Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case

ESAIM: COCV ESAIM: Con ol, Op imisa ion and Calculus o Va ia ions July 2006, Vol. 12, 466–483 www.edpsciences.o g/coc DOI: 10.1051/coc :2006011 EXACT CONTROLLABILITY TO THE TRAJECTORIES OF THE HEAT EQUATION WITH FOURIER BOUNDARY CONDITIONS: THE SEMILINEAR CASE ∗ En ique Fe n´ andez-Ca a1, Manuel Gonz´ alez-Bu gos1, Se gio Gue e o2 and Jean-Pie e Puel3 Abs ac . This pape is conce ned wi h he global exac con ollabili y o he semilinea hea equa ion (wi h nonlinea e ms in ol ing he s a e and he g adien ) comple ed wi h bounda y condi ions o he o m ∂y ∂n + (y) = 0. We conside dis ibu ed con ols, wi h suppo in a small se . The null con ollabili y o simila linea sys ems has been analyzed in a p e ious i s pa o his wo k. In his second pa we show ha , when he nonlinea e ms a e locally Lipschi z-con inuous and sligh ly supe linea , one has exac con ollabili y o he ajec o ies. Ma hema ics Subjec Classi ica ion. 35K20, 93B05. Recei ed Feb ua y 17, 2005. Re ised May 30, 2005 and June 13, 2005. 1. In oduc ion Le Ω ⊂RN(N≥1) be a bounded connec ed open se whose bounda y ∂Ω is egula enough ( o ins ance ∂Ω∈C2). Le ω⊂Ω be a (small) nonemp y open subse and le T>0. We will use he no a ion Q=Ω×(0,T) and Σ = ∂Ω×(0,T) and we will deno e by n(x) he ou wa d uni no mal o Ω a he poin x∈∂Ω. We will conside he semilinea hea equa ion wi h nonlinea Fou ie (o Robin) bounda y condi ions ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ y −∆y+F(y,∇y)= 1ωin Q, ∂y ∂n + (y)=0 onΣ, y(x, 0) = y0(x)inΩ. (1) He e, we assume ha ∈L2(ω×(0,T)) (a leas ), 1ωis he cha ac e is ic unc ion o ω,y0∈L∞(Ω) and F:R×RN→ Rand :R→ Ra e gi en unc ions. In (1), y=y(x, ) is he s a e and = (x, )is he con ol; i is assumed ha we can ac on he sys em only h ough ω×(0,T). Keywo ds and ph ases. Con ollabili y, hea equa ion, Fou ie bounda y condi ions, semilinea . ∗This wo k has been pa ially suppo ed by D.G.E.S. (Spain), G an s BFM2000–1317 and BFM2003–06446. 1Dp o. E.D.A.N., Uni e si y o Se illa, Ap do. 1160, 41080 Se illa, Spain; [email p o ec ed];[email p o ec ed]; [email p o ec ed] 2Labo a oi e Jacques-Louis Lions, Uni e si ´e Pie e e Ma ie Cu ie, boˆı e cou ie 187, 75035 Cedex 05, Pa is, F ance; [email p o ec ed] 3Labo a oi e de Ma h´ema iques Appliqu´ees, Uni e si ´e de Ve sailles – S . Quen in, 45 a enue des ´ E a s-Unis, 78035 Ve sailles, F ance; [email p o ec ed]e. c EDP Sciences, SMAI 2006 A icle published by EDP Sciences and a ailable a h p://www.edpsciences.o g/coc o h p://dx.doi.o g/10.1051/coc :2006011 E. FERN´ ANDEZ-CARA ET AL. 467 Fo he exis ence, uniqueness, egula i y and gene al p ope ies o he solu ions o p oblems like (1), see o ins ance [1, 2, 7]. An illus a i e in e p e a ion o he da a and a iables in (1) is he ollowing. The unc ion y=y(x, ) can be iewed as he ela i e empe a u e o a medium (wi h espec o he ex e io su ounding ai ) subjec o anspo and chemical eac ions. The pa abolic equa ion in (1) means, among o he hings, ha a hea sou ce 1ωis applied on a pa o he body. On he bounda y, −∂y ∂n can be