Controllability of linear and semilinear non-diagonalizable parabolic systems

ESAIM: COCV 21 (2015) 1178–1204 ESAIM: Con ol, Op imisa ion and Calculus o Va ia ions DOI: 10.1051/coc /2014063 www.esaim-coc .o g CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS ∗ En ique Fe n´ andez-Ca a1, Manuel Gonz´ alez-Bu gos1and Luz de Te esa2 Abs ac . This pape is conce ned wi h he con ollabili y o some (linea and semilinea ) non- diagonalizable pa abolic sys ems o PDEs. We will show ha he well known null con ollabili y p op- e ies o he classical hea equa ion a e also sa is ied by hese sys ems a leas when he e a e as many scala con ols as equa ions and some (maybe echnical) condi ions a e sa is ied. We will also show ha , in some pa icula si ua ions, he numbe o con ols can be educed. The minimal amoun is hen de e mined by a Kalman ank condi ion. Ma hema ics Subjec Classi ica ion. 93B05, 35K20. Recei ed Ma ch 15, 2012. Re ised Ap il 29, 2014. Published online July 6, 2015. 1. In oduc ion This pape deals wi h he con ollabili y p ope ies o some non-diagonalizable pa abolic sys ems. Le Ω⊂RNbe a non-emp y egula and bounded domain, le us fix T>0andle usse Q:= Ω×(0,T) and Σ:= ∂Ω ×(0,T). We will fi s conside he linea sys em ⎧ ⎨ ⎩ y −AΔy =M(x, )y+B 1ωin Q, y=0 on Σ, y(x, 0) = y0(x)inΩ, (1.1) whe e ω⊂Ωis a (small) open subdomain, A∈L(Rn),M∈L∞(Q;L(Rn)),B∈L(Rn;Rn)andy0∈L2(Ω;Rn). He e, =( 1,..., n)∗is he con ol, o be de e mined o example in L2(ω×(0,T); Rn), while y= (y1,...,y n)∗is he s a e. O cou se, he mos in e es ing si ua ion appea s when we a e able o con ol he sys em o la ge nand small n, since his means ha we go e n he beha io o many equa ions wi h ew Keywo ds and ph ases. Null con ollabili y, pa abolic, non-diagonalizable. ∗Suppo ed by G an MTM2010-15592 o D.G.E.S. (Spain) and p ojec IN101013 o D.G.A.P.A. (Mexico). 1Dp o, E.D.A.N., Uni e sidad de Se illa, Ap do. 1160, 41080 Se illa, Spain. [email p o ec ed] & [email p o ec ed] 2Ins i u o de Ma em´a icas, Uni e sidad Nacional Au ´onoma de M´exico, Ci cui o Ex e io , C.U. 04510 D.F. M´exico, Mexico. [email p o ec ed] A icle published by EDP Sciences c EDP Sciences, SMAI 2015 CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1179 con ols. Howe e , we will see ha he coupling h ough he non-diagonalizable ma ix Ain he highe o de e ms o he ope a o in oduces se ious difficul ies o con ol wi h n<n. The ollowing assump ion will be assumed h oughou his pape : ∃a0>0 such ha Aξ ·ξ≥a0|ξ|2∀ξ∈Rn. (1.2) No ice ha , i (1.2) is sa isfied, o e e y ∈L2(ω×(0,T); Rn) and e e y y0∈L2(Ω;Rn), (1.1) possesses a unique weak solu ion y,wi h y∈L2(0,T;H1 0(Ω;Rn)) ∩C0([0,T]; L2(Ω;Rn)); see Sec ion 2. Fo maybe echnical easons, we will also assume ha The dimensions o he Jo dan blocks o he canonical o m o Aa e ≤4. (1.3) I will be said ha (1.1)isnull-con ollable a ime Ti , o any y0∈L2(Ω;Rn), he e exis s ∈L2(ω× (0,T); Rn) such ha he associa ed solu ion sa isfies y(x, T )=0 in Ω. (1.4) Since (1.1) is linea , his is equi alen o he exac con ollabili y o he ajec o ies a ime T, ha is osay, o he ollowing p ope y: o any ajec o y y(i.e. any weak solu ion o (1.1) co esponding o an ini ial s a e y0∈L2(Ω;Rn) and he con ol ≡0) and any y0∈L2(Ω;Rn), he e exis s ∈L2(ω×(0,T); Rn) such ha he associa ed solu ion sa isfies y(x, T )=y(x, T )inΩ. (1.5) Consequen ly, he null con ollabili y o (1.1) also implies app oxima e con ollabili y, i.e. he ac ha , o any y0,y d∈L2(Ω;Rn)andanyε>0, he e exis s ∈L2(ω×(0,T); Rn) such ha he associa ed solu ion sa isfies y(·,T)−ydL2≤ε. The con ollabili y p ope ies o simila scala p oblems a e nowadays well known; see o in- s ance [18,19,22,26,31,32]. To be p ecise, le us conside he ollowing con ol sys em ⎧ ⎨ ⎩ z −Δz =u1ωin Q, z=0 on Σ, z(x, 0) = z0(x)in Ω. (1.6) Then, o e e y Ω,ωand T,(1.6) is null-con ollable a ime T;see[26,31]. To ou knowledge, almos all he pape s in he li e a u e de o ed o he con ollabili y o pa abolic non-scala sys ems o PDEs deal wi h dis ibu ed con ols; see o ins ance [3–5,14,27–29,35]. In hese pape s, mos esul s ha e been es ablished o 2 ×2 sys ems, wi h he con ol exe ed only on one equa ion. The bes achie emen s in his con ex seem o be hose in [4,5,28]. In [28], he au ho s s udy a cascade pa abolic sys em o nequa ions (n≥2) con olled wi h one single dis ibu ed con ol. In [4,5], he au ho s p o ide necessa y and sufficien condi ions o he con ollabili y o n×npa abolic linea sys ems wi h cons an o ime-dependen coefficien s. The analysis o simila bounda y con ollabili y p oblems has been he objec i e o [7,13,20]. A e iew o all hese esul s can be ound in [8]. I is an in e es ing ac ha , in he amewo k o he con ollabili y o coupled pa abolic sys ems, new (and possibly coun e -in ui i e) phenomena a ise: minimal ime o con ollabili y and dependence o he con ollabili y esul on he posi ion o he con ol domain ω;see[9–11,15]. 