e tA z 0 , ϕ i and that F can b e extended fro m R to C − = { w ∈ C | Im( w ) ≤ 0 } , thank s to the low er b oundedness (in R ) o f ( iλ k ) k ∈ N . Moreover, F is complex ana lytic in the int erior of C − and contin uous up to its bo undary . Since F is zer o on a subset of the b oundary of C − of po sitive (one-dimensio na l) measure, then it follows fro m Priv alo v’s uniqueness theorem (see [37, V ol. I I, Theorem 1.9, p. 203]) that F v anis hes identically . By the arbitr ariness of ϕ ∈ L 2 ( ω ) it follows, as requir ed, that e tA z 0 v anis hes on ω for t ∈ (0 , θ ). Applying Holmgren’s uniqueness theorem (see [17, Theor em 8.6.8] and also [36]), we deduce that z 0 v anis hes on Ω, proving Hypo thesis 4.1. ⋄ 5. Strong st ability Condition (6) means that the feedback control B u = − B B ∗ z is, to some extent, active on every interv al of the length T . F ro m an a pplication p o int o f view it is also in teresting to study the case when there a re interv als of a r bitrary leng th where no feedback cont rol is active, in the spirit of the results in, e. g ., [14, 15, 25, 30]. A natura l q uestion is then to ask whic h conditions imp osed on A , B , and on the distribution and length of these interv als suffice to ensure stability . Below we give an abstract re s ult ensur ing the strong asymptotic stabilit y of the c lo sed-lo op system (5) using observ abilit y estimates for the open- lo op system. Stressing the imp orta nce , in order to apply such r esult, of having explicit estimates for c o ntrol costs (i.e., the constants c app ear ing in inequalities of the t ype (20)), we then show on several exa mples how this can lead to stabilizing conditions. Definition 5. 1. We say that α ( · ) ∈ L ∞ ([0 , T ] , [0 , 1 ]) is of class K ( A, B , T , c ) if Z T 0 α ( t ) k B ∗ e sA z 0 k 2 U dt ≥ c k z 0 k 2 H , for al l z 0 ∈ H . (56) With this definition, we can state the following abstra ct result. 15 Theorem 5.2. Su pp ose that ( a n , b n ) , n ∈ N , is a se quenc e of disjoint intervals in [0 , ∞ ) , t hat c n , n ∈ N , is a se quenc e of p o sitive r e al num b e rs and t hat α ( · ) ∈ L ∞ ([0 , ∞ ) , [0 , 1]) is such that its r est riction α ( a n + · ) | [0 ,b n − a n ] to the interval ( a n , b n ) is of class K ( A, B , b n − a n , c n ) for al l n ∈ N . Mor e over, assume that sup n ∈ N ( b n − a n ) < ∞ and P ∞ n =1 c n = ∞ . Then the mild solution of (5) s at isfies k z ( t ) k H → 0 as t → ∞ . Pr o of. First of all notice that (56) implies that c ≤ T k B ∗ k 2 . Hence, the sum of the c n corres p o nding to interv a ls ( a n , b n ) co ntained in a g iven b ounded interv al [ τ 0 , τ 1 ] is finite and ca n b e approximated ar bitrarily well by the sum of finitely many of suc h c n . Therefore, we can extr act a lo ca lly finite subsequence of in terv a ls, still deno ted by ( a n , b n ), n ∈ N , such that P ∞ n =1 c n = ∞ and, up to