iewed as he no mal hea flux, inwa ds di ec ed, up o a posi i e coefficien . Thus, he equali y −∂y ∂n = (y) means ha his flux is a (nonlinea ) unc ion o he empe a u e. Acco dingly, i is easonable o assume ha is nondec easing and (0) = 0. A simplified linea model which was conside ed in a p e ious pape [10] is he ollowing: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ y −∆y+a(x, )y+B(x, )·∇y= 1ωin Q, ∂y ∂n +β(x, )y=0 onΣ, y(x, 0) = y0(x)inΩ. (2) He e, i is assumed ha he coefficien s a,Band βsa is y a∈L∞(Q),B∈L∞(Q)N,β∈L∞(Σ) (3) and, o he easons abo e, i is also na u al o assume ha β≥0 (al hough his assump ion was no used in [10]). The main goal o his pape is o analyze he con ollabili y p ope ies o he nonlinea sys em (1). Mo e p ecisely, we will y o each exac ly uncon olled solu ions o (1), i.e. unc ions y=y(x, ) sa is ying ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ y −∆y+F(y,∇y)=0 inQ, ∂y ∂n + (y)=0 onΣ, y(x, 0) = y0(x)inΩ. (4) I will be said ha (1) is (globally) exac ly con ollable o he ajec o ies a ime Ti , o any solu ion o (4) wi h “sui able” egula i y and any y0∈L∞(Ω), he e exis con ols ∈L2(ω×(0,T)) and associa ed solu ions y∈C0([0,T]; L2(Ω)) such ha y(x, T )=y(x, T )inΩ.(5) He e, by sui able egula i y we mean he ollowing: y∈L2(0,T;H1(Ω)) ∩C0([0,T]; L2(Ω)) ∩L∞(Q), y0∈L∞(Ω).(6) The con ollabili y p ope ies o semilinea ime-dependen sys ems ha e been s udied in ensi ely hese las yea s. See o ins ance [8,11–13,15,16], whe e nonlinea i ies o he o m (y) a e conside ed. See also he gene al ea ise [14]. In pa icula , o pa abolic sys ems comple ed wi h Di ichle bounda y condi ions, nonlinea e ms (y,∇y) depending on bo h he s a e and he g adien ha e been aken in o accoun in [6, 9]. Fo he simila linea sys em (2), he null con ollabili y was analyzed mo e in de ail in [10]. In he case o (1), some pa ial esul s ha e been gi en in [5]. 468 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION Ou main esul conce ns he global exac con ollabili y o he ajec o ies o (1). I is he ollowing: Theo em 1. Le us assume ha Fand a e locally Lipschi z-con inuous and sa is y lim |s|→∞ |F(s, p)−F( , p)| |s− |log3/2(1 + |s− |)=0,(7) uni o mly in ( , p)∈[−K, K]×RN∀K>0, ⎧ ⎪ ⎨ ⎪ ⎩ ∀L>0,∃M>0such ha |F(s, p)−F( , p)|≤M|s− |,|F(s, p)−F(s, q)|≤M|p−q| ∀(s, , p, q)∈[−L, L]2×RN×RN (8) and lim |s|→∞ | (s)− ( )| |s− |log1/2(1 + |s− |)=0 (9) uni o mly in ∈[−K, K]∀K>0. Then, o each T>0, he nonlinea sys em (1) is exac ly con ollable o he ajec o ies a ime Twi h L∞con ols. Rema k 1. Condi ions (7)–(9) a e sa isfied i Fand a e globally Lipschi z con inuous. No ice ha (7) means ha he unc ion Fcan only be sligh ly supe linea in s, uni o mly in p. In he simila case o Di ichle bounda y condi ions, i is known ha condi ions like hese a e sha p. Indeed, o ins ance, when Fdoes no depend on pand |F(s)−F( )|∼|s− |logβ(1 + |s− |),β>2, due o blow-up phenomena, he sys em ails o be con ollable whene e ω= Ω (see [11]). On he o he hand, (9) is also a sligh ly supe linea g ow h assump ion o . I would be in e es ing o know whe he a mo e supe linea leading o blow