1180 E. FERN´ ANDEZ-CARA ET AL. Le us ecall one o he main esul s p o ed in [4]. Conside he p oblem ⎧ ⎨ ⎩ y −Δy =My+B 1ωin Q y=0 on Σ y(·,0) = y0in Ω (1.7) whe e M∈L(Rn;Rn), B∈L(Rn;Rn)(wi hn, n≥1) and y0∈L2(Ω;Rn). Le [M|B] be he ollowing ma ix in L(Rn×n;Rn): [M|B]=[B|MB|M2B|... |Mn−1B].(1.8) Then he ollowing holds: The linea sys em (1.7)is null con ollable i and only he so called Kalman’s ank condi ion ank [M|B]=n is sa is ied. In ha case, null con ollabili y holds a ime T>0. In his pape , ou fi s main esul is he ollowing. Theo em 1.1. Le A,M=M(x, )and Bbe as abo e. Assume ha (1.2)and (1.3)a e sa is ied and, also, n≥n, ank B=n. (1.9) Then (1.1)is null con ollable. In p ac ice, (1.9) means ha he e a e many scala con ols in he sys em (a leas as many as scala s a es) and, mo eo e , hei “ac ion” h ough Bcan ha e any di ec ion in he n-dimensional space Rn. The p oo elies on a (new) global Ca leman inequali y ha can be ob ained o he solu ions o he ela ed adjoin sys ems ⎧ ⎨ ⎩ −ϕ −A∗Δϕ =M(x, )∗ϕin Q ϕ=0 on Σ ϕ(x, T )=ϕT(x)inΩ (1.10) whe e ϕT∈L2(Ω;Rn); see Lemma 2.3. We ha e ied o explain ha assump ion (1.3) is necessa y o his a gumen in Rema k 2.4. A p esen , we ha e no been able o exclude i om he hypo heses. Rema k 1.2. Obse e ha assump ions (1.3)and(1.9) a e sufficien o ensu e he null con ollabili y p ope y o (1.1) o any diffusion and coupling ma ices Aand M=M(x, ). Assump ion (1.9) is no necessa y o p o ing he null con ollabili y o sys em (1.1). Indeed, in [6,27] he au ho s p o e he (local) null con ollabili y esul o phase-field models by one con ol o ce (n=2,n=1). Rema k 1.3. I n<nand Mis a L∞ma ix- alued unc ion, e en when Ais a mul iple o he iden i y, new phenomena can a ise. Mo e p ecisely, in [15] he au ho s p o e ha he app oxima e con ollabili y p ope y o a 2 ×2 linea sys em wi h A=Idepends on he posi ion o he con ol se ω. On he o he hand, i is es ablished in [10] ha , in he same amewo k, he null con ollabili y esul holds when he con ol ime T>0 is g ea e han a minimal ime T0which depends on he coefficien s o M. The null con ollabili y esul ails when T<T 0. Now, le us in oduce a locally Lipschi z-con inuous unc ion :Rn→ Rnand le us conside he semilinea sys em ⎧ ⎨ ⎩ y −AΔy = (y)+B 1ωin Q, y=0 on Σ, y(x, 0) = y0(x)inΩ. (1.11) CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1181 Again, we can speak o he null, exac o he ajec o ies o app oxima e con ollabili y p ope ies o (1.11). Fo ins ance, i will be said ha (1.11) is exac ly con ollable o he ajec o ies a ime Ti , o any y0∈ L2(Ω;Rn) and any weak solu ion yco esponding o ≡0, he e exis s a con ol ∈L2(ω×(0,T); Rn)and an associa ed solu ion y o (1.11) such ha (1.5) holds. Now, his p ope y is no equi alen bu s onge han null con ollabili y. On he o he hand, i is no difficul o see ha i also implies app oxima e con ollabili y. Ou second main esul in his pape is he ollowing: Theo em 1.4. Le Aand Bbe as in Theo em 1.1. Asume ha (1.2),(1.3)and (1.9)hold and :Rn→ Rnis globally Lipschi z-con inuous. (1.12) Then (1.11)is exac ly con ollable o he ajec o ies a any ime T. This esul can be deduced om Theo em 1.1 (o , mo e p ecisely, om he Ca leman inequali y in Lem. 2.3 below) using a gumen s ha a e nowadays well known; see [23]. Fo comple eness, we will p o ide he p oo in Sec ion 3. We will also see ha he asump ion (1.12) can be weakened so ha , in pa icula , some sligh ly supe linea sys ems a e con ollable and, in ac , he ac ion o he con ol can se e o a oid blow-up be o e =T. Le us come back o (1.1) and le us conside he pa icula case in which Mis cons an . In his si ua ion, i is possible o ob ain con ollabili y esul s also o n≤n,p o idedBsa isfies app op ia e condi ions. Mo e p ecisely, le us deno e by λ1,λ 2,... he eigen alues o he Di ichle Laplacian in Ωand le us ecall he no a ion (1.8). We hen ha e he ollowing heo em, which is he hi d main esul in his pape : Theo em 1.5. Le us assume ha A, M ∈L(Rn),B∈L(Rn;Rn)and (1.2)and (1.3)hold. Then (1.1)is null con ollable a ime Ti and only i he ollowing condi ion is sa is ied: ank [λiA−M|B]=n∀i≥1.