a reo rdering, b n ≤ a n +1 for all n ∈ N . Using the ener gy inequality (13) we get V ( z ( a n +1 )) ≤ V ( z ( b n )) while Lemma 2 .1, with a = a n and b = b n , implies that V ( z ( b n )) − V ( z ( a n )) ≤ − 1 2(1 + ( b n − a n ) 2 k B k 4 ) Z b n − a n 0 α ( a n + t ) k B ∗ e tA z ( a n ) k 2 U dt. Thu s, since α ( a n + · ) | [0 ,b n − a n ] is of class K ( A, B , b n − a n , c n ), we hav e V ( z ( a n +1 )) − V ( z ( a n )) ≤ − c n 1 + ( b n − a n ) 2 k B k 4 V ( z ( a n )) . (57) Using the estimate (57) recursively , w e obtain V ( z ( a n +1 )) ≤ n Y j =1  1 − c j 1 + ( b j − a j ) 2 k B k 4  V ( z 0 ) . Since log ∞ Y j =1  1 − c j 1 + ( b j − a j ) 2 k B k 4  = ∞ X j =1 log  1 − c j 1 + ( b j − a j ) 2 k B k 4  ≤ − ∞ X j =1 c j 1 + ( b j − a j ) 2 k B k 4 ≤ − 1 1 + k B k 4 sup ∞ j =1 ( b j − a j ) 2 ∞ X j =1 c j = −∞ , then V ( z ( a n +1 )) tends to zero as n g o es to infinity .  As a dire c t a pplication of this a bstract r esult, we consider again the Sc hr¨ oding er equation in one space dimension. Example 5.1 . (Sch r¨ oding er eq uation) Consider iy t ( t, x ) + y xx ( t, x ) + i α ( t ) d ( x ) 2 y ( t, x ) = 0 , ( t, x ) ∈ (0 , ∞ ) × (0 , 1) , (5 8) y ( t, 0) = y ( t, 1) = 0 , t ∈ (0 , ∞ ) , (59) y (0 , x ) = y 0 ( x ) , t ∈ (0 , 1) , (60) with d ( · ) ∈ L ∞ (0 , 1) and α ( · ) ∈ L ∞ ([0 , ∞ ) , [0 , 1]). Assume that d = χ ω with ω a nonempty subinterv al of (0 , 1) and assume that ( a n , b n ), n ∈ N , is a sequence of disjoint interv als in [0 , ∞ ) such that sup n ∈ N ( b n − a n ) < ∞ and α ( · ) | ( a n ,b n ) ≡ 1. As in Example 3.2, we write system (58 )–(60) in the form (5) with H = U = L 2 (0 , 1), the skew-adjoint op er ator A g iven by Az = iz xx acting on D ( A ) = H 2 (0 , 1) ∩ H 1 0 (0 , 1), and the multiplication oper ator B : z 7→ χ ω z , so that for y 0 ∈ H , the mild solution z ( · ) of (5) with this choice o f A, B corres po nds to the weak solu- tion y ( · ) of (58)–(60). 16 F ALK M . HANTE, MARIO SIGALOTTI AND M ARIUS TUCSNAK It is well known that for any int erv al ( a n , b n ), n ∈ N , there exists a po sitive constant c n such that Z b n a n Z ω | e tA z ( x ) | 2 dx dt ≥ c n k z k 2 H , z ∈ H , (61) that is, α ( a n + · ) | [0 ,b n − a n ] is of class K ( A, B , b n − a n , c n ) (se e , for instance, [33, Remark 6.5.4]). Mor eov er, r ewriting (61) a s Z b n a n Z ω      X k ∈ N h φ k , z i L 2 (0 , 1) φ k ( x ) e in 2 π 2 t      2 dx dt ≥ c n k z k 2 H , with φ k ( x ) = √ 2 sin( nπ x ) we get from [32, Coro llary 3.2 ] that c n can b e taken satisfying c n ≥ C e − 2 b n − a n for some po s itive constant C indep endent o f n . Hence, Theorem 5.2 gua rantees that the mild solution of (5) conv erges strongly to the origin in H if ∞ X n =1 e − 2 b n − a n = ∞ . ⋄ R emark 5.1 . The results in the ab ov e example and, more gener ally , the metho dol- ogy employ ed in this se c tio n, can b e adapted to the