(1.13) The same esul was es ablished in [5] in he case in which Ais diagonalizable. The e, an obse abili y inequali y o a linea (adjoin ) sys em is shown o be implied by a p ope y sa isfied by he solu ions o a high o de scala PDE. The p oo o Theo em 1.5 uses simila a gumen s; he de ails a e gi en in Sec ion 4. No ice ha , in o de o see whe he o no (1.13) holds, one only has o check a fini e amoun o inequali ies. This is because he λigo o +∞as i→+∞and, consequen ly, o ila ge enough, hey a e ou side he solu ion se o any algeb aic equa ion o he o m de Z(λ)=0, whe e Z(λ)isamino o [λiA−M|B]. An example o sys em ha ulfills he assump ions o Theo em 1.5 is he ollowing linea ized wo-phase solidifica ion model, see [33,34]: ⎧ ⎨ ⎩ θ −Δθ =1u +2w +m11θ+ 1ω, u −Δu =βθ +m22u+m23w, w −Δw =βθ +m32u+m33w. (1.14) He e, we assume ha β, heiand he mij a e posi i e cons an s. The unknowns θ,uand wcan be in e p e ed as he empe a u e and wo phase-field unc ions associa ed wi h wo diffe en kinds o solidifica ion p ocesses. I is no difficul o see ha (1.14)canbew i enin he o m(1.1), by eplacing u and in he fi s PDE. The esul is: ⎛ ⎝θ u w ⎞ ⎠−A⎛ ⎝Δθ Δu Δw ⎞ ⎠=M⎛ ⎝θ u w⎞ ⎠+⎛ ⎝ 1ω 0 0⎞ ⎠, 1182 E. FERN´ ANDEZ-CARA ET AL. whe e A=⎛ ⎝112 01 0 00 1 ⎞ ⎠,M=⎛ ⎝m11 +β(1+2)1m22 +2m32 1m23 +2m33 βm 22 m23 βm 32 m33 ⎞ ⎠. A simple compu a ion shows ha he condi ion (1.13)inTheo em1.5 is in his case independen o λiand is sa isfied i and only i m22 +m23 =m32 +m33. In he sequel, C,C0,C1, ... and Ra e used o deno e gene ic posi i e cons an s. F equen ly, i will be con enien o speci y he pa icula da a on which hey depend. The es o he pape is o ganized as ollows. In he nex sec ion, we p esen he p oo o Theo em 1.1. As men ioned abo e, he main ool o his p oo is a Ca leman inequali y o he solu ions o (1.10). This is es ablished by combining ca e ully app op ia e Ca leman es ima es o simila scala p oblems. In Sec ion 3, we gi e he p oo o Theo em 1.4. As o simila scala p oblems, his elies on a fixed-poin a gumen . Mo e p ecisely, we ew i e he con ollabili y p oblem o (1.11) as a fixed-poin equa ion o an adequa e mapping. I will be seen ha (1.12) (o some o he assump ion o his kind) is needed o bound uni o mly he solu ions, which jus ifies i s inclusion in he esul . In Sec ion 4, we gi e he p oo o Theo em 1.5. As men ioned abo e, he main ideas o he p oo ha e been adap ed om [4,5]; he main es ima es (again leading o app op ia e obse abili y inequali ies) a e es ablished no ing ha he componen s o he solu ions o he adjoin sys em sol e a scala PDE ha is o he n- h o de in and −Δ. Finally, Sec ion 5deals wi h some final commen s and open ques ions. 2. P oo o Theo em 1.1 In he eminde o his pape , we will deno e by dmax he maximal dimension o a Jo dan block o he canonical o m o A. By hypo hesis, dmax ≤4. The s a ing poin is a basic global Ca leman inequali y o he solu ions o scala ( eal- alued and complex- alued) pa abolic equa ions. Thus, le α0=α0(x) be a unc ion sa is ying α0∈C2(Ω),α 0>0inΩ, α0=0 on ∂Ω, |∇α0|>0inΩ ω. (2.1) Such a unc ion exis s, see [26]. Le us se ξ(x, )= eλα0(x) (T− ),α(x, )=e(λ+μ)α0L∞−eλα0(x) (T− ),ρ(x, )=e α(x, ),(2.2) whe e λ>0andμ>0. The ollowing no a ion will be used in o de o ab idge he es ima es: Im(s, λ;ψ):=Q ρ−2s(sξ)m−4(|ψ |2+|Δψ|2) +(sξ)m−2λ2|∇ψ|2+(sξ)mλ4|ψ|2 and Im,ω(s, λ;ψ):=ω×(0,T ) ρ−2s(sξ)mλ4|ψ|2 CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1183 o all s, λ > 0, o any in ege mand o any sufficien ly egula unc ion ψ=ψ(x, )wi h aluesinR,Co Rn. Le us conside he linea backwa ds in ime scala sys em ⎧ ⎨ ⎩ −ψ −Δψ =gin Q, ψ=0 on Σ, ψ(x, T )=ψT(x)inΩ, (2.3) whe e g∈L2(Q)andψT∈L2(Ω). In he ollowing esul , due o Fu siko and Imanu ilo [26], we ecall he basic global Ca leman es ima es o he solu ions o (2.3). Lemma 2.1. Fo any in ege m, he e exis cons an s sm,λmand Cmsuch ha , o any s≥smand any λ≥λm, he solu ions o (2.3)sa is y Im(s, λ;ψ)≤CmIm,ω(s, λ;ψ)+Q ρ−2s(sξ)m−3|g|2.(2.4) Fu he mo e, λmand Cmonly depend on m,Ωand ωand smcan be aken o he o m sm=σm(T+T2), whe e σmonly depends on m,Ωand ω. Fo a de ailed jus ifica ion o he exis ence and p ope ies o λm,smand Cm, see he p oo o Lemma 1.3 in [24]. Secondly, le us conside he simila complex- alued sys em ⎧ ⎨ ⎩ −ψ −(a+ib)Δψ =gin Q, ψ=0 on Σ, ψ(x, T )=ψT(x)inΩ, (2.5) whe e now a, b ∈R,a>0, g∈L2(Q;C)andψT∈L2(Ω;C). I is also possible o deduce global Ca leman es ima es o he solu ions o (2.5). They a e gi en in he ollowing lemma, whose p oo is essen ially gi en in Fu [25]. Lemma 2.2. Fo any in ege m, he e exis cons an s sm,λmand Cmsuch ha , o any s≥smand any λ≥λm, he solu ions o (2.5)sa is y (2.4).Fu he mo e,λmand Cmonly depend on m,Ω,ω,aand band smcan be aken o he o m sm=σm(T+T2),whe eσmonly depends on m,Ω,ω,aand b. Again, he exis ence and p ope ies o λm,smand Cma e jus ified by he a gumen s in [24]. Le us se g=g1+ig2and ψT=ηT+iζTin (2.5). By w i ing he solu ions in he o m ψ=η+iζ,wesee ha hey can also be ega ded as solu ions o he 2 ×2sys em −η ζ −a−b ba Δη Δζ =g1 g2, oge he wi h Di ichle bounda y and ini ial condi ions o ηand ζ. Le us deno e by M∞ he no m o Min L∞(Q;L(Rn)). Now, we p esen a Ca leman es ima e o he solu ions o he non-scala (adjoin ) p oblem (1.10): Lemma 2.3. Le he assump ions in Theo em 1.1 be sa is ied. Fo any in ege m, he e exis cons an s s m,λ m and C msuch ha , o any s≥s mand any λ≥λ m, he solu ions o (1.10)sa is y Im−3(s, λ;ϕ)≤C mIm,ω(s, λ;ϕ).(2.6) Fu he mo e, C monly depends on m,Ω,ωand A,λ monly depends on m,Ω,ω,Aand M∞and s mcan be aken o he o m s m=σ m(T+T2),whe eσ monly depends on m,Ω,ω,Aand M∞. 1184 E. FERN´ ANDEZ-CARA ET AL. P oo . In his p oo , we will deno e by C0a gene ic posi i e cons an only depending on Ω,ωand A. Fi s , no ice ha i can be assumed ha Ais w i en in he canonical o m. Indeed, he e exis s a non-singula ma ix P∈L(Cn) such ha A=PJP−1 o some J∈L(Cn)o he o m J=diag(J1,...,J s), whe e he Jia e he Jo dan blocks associa ed o he eigen alues μio A. By hypo hesis, we ha e (1.2)and his implies ha , o all i, Ji=⎡ ⎢ ⎣ μi1 μi1 μi1 μi ⎤ ⎥ ⎦(2.7) o a ma ix wi h he same shape and smalle dimension, wi h Re μi>0. The solu ions o (1.10) can be pu in co espondance wi h he solu ions o ⎧ ⎪ ⎨ ⎪ ⎩ −ψ −J∗Δψ =P∗M(x, )∗(P∗)−1ψin Q, ψ=0 on Σ, ψ(x, T )=P∗ϕT(x)inΩ, (2.8) h ough he change o a iable ϕ=(P∗)−1ψand, ob iously, i suffices o p o e (2.6) o ψ. Fo ins ance, le us assume ha , in (2.8), he fi s ou PDEs co espond o he same block and le us w i e hem in he o m −ψ1, −μΔψ1= n  j=1 ˜ M1j(x, )ψj, −ψ2, −μΔψ2= n  j=1 ˜ M2j(x, )ψj+Δψ1, −ψ3, −μΔψ3= n  j=1 ˜ M3j(x, )ψj+Δψ2, −ψ4, −μΔψ4= n  j=1 ˜ M4j(x, )ψj+Δψ3,(2.9) whe e he ˜ Mij (x, ) s and o he componen s o he ma ix P∗M(x, )∗(P∗)−1. Le us w i e (2.4) o ψ1,ψ2,ψ3and ψ4 espec i ely wi h m= 3, 2, 1 and 0. The ollowing is ound o all la ge sand λ: I3(s, λ;ψ1)≤C0I3,ω(s, λ;ψ1)+M2 ∞Q ρ−2s|ψ|2 , I2(s, λ;ψ2)≤C0I2,ω(s, λ;ψ2)+M2 ∞Q ρ−2s(sξ)−1|ψ|2+Q ρ−2s(sξ)−1|Δψ1|2 , I1(s, λ;ψ3)≤C0I1,ω(s, λ;ψ3)+M2 ∞Q ρ−2s(sξ)−2|ψ|2+Q ρ−2s(sξ)−2|Δψ2|2 , I0(s, λ;ψ4)≤C0I0,ω(s, λ;ψ4)+M2 ∞Q ρ−2s(sξ)−3|ψ|2+Q ρ−2s(sξ)−3|Δψ3|2 .(2.10) CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1185 The e o e, an app op ia e linea combina ion o he le hand sides can be used o con ol and abso b all he second-o de e ms in he igh . Indeed, i is clea ha (sξ)n1≤C(sξ)n2whene e n1≤n2. Acco dingly, i s≥σ0(T+T2) o someσ0only depending on Ωand ω,weha e: I3(s, λ;ψ1)+I2(s, λ;ψ2)+I1(s, λ;ψ3)+I0(s, λ;ψ4) ≤C0I3,ω(s, λ;ψ1)+I2,ω(s, λ;ψ2)+ I1,ω(s, λ;ψ3)+I0,ω(s, λ;ψ4)+C0M2 ∞Q ρ−2s|ψ|2. In o de o abso b as many e ms as possible in he las in eg al in he igh hand side, we do as ollows: (1) We ake s≥s3+C0M2/3 ∞T2and λ≥λ3; his makes i possible o skip |ψ1|2. (2) Then, we ake sand λas be o e and also sa is ying s≥s2+C0M∞T2and λ≥λ2;in hisway,wecan supp ess |ψ2|2. (3) Then, wi h sand λas in he p e ious s ep and also sa is ying s≥s1+C0M2 ∞T2and λ≥λ2, we can also skip |ψ3|2and, finally, (4) We choose sand λas in he p e ious s ep and also sa is ying s≥s0and λ≥λ0+C0M1/2 ∞,ino de oskip |ψ4|2. Hence, he e exis s 3and λ 3(as in he s a emen ) such ha , o all s≥s 3and λ≥λ 3, one has: I3(s, λ;ψ1)+I2(s, λ;ψ2)+I1(s, λ;ψ3)+I0(s, λ;ψ4) ≤C0I3,ω(s, λ;ψ1)+I2,ω(s, λ;ψ2)+ I1,ω(s, λ;ψ3)+I0,ω(s, λ;ψ4)+C0M2 ∞Q ρ−2s j≥5 |ψj|2.(2.11) Ob iously, simila es ima es can also be ob ained o he ψico esponding o any o he Jo dan block o equal o lowe dimension. I is also clea ha , i we choose sand e en ually λas indica ed, a e addi ion, we ge in he le hand side e ms ha can abso b all he ze o-o de e ms in he igh . The e o e, I0(s, λ;ψ)≤C0I3,ω(s, λ;ψ) o all s≥s 3,λ≥λ 3. This p o es he lemma o m=3. Wi h simila compu a ions, i is possible o p o e (2.6) o any o he in ege m. We skip he de ails, ha can be easily deduced om he p e ious a gumen .  Rema k 2.4. F om he p oo o his lemma, we see ha he bes possible choices o s mand λ ma e as ollows ( ecall ha dmax deno es he maximal dimension o a Jo dan block o A): •I dmax =1(i.e. Ais diagonalizable), hen we can choose s m=sm+C0M2/3 ∞T2and λ m=λm. •I dmax =2,wecan akes m=sm+C0M∞T2and λ m=λm. •I dmax = 3, hen we need s m=sm+C0M2 ∞T2and λ m=λm. •Finally, i dmax =4,weha e o akes m=smand λ m=λm+C0M2 ∞. Rema k 2.5. I is also clea ha a p oo o (2.6) o he same kind canno wo k when dmax ≥5. Indeed, i ( o ins ance) he fi s block is o dimension 5, he associa ed componen s ψ1, ..., ψ5a e coupled h ough second-o de e ms and we mus add o (2.9) a fi h PDE: −ψ5, −μΔψ5= n  j=1 ˜ M5j(x, )ψj+Δψ4. 1186 E. FERN´ ANDEZ-CARA ET AL. I we look o an es ima e o ψ5and we y o inco po a e a new Ca leman inequali y o (2.10), in iew o he e m Δψ4in he igh hand side, we a e o ced o ake m=−1. Bu hen he ze o-o de e m o ψ5 ha we ob ain in he le is Q ρ−2s(sξ)−1λ4|ψ5|2 and his is no sufficien o con ol he simila ze o-o de e m in he igh coming om he fi s Ca leman inequali y in (2.10). We can now achie e he p oo o Theo em 1.1. In he emainde o his sec ion, R( esp. C) deno es a ious posi i e cons an s only depending on Ω,ω,A and M∞( esp. Ω,ω,A,M∞and T). Fi s , ecall ha , in iew o classical a gumen s, he null con ollabil y o (1.1) is equi alen o he obse abili y o (1.10), ha is, o he es ima e ϕ(·,0)2 L2≤Cω×(0,T ) |B∗ϕ|2(2.12) o any solu ion o (1.10); o a de ailed explana ion, see o ins ance [24]. In iew o he assump ion (1.9), his is also equi alen o he simple es ima e ϕ(·,0)2 L2≤Cω×(0,T ) |ϕ|2.