ca se of some un bounded co n trol op erator s a nd thus to bounda ry stabilization pro blems. As a n exa mple, consider the system iy t ( t, x ) + y xx ( t, x ) = 0 , ( t, x ) ∈ (0 , ∞ ) × (0 , 1) , (62) y x ( t, 0) = − iα ( t ) y ( t, 0) , t ∈ (0 , ∞ ) , (63) y ( t, 1) = 0 , t ∈ (0 , ∞ ) , (64) y (0 , x ) = y 0 ( x ) , x ∈ (0 , 1) . (65) with a piecewise consta nt α : [0 , ∞ ) → [0 , 1] s atisfying conditions a s in Ex ample 5.1 for some sequences ( a n ) n ∈ N , ( b n ) n ∈ N . Clearly , Theor em 5.2 do es not a pply in this case. How ev er, we can retrieve similar conditions o n the in terv als ( a n , b n ) in order to hav e the strong stability pro pe r ty as in Example 5.1. Indeed, it suffices to chec k the e x act observ abilit y for the undamp ed dyna mics and to s how that a n energ y estimate such in Lemma 2.1 holds for the c onstant damping ca se. The op erator A : D ( A ) → L 2 (0 , 1) corre s po nding to the undamp ed case (i.e., α = 0 in (63)) is D ( A ) = { ϕ ∈ H 2 (0 , 1) | ϕ x (0) = 0 , ϕ (1) = 0 } , Aϕ = i ϕ xx ( ϕ ∈ D ( A )) , whereas the control op era tor is giv en b y B = δ 0 , where δ 0 is the Dirac mass concentrated at the orig in. Using the r esults in [32], it is not difficult to chec k that for α = 0 there exist C 1 , C 2 > 0 such that C 1 e C 2 T Z T 0 | y ( t, 0) | 2 d t > k y 0 k 2 L 2 (0 , 1) ( T > 0 , y 0 ∈ D ( A )) . The las t formula is, a ccording to the ab ove definitions of A a nd B , equiv a lent to the inequality C 1 e C 2 T Z T 0 | B ∗ e sA y 0 | dt ≥ c k y 0 k L 2 (0 , 1) ( y 0 ∈ D ( A )) , (66) 17 so that we hav e indeed the ex act obse r v a bilit y in any time T > 0 for the undamp ed dynamics. T o chec k an energy estimate similar to the one in Lemma 2.1, o ne can first prove (13) for y 0 in the domain of the generator (which is done via integration by par ts). One can then check (using, for insta nce, a transfer function like in Guo and Sha o [11]) tha t the system ( A, B , B ∗ ) is well-po sed in the sense o f Salamon and W eiss (see [35]). ⋄ Sufficien t conditions for stro ng stability as those obtained in Theor e m 5.2 can be specified mo re precisely in the case of integral “excitations ”. Theorem 5.3. S upp o se t hat ther e exist c o nstants ρ, T 0 > 0 and a p ositive, c on- tinuous function c : (0 , ∞ ) → R such that for al l T ∈ (0 , T 0 ] , if for some ˜ α ∈ L ∞ ([0 , T ] , [0 , 1 ]) Z T 0 ˜ α ( t ) dt ≥ ρT then ˜ α ( · ) is of class K ( A, B , T , c ( T )) . L et ( a n , b n ) , n ∈ N , b e a se q uenc e of disjoint intervals in [0 , ∞ ) and α ∈ L ∞ ([0 , ∞ ) , [0 , 1]) . Assume that R b n a n α ( t ) dt ≥ ρ ( b n − a n ) and P ∞ n =1 c ( b n − a n ) = ∞ . Then the mild solut ion of (5) satisfies k z ( t ) k H → 0 as t → ∞ . Pr o of. In the ca se where sup n ∈ N ( b n − a n ) ≤ T 0 the conclusio n follows directly from Theorem 5.2. Now assume that for infinitely many n ∈ N , b n − a n > T 0 . Let n