(2.13) The e o e, le us check ha he Ca leman inequali y (2.6), oge he wi h he usual pa abolic ene gy es ima es, imply (2.13) o someC. Indeed, le us ake ( o example) m=3,s=s 3and λ=λ 3in (2.6). In iew o he ene gy es ima es −d d ϕ2 L2+2a0∇ϕ2 L2≤Rϕ2 L2, we find ha ϕ(·,0)2 L2≤2 TeRT 3T/4 T/4 ϕ(·, )2 L2d . (2.14) The le hand side o (2.6) is bounded om below as ollows: I0(s 3,λ  3;ϕ)≥Q ρ−2s 3λ4 3|ϕ|2 ≥R3T/4 T/4 e−2s 3α∗( )ϕ(·, )2 L2d , (2.15) whe e α∗( ):=max Ω α(x, )≤R T2· Since s 3has he o m s 3=R(T+T2), he ollowing is ob ained: I0(s 3,λ  3;ϕ)≥e−R(1+ 1 T)3T/4 T/4 ϕ(·, )2 L2d and his inequali y, oge he wi h (2.14), yields: I0(s 3,λ  3;ϕ)≥e−R(1+T+1 T)ϕ(·,0)2 L2. CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1193 whe e he pij a e he componen s o Id ∂ +A∗Δ+M∗.Thus, P(∂ ,Δ)ϕ1= σ∈Pnpσ(2),2...p σ(n),n(pσ(1),1ϕ1) =− σ∈Pnpσ(2),2...p σ(n),n( n  j=2 pσ(1),jϕj) =− n  j=2  σ∈Pnpσ(1),jpσ(2),2...p σ(n),nϕj:= − σ∈Pn ˜pjϕj. Bu all he ope a o s in his las sum anish, since each o hem can be w i en as he de e minan o a squa e ma ix wi h wo columns ha a e iden ical. Consequen ly, we ce ainly ha e P(∂ ,Δ)ϕ1=0. Ob iously, his a gumen also holds o ϕ2,...,ϕ n.Thus,wecanw i e(4.10) o any componen o B∗ϕ. This gi es he ollowing inequali y o all j, k ≥0andall=1,...,n : Q ρ−2s|(B∗((−Δ)k∂j ϕ))|2=Q ρ−2s|(−Δ)k∂j (B∗ϕ)|2 ≤C(k,j)ω×(0,T ) |(B∗φ)|2. Le us fix k≥0. By in oducing ˆρ( ):=max Ω ρ(x, ), and ecalling (4.4), we ge Q ˆρ( )−2s|(−Δ)kK∗ϕ|2≤C n  =1 Q ρ−2s|(B∗((−Δ)k∂j ϕ)|2 ≤C n  =1 C(k,j)ω×(0,T ) |(B∗ϕ)|2 ≤Cω×(0,T ) |B∗ϕ|2. This yields (4.3). 4.3. Conclusion and end o he p oo o Theo em 1.5 In iew o (4.2)and(4.3) o k=2(n−1)2, he ollowing holds o any solu ion o (1.10) associa ed o a final da a ϕTsa is ying (4.13): Q ˆρ( )−2s|ϕ|2=T 0 ˆρ( )−2sΩ |ϕ(x, )|2dxd ≤R(k)Q ˆρ( )−2s|(−Δ)k(K∗ϕ)|2 ≤CR(k)ω×(0,T ) |B∗ϕ|2. 1194 E. FERN´ ANDEZ-CARA ET AL. In pa icula , Q ˆρ( )−2s|ϕ|2≤Cω×(0,T ) |B∗ϕ|2. By densi y, i is ob ious ha his inequali y emains ue o he solu ions o (1.10) co esponding o a bi a y final da a in L2(Ω;Rn). Hence, a guing as in he final pa o he p oo o Theo em 1.1 in Sec ion 2,wege (4.1), which shows ha , unde he assump ions o Theo em 1.5,(1.1) is null-con ollable. 4.4. P oo o Lemma 4.1 We will adap he a gumen s in [5]. The ac ha Ais no diagonalizable in oduces some non i ial compli- ca ions and he compu a ions and es ima es a e mo e in ol ed; bu he idea is simila . We will p o e (4.10) by induc ion on kand j. 4.4.1. S ep 1: P oo o (4.10) o k=j=0 Le us see ha , i sand λa e la ge enough, one has I12(s, λ;φ)≤C(0,0) ω(0,0)×(0,T) ( (T− ))−m(0,0)ρ−2s|φ|2(4.14) o some m(0,0), C(0,0) and ω(0,0). Again,i canbeassumed ha Ais in he Jo dan canonical o m. We hen ha e o some p≥1 Id ∂ +A∗Δ+M∗=⎡ ⎢ ⎢ ⎢ ⎣ H1(∂ ,Δ)M∗ 21 ... M∗ p1 M∗ 12 H2(∂ ,Δ)... M∗ p2 . . .. . ..... . . M∗ 1pM∗ 2p...H p(∂ ,Δ) ⎤ ⎥ ⎥ ⎥ ⎦, whe eweha ein oduced henon-scala ope a o sHi(∂ ,Δ):=Id ∂ +J∗ iΔ+M∗ ii, heJ∗ ia e Jo dan blocks, i.e. each o hem is o he o m (2.7) o someμi∈Cand he Mij p o ide he co esponding block decomposi ion o M. The PDE (4.12) can be w i en in he o m p  i=1 de Hi(∂ ,Δ)φ=F(φ).(4.15) In he e ms in F(φ) we find he composi ion o a mos p−2 ope a o s o he kind de Hj(∂ ,Δ) applied o φ. Le us in oduce he unc ions ψi,wi h ψ1=φ, ψ2=de H1(∂ ,Δ)ψ1, ..., ψ p=de Hp−1(∂ ,Δ)ψp−1. Then we can ew i e (4.15) as he ollowing sys em o he ψi: ⎧ ⎪ ⎨ ⎪ ⎩ de Hp(∂ ,Δ)ψp=F(φ), de Hp−1(∂ ,Δ)ψp−1=ψp, ... ... ... de H1(∂ ,Δ)φ=ψ2. (4.16) Recall ha , by hypo hesis, we also ha e φ=ψ2=...=ψp=0 on Σ. (4.17) We will now p o ide global es ima es o ψpand i s de i a i es in e ms o local es ima es o ψpand (lowe o de ) es ima es o F(φ); hen, global es ima es o ψp−1and i s de i a i es in e ms o local es ima es o ψp−1 CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1195 and lowe o de es ima es o ψp;e c. And, finally, global es ima es o φand i s de i a i es in e ms o local es ima es o φand (lowe o de ) es ima es o ψ2. An app op ia e combina ion o hese es ima es will lead o an inequali y whe e we find, in he le hand side, global weigh ed in eg als o φ,ψ2,... and,in he igh , only local in eg als and a global weigh ed in eg al o |F(φ)|2. Thus, le us conside he fi s PDE in (4.16). Fo ins ance, assume ha Jpis a Jo dan block o dimension associa ed o he complex eigen alue αwi h α>0andle usdeno ebyη1,...,η he diagonal componen s o Mpp; by assump ion, ≤4. Then, his PDE can be ew i en in he o m  i=1 (∂ +αΔ +ηi)ψp=F(φ)−G(ψp),(4.18) whe e G(ψp) is a linea combina ion o pa ial de i a i es o ψp. Le us in oduce he new a iables ζ1=ψp,ζ 2=(∂ +αΔ +η )ζ1, ..., ζ =(∂ +αΔ +η2)ζ −1. Now, we can ew i e (4.18) as a fi s -o de sys em o he ζi: ⎧ ⎪ ⎨ ⎪ ⎩ (∂ +αΔ +η1)ζ =F(φ)−G(ψp), (∂ +αΔ +η2)ζ −1=ζ , ... ... ... (∂ +αΔ +η )ζ1=ζ2. (4.19) Again, we ha e in o ma ion on he ζion he la e al bounda y: ζ1=...=ζ =0 on Σ. (4.20) In he “wo s ” possible case, we ha e =4and(4.19)and(4.20) espec i ely ead ⎧ ⎪ ⎨ ⎪ ⎩ (∂ +αΔ +η1)ζ4=F(φ)−G(ψp), (∂ +αΔ +η2)ζ3=ζ4, (∂ +αΔ +η3)ζ2=ζ3, (∂ +αΔ +η4)ζ1=ζ2, (4.21) and ζ1=ζ2=ζ3=ζ4=0 on Σ. (4.22) No ice ha |G(ψp)|2is bounded by a sum o squa es o de i a i es o ψp. Mo e p ecisely, we ha e |G(ψp)|2≤ CIG(ψp), wi h IG(ψp):= 3  a=0 |(−Δ)aψp|2+ 4  j=1 2  b=0 |(−Δ)b(∂ +αΔ +ηj)ψp|2 + 4  j,k=1 1  c=0 |(−Δ)c(∂ +αΔ +ηj)(∂ +αΔ +ηk)ψp|2. In iew o he Ca leman es ima es (2.4) (es ablished in Lem. 2.2) applied o he unc ions ζi,weha e: I3(s, λ;ζ4)≤CQ ρ−2s|F(φ)|2+IG(ψp)+CI3,ω0(s, λ;ζ4), λ4I6(s, λ;ζ3)≤CQ ρ−2s(sξ)3λ4|ζ4|2+Cλ4I6,ω0(s, λ;ζ3), λ8I9(s, λ;ζ2)≤CQ ρ−2s(sξ)6λ8|ζ3|2+Cλ8I9,ω0(s, λ;ζ2), λ12I12(s, λ;ζ1)≤CQ ρ−2s(sξ)9λ12|ζ2|2+Cλ12I12,ω0(s, λ;ζ1), o any λ≥λ0and s≥s0=σ0(T+T2). 1196 E. FERN´ ANDEZ-CARA ET AL. Consequen ly, an app op ia e linea combina ion o he e ms in he le hand sides abso bes he global weigh ed in eg als o |ζ4|2,|ζ3|2and |ζ2|2.Mo ep ecisely, o s≥s0and λ≥λ0,wege : I3(s, λ;ζ4)+λ4I6(s, λ;ζ3)+λ8I9(s, λ;ζ2)+λ12I12(s, λ;ζ1) ≤C(I3,ω0(s, λ;ζ4)+λ4I6,ω0(s, λ;ζ3)+λ8I9,ω0(s, λ;ζ2)+λ12I12,ω0(s, λ;ζ1)) +CQ ρ−2s|F(φ)|2+CQ ρ−2sIG(ψp).(4.23) The nex ask will be o add some ex a e ms on he le hand side o he p e ious inequali y. To his end, we eason as ollow. We apply −Δ o he second PDE in (4.21) and we use Lemma 2.1 o he esul ing equa ion wi h m= 2. No ice ha his is possible, since −Δζ3=0onΣ. We find: I2(s, λ;Δζ3)≤Cω0×(0,T) ρ−2s(sξ)2λ4|Δζ3|2)+CQ ρ−2s(sξ)−1|Δζ4|2 ≤Cλ4I6(s, λ;ζ3)+I3(s, λ;ζ4). Obse e ha he p e ious a gumen can be applied, his ime, o he hi d and ou h PDE in (4.21). Applying −Δand using Lemma 2.1 o he co esponding equa ions wi h m=5andm= 8, we deduce: λ4I5(s, λ;Δζ2)≤Cω0×(0,T ) ρ−2s(sξ)5λ8|Δζ2|2)+CQ ρ−2s(sξ)2λ4|Δζ3|2 ≤Cλ8I9(s, λ;ζ2)+λ4I6(s, λ;ζ3) and λ8I8(s, λ;Δζ1)≤Cω0×(0,T) ρ−2s(sξ)8λ12|Δζ1|2)+CQ ρ−2s(sξ)5λ8|Δζ2|2 ≤Cλ12I12(s, λ;ζ1)+λ8I8(s, λ;ζ2). Then, we can add all hese new e ms o he le hand side o (4.23) and, o a new posi i e cons an C, ob ain I3(s, λ;ζ4)+λ4I6(s, λ;ζ3)+λ8I9(s, λ;ζ2)+λ12I12(s, λ;ζ1) +I2(s, λ;Δζ3)+λ4I5(s, λ;Δζ2)+λ8I8(s, λ;Δζ1) ≤C(I3,ω0(s, λ;ζ4)+λ4I6,ω0(s, λ;ζ3)+λ8I9,ω0(s, λ;ζ2)+λ12I12,ω0(s, λ;ζ1)) +CQ ρ−2s|F(φ)|2+CQ ρ−2sIG(ψp).(4.24) We can con inue he p e ious p ocess and add be e global e ms on he le hand side o (4.24). Thus, i we apply (−Δ)2 o he hi d PDE in (4.21) and we use again Lemma 2.1 o he co esponding equa ions wi h m=1,wege I1(s,λ;(−Δ)2ζ2)≤Cω0×(0,T) ρ−2ssξλ4|(−Δ)2ζ2|2)+CQ ρ−2s(sξ)−2|Δζ3|2 ≤Cλ4I5(s, λ;Δζ2)+I2(s, λ;Δζ3). The p e ious a gumen , his ime applied o he las PDE in (4.21), also gi es λ4I4(s, λ;(−Δ)2ζ1)≤Cλ8I8(s, λ;Δζ1)+λ8I5(s, λ;Δζ2). CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1197 Pu ing he p e ious inequali ies in (4.24), we ob ain I3(s, λ;ζ4)+λ4I6(s, λ;ζ3)+λ8I9(s, λ;ζ2)+λ12I12(s, λ;ζ1) +I2(s, λ;Δζ3)+λ4I5(s, λ;Δζ2)+λ8I8(s, λ;Δζ1) +I1(s, λ;(−Δ)2ζ2)+λ4I4(s, λ;(−Δ)2ζ1) ≤C(I3,ω0(s, λ;ζ4)+λ4I6,ω0(s, λ;ζ3)+λ8I9,ω0(s, λ;ζ2)+λ12I12,ω0(s, λ;ζ1)) +CQ ρ−2s|F(φ)|2+CQ ρ−2sIG(ψp).(4.25) Finally, le us ake (−Δ)3 o he las PDE o (4.21). Again, om he egula i y assump ions on φwe ha e (−Δ)3ζ1=0onΣ. So, we can apply Lemma 2.1 wi h m= 0 o he esul ing PDE and deduce I0(s, λ;(−Δ)3ζ1)≤Cλ4I4(s, λ;(−Δ)2ζ1)+I1(s, λ;(−Δ)2ζ2). This inequali y oge he wi h (4.25)p o ides I3(s, λ;ζ4)+λ4I6(s, λ;ζ3)+λ8I9(s, λ;ζ2)+λ12I12(s, λ;ζ1) +I2(s, λ;Δζ3)+λ4I5(s, λ;Δζ2)+λ8I8(s, λ;Δζ1) +I1(s, λ;(−Δ)2ζ2)+λ4I4(s, λ;(−Δ)2ζ1) +I0(s, λ;(−Δ)3ζ1) ≤C(I3,ω0(s, λ;ζ4)+λ4I6,ω0(s, λ;ζ3)+λ8I9,ω0(s, λ;ζ2)+λ12I12,ω0(s, λ;ζ1)) +CQ ρ−2s|F(φ)|2+CQ ρ−2sIG(ψp), o a new posi i e cons an C. Le us deno e by J o (s, λ;ζ) he sum o he e ms on he le hand side o he p e ious inequali y (wi h ζ=(ζ1,ζ 2,ζ 3,ζ 4)). Then, he p e ious inequali y can be w i en as J o (s, λ;ζ)≤C(I3,ω0(s, λ;ζ4)+λ4I6,ω0(s, λ;ζ3)+λ8I9,ω0(s, λ;ζ2) +λ12I12,ω0(s, λ;ζ1)) + CQ ρ−2s|F(φ)|2+Q ρ−2sIG(ψp),(4.26) wi h s≥s1=σ1(T+T2)andλ≥λ1. Le ω(0,0) be an open se sa is ying ω0⊂⊂ ω(0,0) ⊂⊂ ω1.F omnowon,wefixs≥s1and λ≥λ1and we y o eplace he local e ms in (4.26) co esponding o ζ2,ζ3and ζ4by a e m o he o m (ψp=ζ1) Cλ1ω(0,0)×(0,T) ρ−2s(sξ)2|ψp|2, whe e 1and 2a e nonnega i e in ege s. We need some leng hy compu a ions, bu using he cascade s uc u e o he sys em (4.21), he p ocess is clea . Fo ins ance, le us see wha can be done wi h he local e m co esponding o ζ4. Fi s , we in oduce an open subse ω0,wi hω0⊂⊂ ω0⊂⊂ ω(0,0), and a cu -off unc ion χ=χ(x)wi h χ∈C∞ 0(ω0), χ≥0andχ≡1inω0and we w i e ζ4=(∂ +αΔ +η2)ζ3(see (4.21)) I3,ω0(s, λ;ζ4)≤λ4ω0×(0,T) ρ−2s(sξ)3χ|ζ4|2 =λ4ω0×(0,T) ρ−2s(sξ)3χ(∂ +αΔ +η2)ζ3ζ4=I1+I2+I3. 