b e such that b n − a n > T 0 and split I n = ( a n , b n ) into finitely many pairwise disjoin t subin terv als I 1 n , . . . , I r n of common length l n ∈ [ T 0 / 2 , T 0 ]. Since P r j =1 R I j n α ( t ) dt ≥ ρ ( b n − a n ) = rρ l n , then ther e exists j ∈ { 1 , . . . , r } suc h that R I j n α ( t ) dt ≥ ρl n = ρ | I j n | . Denote I j n by ( a ′ n , b ′ n ). If n is such that b n − a n ≤ T 0 , set a ′ n = a n and b ′ n = b n . Again a pplying Theorem 5.2 to the sequence of interv als ( a ′ n , b ′ n ), n ∈ N , we can co nclude by showing that P ∞ n =1 c ( b ′ n − a ′ n ) = ∞ , since sup n ∈ N ( b ′ n − a ′ n ) < ∞ . The unboundedness o f P ∞ n =1 c ( b ′ n − a ′ n ) follows fro m the rema rk that, for infinitely many n ∈ N , c ( b ′ n − a ′ n ) ≥ min T ∈ [ T 0 / 2 ,T 0 ] c ( T ) > 0.  Example 5.2 . (W ave equation) Le t t 7→ z ( t ) = ( v ( t, · ) , v t ( t, · )) be a solution of the wa v e equation (22)–(25) where , as in Exa mple 3.1, Ω is a b ounded do ma in in R N , N ≥ 1 , d ∈ L ∞ (Ω) satisfies | d ( x ) | ≥ d 0 > 0 for almost all x ∈ Ω . (67) Consider T , ρ > 0 and some α ( · ) ∈ L ∞ ([0 , T ] , [0 , 1 ]) satisfying Z T 0 α ( t ) dt ≥ T ρ. (68) Using the s a me notation as in Example 3.1 a nd, in par ticula r, fixing a n initial condition and defining the set A ǫ n as in (27), w e ha ve, accor ding to (28), meas( A ǫ n ) ≥ T  1 − 2 ǫ π  − 2 ǫ min n ∈ N λ n for any ǫ ∈ (0 , 1). Without lo s s of generality , w e ca n assume that min n ∈ N λ n = λ 1 . F or T small enough, c hoo sing ¯ ǫ = ρλ 1 6 T , we ge t meas( A ǫ n ) ≥ T  1 − ρ 2  , leading to Z A ¯ ǫ n α ( t ) dt ≥ T ρ 2 , 18 F ALK M . HANTE, MARIO SIGALOTTI AND M ARIUS TUCSNAK bec ause of (68). The definition of A ¯ ǫ n yields the observ abilit y estimate Z T 0 α ( t )( − a n sin( p λ n t ) + b n cos( p λ n t )) 2 dt ≥ ¯ ǫ 2 ( a 2 n + b 2 n ) Z A ¯ ǫ n α ( t ) dt ≥ ρ 3 λ 2 1 72 T 3 ( a 2 n + b 2 n ) . Reasoning as in E xample 3.1, we obta in that the function c ( T ) app ear ing in the statement of Theor em 5.3 for the system (2 2)–(25) with uniform da mping (67) ca n be c hosen of o rder T 3 for T small. In particular, Theor e m 5.3 states that a sufficient condition for the strong as- ymptotic stability of the solutio ns of (22)–(25) with uniform damping (67 ) is that α ( · ) ∈ L ∞ ([0 , ∞ ) , [0 , 1]) s atisfies Z b n a n α ( t ) dt ≥ ρ ( b n − a n ) , n ∈ N , for so me po sitive constant ρ and some sequence ( a n , b n ), n ∈ N , of dis jo in t interv a ls in [0 , ∞ ) such that ∞ X n =1 ( b n − a n ) 3 = ∞ . F or the particular case of the w av e equation, the sufficient condition obtained here weakens the one considered in [14] where α ( · ) is b ounded aw ay from 0 by a constant m n on each in terv al ( a n , b n ) in or der to guar antee as ymptotic stability (cf. also Remark 3.1). ⋄ Example 5.3 . (Finite-dimensional linear systems) Let us characterize the function c ( · ) a ppea