1198 E. FERN´ ANDEZ-CARA ET AL. Le us ake μin (2.2)insuchawayweha e e(λ+μ)α0L∞−eλα0(x)≥eλα0(x)≥1,∀x∈Ω. Then, we also ha e |∇ ρ−2s(sξ)|≤Cλρ−2s(sξ)+1,|Δρ−2s(sξ)|≤Cλ2ρ−2s(sξ)+2, ∂ (ρ−2s(sξ))≤Cρ−2s(sξ)+2 (4.27) o any (x, )∈Qand s≥s1and λ≥λ1. Using he p e ious inequali ies, we deduce I1=−λ4ω0×(0,T) ∂ ρ−2s(sξ)3χζ3ζ4−λ4ω0×(0,T ) ρ−2s(sξ)3χζ3∂ ζ4 ≤Cλ4ω0×(0,T) ρ−2s(sξ)5χζ3ζ4+λ4ω0×(0,T ) ρ−2s(sξ)3χζ3∂ ζ4 ≤εI3(s, λ;ζ4)+C ελ8ω0×(0,T ) ρ−2s(sξ)7|ζ3|2, wi h ε>0. We can also bound I2=αλ4ω0×(0,T) Δρ−2s(sξ)3χζ4ζ3 =αλ4ω0×(0,T ) [Δρ−2s(sξ)3ζ4+2∇ρ−2s(sξ)3·∇ ζ4]χζ3 +αλ4ω0×(0,T ) ρ−2s(sξ)3Δζ4χζ3+... whe e he do s con ain e ms o lowe o de . Now, we apply he Cauchy-Schwa z inequali y and inequali y (4.27). So, I2≤εI3(s, λ;ζ4)+C ελ8ω0×(0,T ) ρ−2s(sξ)7|ζ3|2. Finally, I3≤εI3(s, λ;ζ4)+C ελ4ω0×(0,T ) ρ−2s(sξ)3|ζ3|2. Pu ing he p e ious inequali ies oge he , we ge I3,ω0(s, λ;ζ4)≤εI3(s, λ;ζ4)+C ελ4ω0×(0,T ) ρ−2s(sξ)7|ζ3|2. The p e ious inequali y is alid o any ε>0, s≥s1and λ≥λ1. Coming back o (4.26), i we ake εsmall enough, we ob ain J o (s, λ;ζ)≤C(λ8I7,ω0(s, λ;ζ3)+λ8I9,ω0(s, λ;ζ2)+λ12I12,ω0(s, λ;ζ1)) +CQ ρ−2s|F(φ)|2+Q ρ−2sIG(ψp) o a new posi i e cons an C. CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1199 Obse e ha he p e ious easoning can be applied wice in o de o elimina e he local e ms co esponding o ζ3and ζ2. The esul ing inequali y is J o (s, λ;ζ)≤Cλ1ω1×(0,T) ρ−2s(sξ)2|ψp|2+CQ ρ−2s|F(φ)|2+Q ρ−2sIG(ψp),(4.28) wi h s≥s1=σ1(T+T2)andλ≥λ1. In inequali y (4.28)ω1is a new open subse sa is ying ω0⊂⊂ ω1⊂⊂ ω(0,0) and 1and 2a e nonnega i e in e ge s. Now, aking in o accoun ha he ope a o s ∂ +αΔ +ηicommu e, we see ha (4.18)( o =4)canbe ew i en equi alen ly in he o m 4  i=1 (∂ +αΔ +ησ(i))ψp=F(φ)−G(ψp), whe e σis any pe mu a ion in P4. Hence, we can in oduce he new a iables ζσ 1=ψp,ζ σ 2=(∂ +αΔ +ησ(4))ζσ 1, ..., ζσ 4=(∂ +αΔ +ησ(2))ζσ 3 and we can also w i e (4.18) as a simila fi s -o de sys em o he ζσ i: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (∂ +αΔ +ησ(1))ζσ 4=F(φ)−G(ψp). (∂ +αΔ +ησ(2))ζσ 3=ζσ 4, (∂ +αΔ +ησ(3))ζσ 2=ζσ 3, (∂ +αΔ +ησ(4))ζσ 1=ζσ 2. (4.29) Again, we ha e: ζσ 1=ζσ 2=ζσ 3=ζσ 4=0 on Σ. (4.30) A guing as be o e, we ob ain an es ima e like (4.28) whe e, now, we ha e in he le global weigh ed in eg als o ζσ 1=ψp,ζσ 2,ζσ 3and ζσ 4and, in he igh , e ms conce ning F(φ)andIG(ψp): J o (s, λ;ζσ)≤Cλ1ω1×(0,T) ρ−2s(sξ)2|ψp|2 +CQ ρ−2s|F(φ)|2+Q ρ−2sIG(ψp). In his inequali y we ha e used he no a ion ζσ=(ζσ 1,ζσ 2,ζσ 3,ζσ 1). Le us deno e by I o (s, λ;ψp) he sum o all hese le hand sides, ob ained o all σ∈P 4.Then I o (s, λ;ψp)≤Cλ1ω1×(0,T) ρ−2s(sξ)2|ψp|2 +CQ ρ−2s|F(φ)|2+Q ρ−2sIG(ψp). Obse e ha all he e ms in IG(ψp) excep |(−Δ)3ψp|2a e also in he le mul iplied by weigh s o he o m (sξ)aρ−2swi h a>0. Consequen ly, o sufficien ly la ge s, hese e ms a e abso bed and we find: I o (s, λ;ψp)≤Cλ1ω1×(0,T ) ρ−2s(sξ)2|ψp|2 +CQ ρ−2s|F(φ)|2+CQ ρ−2s|(−Δ)3ψp|2. 1200 E. FERN´ ANDEZ-CARA ET AL. Finally he global e m co esponding o ρ−2s|(−Δ)3ψp|2can be abso bed by he e m I0(s, λ;(−Δ)3ψp) (appea ing in he exp ession o I o (s, λ;ψp)) aking λla ge enough. Hence, I o (s, λ;ψp)≤Cλ1ω1×(0,T) ρ−2s(sξ)2|ψp|2 +CQ ρ−2s|F(φ)|2,∀s≥s2=σ2(T+T2),λ≥λ2, o some n(0,0),m(0,0) ≥1. F om now on, we fix λ=λ2. Le us now conside he second PDE in (4.16). A guing in he same way (and assuming again ha we a e in he wo s possible si ua ion, associa ed o a block o dimension 4), we deduce he ollowing es ima e o ψp−1: I12(s, λ;ψp−1)≤CQ ρ−2s|ψp|2+ω1×(0,T ) (sξ)2ρ−2s|ψp−1|2. The co esponding simila es ima e also holds o ψp−2,e c. Thus, a e addi ion and aking in o accoun ha ψ1=φand he global in eg als o ψp,..., ψ2in he igh hand side a e smalle han he e ms in he le , we ge an es ima e o all he ψi: p  i=1 I12(s, λ;ψi)≤CQ ρ−2s|F(φ)|2+ω1×(0,T ) (sξ)2ρ−2s|φ|2+ p  i=2 ω1×(0,T ) (sξ)2ρ−2s|ψi|2. Again, using he cascade s uc u e o sys em (4.16), all he local in eg als in he igh can be abso bed by he le hand side, wi h he excep ion o he local weigh ed in eg al o |φ|2. All we ha e o do is o enla ge he open se ω1and a gue like in he passage om (4.26) o(4.28). The e o e, he ollowing is ob ained: p  i=1 I12(s, λ;ψi)≤CQ ρ−2s|F(φ)|2+ω(0,0)×(0,T) (sξ)m(0,0)ρ−2sχ|φ|2. Howe e , we see ha , aking in o accoun ha he ope a o s de Hi(∂ ,Δ)commu e,(4.16) can also be w i enin he o m p  i=1 de Hσ(i)(∂ ,Δ)φ=F(φ), whe e σis any pe mu a ion in Pn. This means ha ano he equi alen o mula ion o (4.12)is ⎧ ⎪ ⎨ ⎪ ⎩ de Hσ(p)(∂ ,Δ)ψσ p=F(φ), de Hσ(p−1)(∂ ,Δ)ψσ p−1=ψσ p, ... ... ... de Hσ(1)(∂ ,Δ)φ=ψσ 2, and we can also ge an es ima e o he same o m whe e, now, we ha e in he le global weigh ed in eg als o φ,ψσ 2, ..., ψσ p. Recall ha F(φ) is a sum o e ms whe e, a mos , p−2 ope a o s o he kind de Hj(∂ ,Δ) a e applied o φ.Sinceσis a bi a y in Pn, using all hese possible es ima es oge he and a guing as abo e, i becomes also clea ha he e ms con aining |F(φ)|2can be con olled by he e ms in he le . This p o es (4.14). CONTROLLABILITY OF LINEAR AND SEMILINEAR NON-DIAGONALIZABLE PARABOLIC SYSTEMS 1201 4.4.2. S ep 2: Induc ion on kand j Now, le us assume ha (4.10) is ue o any k=0,1,...,k,anyj=0,1,...,j and any solu ion o (4.12) sa is ying he assump ions o he lemma and le us p o e (4.10) ( o ins ance) wi h k eplaced by k+1; he p oo wi h he same kand j eplaced by j+ 1 is essen ially he same. Since ˆ φ:= (−Δ)φalso sa isfies (4.11)and(4.12), we ha e by hypo hesis I12(s, λ;(−Δ)k∂j ˆ φ)≤C(k,j)ω(k,j)×(0,T ) ( (T− ))−m(k,j)ρ−2s|ˆ φ|2, ha is, I12(s, λ;(−Δ)k+1∂j φ)≤C(k,j)ω(k,j)×(0,T ) ( (T− ))−m(k,j)ρ−2s|Δφ|2.(4.31) Le us se ˜m=m(k,j)and˜ω=ω(k,j), le ω∗be an open se sa is ying ˜ω⊂⊂ ω∗⊂⊂ ωand le χ∗=χ∗(x) be a new cu -off unc ion, wi h χ∗∈C∞ 0(ω∗), χ∗≥0andχ∗≡1in˜ω. Then, o some in ege m∗≥m(k,j), one has ω(k,j)×(0,T ) ( (T− ))−m(k,j) ρ−2s|Δφ|2 ≤ω∗ ×(0,T) ( (T− ))−m(k,j) ρ−2sχ∗|Δφ|2 =ω∗ ×(0,T) ( (T− ))−m(k,j) ρ−2sχ∗φ(−Δ)2φ+... ≤ω∗ ×(0,T) ( (T− ))−m∗ ρ−2sχ∗ |φ|2+CI12(s, λ;Δφ)+... ≤ω∗ ×(0,T) ( (T− ))−m∗ ρ−2sχ∗ |φ|2 +Cω∗ ×(0,T) ( (T− ))−m(1,0) ρ−2s|φ|2+..., whe e he do s deno e again lowe o de e ms. I is hus clea ha he e exis m(k+1,j), C(k+1,j)andω(k+ 1,j) such ha I12(s, λ;(−Δ)k+1∂j φ)≤C(k+1,j) ω(k+1,j)×(0,T ) ( (T− ))−m(k+1,j) ρ−2s|φ|2. This ends he p oo o he lemma.  5. Fu he commen s and open ques ions This sec ion is de o ed o make some commen s on ex ensions o he p e ious esul s and, also, o epo some ela ed open p oblems. The fi s ques ion is whe he he hypo hesis (1.3) can be elimina ed o a leas weakened in Theo ems 1.1 and 1.4. This is no clea ; o cou se, as no iced in Rema k 2.4, wi hou new ools o he p oo o (2.13), diffe en om global (scala ) Ca leman es ima es, i seems e y difficul o ob ain he same esul s o dmax ≥5. In he one-dimensional case, when he ma ix Mdoes no depend on , a possible al e na i e is he e o mula ion o he null con ollabili y p oblem o (1.1) as a momen p oblem. This has been done in some ecen pape s (see o ins ance [7,11,13,20]) gi en bounda y con ollabili y cha ac e iza ions o some coupled pa abolic sys ems. Recall ha he analysis and me hods in Sec ion 2can be pe o med o he mo e gene al coupled sys ems ⎧ ⎪ ⎨ ⎪ ⎩ y −ASy =M(x, )y+ N k=1 Wk(x, )∂ky+ 1ωin Q, y=0 on Σ, y(x, 0) = y0(x)inΩ 1202 E. FERN´ ANDEZ-CARA ET AL. and ⎧ ⎪ ⎨ ⎪ ⎩ y −ASy =M(x, )y+ N k=1 ∂k(Wk(x, )y)+ 1ωin Q, y=0 on Σ, y(x, 0) = y0(x)inΩ, whe e Asa isfies (1.2)and dmax ≤2,(5.1) Sis a second-o de pa ial diffe en ial ope a o o he o m Sz := N  i,j=1 aij(x, )∂i∂jz wi h aij =aji ∈W1,∞(Q) o all i, j and N  i,j=1 aij (x, )ηiηj≥a0|ξ|2∀η∈RN,∀(x, )∈Q, a0>0 and Wk∈L∞(Q;L(Rn;Rn)) o all k=1,...,N. This also leads o con ollabili y esul s o semilinea sys ems o a mo e gene al class han (1.11). Mo e p ecisely, le us conside he sys em ⎧ ⎨ ⎩ y −AΔy =F(y,∇y)+ 1ωin Q, y=0 on Σ, y(x, 0) = y0(x)inΩ, (5.2) whe e F:Rn×Rn×N→ Rnis gi en. We hen ha e he ollowing esul : Theo em 5.1. Le Aand Bbe as in Theo em 1.1 and asume ha (1.2)and (5.1)hold and F:Rn×Rn×N→ Rnis globally Lipschi z-con inuous. Then (5.2)is exac ly con ollable o he ajec o ies. The p oo is simila o he p oo o Theo em 1.4. The de ails a e le o he eade ; hey ely on he ideas in [30], see he p oo o Theo em 1.1 in [17]. O cou se, i is again unknown whe he he assump ion dmax ≤2 can be supp essed. Fo gene al linea sys ems o he kind (1.1), i is an open ques ion o cha ac e ize hose n≤nand B∈ L(Rn;Rn) such ha null con ollabili y holds. Up o now, his is known only o cons an ma ices M,as indica ed in Theo em 1.5;see Rema ks 1.2 and 1.3;see also [1,2] o some esul s in his di ec ion and he ecen pape [16], whe e he au ho s ha e in oduced o he echniques ha could shed some ligh o his ques ion. I is also meaning ul o conside bounda y con ollabili y p oblems o sys ems simila o (1.1)and(1.11). Fo ins ance, i makes sense o analyze he null con ollabili y o ⎧ ⎨ ⎩ y −AΔy =M(x, )yin Q, y=B 1γon Σ, y(x, 0) = y0(x)inΩ, (5.3) whe e γ⊂∂Ω is a non-emp y se , wi h con ols o ins ance in L∞(γ×(0,T); Rn). I also makes sense o analyze he exac con ollabili y o he ajec o ies o he semilinea sys em ⎧ ⎨ ⎩ y −AΔy = (y)inQ, y=B 1γon Σ, y(x, 0) = y0(x)inΩ. (5.4)