ring in the statement of Theorem 5.3 in the case of finite-dimensio nal systems, that is, when dim( H ) < ∞ . W e prove below that T b ehav es poly nomially and that its degree for T small c a n b e tak en equal to the sharp es tima te computed by Seidman in the case α ≡ 1 (see [28]). In the finite- dimensional case , the as s umption that A generates a strongly c o n- tin uous co nt raction semig roup is standardly weakened in to the re quirement that A is neutra lly stable , that is, its eig env alues are of non-p ositive r eal part and all Jordan blo cks corr esp onding to pure imag ina ry eigen v a lue s ar e triv ia l. Clearly , a necessa r y condition for ensur ing the conv ergence to the origin of all tra jector ies of the system ˙ x = Ax + αB u for s ome α = α ( t ) ∈ [0 , 1] is that the pair ( A, B ) is stabilizable. W e will make this assumption from now on. Up to a linea r c hange of v ar ia bles, A a nd B can be w r itten as A =  A 1 A 2 0 A 3  , B =  B 1 B 3  , where A 1 is Hurwitz and all the eigenv alues of A 3 are purely imaginary . F rom the neutral stability assumption and up to a further linea r change of co or dina tes, we may a ssume that A 3 is skew-symmetric. F ro m the stabilizability assumption o n ( A, B ), moreov er, we deduce that ( A 3 , B 3 ) is controllable. Setting x = ( x 1 , x 3 ) acc ording to the ab ov e decomp os ition, the system ˙ x = Ax + αB u can b e written as ˙ x 1 = A 1 x 1 + A 2 x 3 + α ( t ) B 1 u, (69) ˙ x 3 = A 3 x 3 + α ( t ) B 3 u. (70) Assume that, for a giv en α ( · ), all solutions o f (70) with u = − B ⊤ 3 x 3 conv erge to the origin. Then all tra jectories of system (69)-(70), with the choice of feedback u = − B ⊤ 3 x 3 , converge to the or igin, since (69) b ecomes an autonomous linear 19 Hurwitz system sub ject to a per tur bation whose norm conv erges to zero as time go es to infinit y . The previous dis c ussion allows us to fo cus on the sp ecial case where A is skew- symmetric and ( A, B ) is controllable. Denote b y K ( A,B ) the minimal non-negative integer such that rank[ B , AB , . . . , A K ( A,B ) B ] = N , (71) where N is the dimension of H . W e hav e the following result. Pr o p osition 5.4 . L et A b e skew-symmetric and ( A, B ) c ontr ollable. Then for ev- ery ρ > 0 t her e ex ists κ > 0 su ch that, for every T ∈ (0 , 1 ] and every α ∈ L ∞ ([0 , T ] , [0 , 1 ]) , if R T 0 α ( t ) dt ≥ ρT then α is of class K ( A, B , T , κT 2 K ( A,B ) +1 ) . Pr o of. Let K = K ( A,B ) and fix ρ > 0. W e should prov e that, for some κ > 0 , given any z 0 ∈ R n and an y α ∈ L ∞ ([0 , T ] , [0 , 1 ]) such that T ∈ (0 , 1] a nd R T 0 α ( t ) dt ≥ ρT , we hav e Z T 0 α ( t ) k B ⊤ e tA z 0 k 2 dt ≥ κT 2 K +1 k z 0 k 2 . Denote b y b 1 , . . . , b r the columns of B and assume, by c o ntradiction, that there exist ( T n ) n ∈ N ⊂ (0 , 1], ( z n 0 ) n ∈ N ⊂ R N with k z n 0 k = 1 , a nd ( α n ) n ∈ N ⊂ L ∞ ([0 , 1] , [0 , 1]) with R T n 0 α n ( t ) dt ≥ ρT n such that lim n →∞ κ n = 0 where κ n = R T n 0 α n ( t ) P r i =1 ( b ⊤ i e tA z n 0 ) 2 dt T 2 K +1 n , n ∈ N . Let β n ( t ) = α n ( T n t ) for n ∈ N a nd t ∈ [0 , 1]. Then R 1 0 β n ( t ) dt ≥ ρ and κ n = R 1 0 β n ( t ) P r i =1 ( b ⊤ i e tT n A z n 0 ) 2 dt T 2 K n , n ∈ N . By compactness , up to e x tracting a subseque nc e , T n → T ∞ in [0 , 1], z n 0 → z ∞ 0 in R N and β n ∗ ⇀ β ∞ in L ∞ ([0 , 1] , [0 , 1]). In particular , k z ∞ 0 k = 1, R 1 0 β ∞ ( t ) dt ≥ ρ and lim n →∞ κ n T 2 K n = Z 1 0 β ∞ ( t ) r X i =1  b ⊤ i e tT ∞ A z ∞ 0  2 dt = 0 . Assume first that T ∞ > 0. Then the a nalytic function t 7→ P r i =1  b ⊤ i e tT ∞ A z ∞ 0  2 annihilates o n the supp or t of β ∞ , whic h has p ositive mea sure, and is th us identically equal to zero, contradicting the controllability of the pair ( A, B ). Let then T ∞ = 0. Rewr ite κ n as κ n = Z 1 0 β n ( t ) r X i =1  c i,n 0 + tc i,n 1 + · · · + t K c i,n K + r i,n ( t )  2 dt, where c i,n j = b ⊤ i A j z n 0 j ! T K − j n , k r i,n k L ∞ (0 , 1) ≤ M T n , for some M > 0 only depending o n A , B , and K . Define the vector C n = ( c 1 ,n 0 , . . . , c 1 ,n K , c 2 ,n 0 , . . . , c 2 ,n K , . . . , c r,n 0 , . . . , c r,n K ) belo nging to R r ( K +1) . Since k z n 0 k = 1, T n ≤ 1, and bec ause of (71), there exists ν > 0 only dep ending on A a nd B such that k C n k ≥ ν . Thus, κ n ≥ ν 2 R 1 0 β n ( t ) P r i =1  c i,n 0 + tc i,n 1 + · · · + t K c i,n K + r i,n ( t )  2 dt k C n k 2 . 20 F ALK M . HANTE, MARIO SIGALOTTI AND M ARIUS TUCSNAK Up to extracting a subsequence, C n / k C n k conv erges in the unit sphere of R r ( K +1) . Denote its limit by ( γ 1 0 , . . . , γ r K ). Then t 7→ r X i =1  c i,n 0 + tc i,n 1 + · · · + t K c i,n K + r i,n ( t )  2 conv erges uniformly o n [0 , 1] to t 7→ P r i =1  γ i 0 + tγ i 1 + · · · + t K γ i K  2 . W e can co n- clude that Z 1 0 β ∞ ( t ) r X i =1  γ i 0 + tγ i 1 + · · · + t K γ i K  2 dt = 0 , leading to a co nt radiction, since β ∞ is nonzer o on a subset of [0 , 1] of po sitive measure and ( γ 1 0 , . . . , γ r K ) is a nonzero vector.  Prop ositio n 5.4 and Theorem 5.3 imply the following. Cor ol lary 5 .5 . L et A b e skew-symmetric and ( A, B ) b e c ontr ol lable. Then for every ρ > 0 , every α ∈ L ∞ ([0 , ∞ ) , [0 , 1]) such that ther e exist a se quenc e ( a n , b n ) , n ∈ N , of disjoint intervals in [0 , ∞ ) with R b n a n α ( t ) dt ≥ ρ ( b n − a n ) and P ∞ n =1 ( b n − a n ) 2 K ( A,B ) +1 = ∞ , and every solution z ( · ) of (5) c orr esp onding to α , we have k z ( t ) k R N → 0 as t → ∞ . ⋄ Ac kno wl edgments. This work was s upp or ted by the ANR grant ArHyCo, Pro - gram ARPE GE, contract n um be r ANR-2008 SEGI 004 0 1 -3001 1459 . The research presented in this a rticle w as mostly carr ied out while F. M. Hant e and M. 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