up in he absence o con ol can also be an obs uc ion o he null con ollabili y o (1). Bu his ques ion does no seem ob ious and emains open. Rema k 2. A esul p o ed in [5] says ha when F≡0, is smoo h nea ze o and (s)s≥0∀s∈R,(10) he nonlinea sys em (1) is null con ollable o la ge T. Tha is o say, unde hese assump ions, o each y0∈L2(Ω) he e exis T(y0)>0andcon ols in L∞(ω×(0,T) such ha he associa ed s a es ysa is y y(x, T (y0)) = 0 in Ω. By inspec ion o he p oo o heo em 1, we see ha he same esul holds o (1) wi h F≡0 whene e is locally Lipschi z-con inuous and sa isfies he good sign condi ion (10). Fo he p oo o Theo em 1, we will fi s es ablish a null con ollabili y esul o (2) (see P op. 1 below). This will be used, oge he wi h an app op ia e fixed poin a gumen , o deduce he desi ed esul . This s a egy was in oduced in [15] in he amewo k o he exac con ollabili y o he semilinea wa e equa ion. See also [8,12] o simila esul s conce ning he app oxima e and null con ollabili y o he semilinea hea equa ion wi h Di ichle o Neumann bounda y condi ions. Ou null con ollabili y esul o (2) is he ollowing: P oposi ion 1. Fo e e y T>0,sys em(2) is null con ollable a ime T, wi h con ols in L∞(ω×(0,T)). Mo e p ecisely, o each y0∈L2(Ω), he eexis s ∈L∞(ω×(0,T)) such ha he associa ed solu ion o (2) sa isfies (5). Fu he mo e, he con ol can be ound sa is ying  L∞(ω×(0,T)) ≤eC(Ω,ω)K(T,a∞,B∞,β∞)y0L2(Ω) ,(11) E. FERN´ ANDEZ-CARA ET AL. 469 whe e K=1+1/T +a2/3 ∞+B2 ∞+β2 ∞+T(1 + a∞+B2 ∞+β2 ∞).(12) Fo he p oo o p oposi ion 1, we fi s in oduce a con ol L2(ω×(0,T)) which leads he solu ion o (2) o ze o a ime T. In a second s ep, a guing as in Sec ion 2 in [4], a egula izing a gumen will lead o he desi ed L∞con ol. The es o his pape is o ganized as ollows. In Sec ion 2, we p o e P oposi ion 1. Sec ion 3 is de o ed o he p oo o Theo em 1. Fo comple eness, we ha e also included an Appendix whe e he p oo o a a he echnical local egula i y esul is gi en in de ail. In he sequel, Cdeno es a gene ic posi i e cons an only depending on Ω and ω. 2. A null con ollabili y esul o he linea sys em In his sec ion we p esen he p oo o P oposi ion 1. Le y0∈L2(Ω) be gi en and le us in oduce wo open se s ωand ω,wi hω ⊂⊂ ω⊂⊂ ω. Then, we can use he main esul in [10] (Th. 2) wi h con ol egion ω ×(0,T) o deduce he exis ence o a con ol  ∈L2(ω ×(0,T)) such ha he associa ed solu ion o (2) e ifies (5) and also he es ima e  L2(ω ×(0,T)) ≤eC(Ω,ω)K(T,a∞,B∞,β∞)y0L2(Ω),(13) whe e Kis o he o m (12). Le us deno e by y he s a e associa ed o  . We now in oduce a cu -off unc ion η=η( ) sa is ying η∈C∞([0,T]),η( )=1in(0,T/4),η( )=0in(3T/4,T) and 0≤η( )≤1,|η( )|≤C in (0,T) and we deno e by χ he solu ion o he sys em ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ χ −∆χ+a(x, )χ+B(x, )·∇χ=0 inQ, ∂χ ∂n +β(x, )χ=0 onΣ, χ(x, 0) = y0(x)inΩ. Then, he unc ion w=y−ηχ sa isfies ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩w −∆w+a(x, )w+B(x, )·∇w=−η( )χ+ 1ω in Q, ∂w ∂n +β(x, )w=0 onΣ, w(x, 0) = 0,w(x, T )=0 inΩ. Ou aim is o cons uc a con ol ∈L∞(ω×(0,T)) which d i es he solu ion o (2) o ze o a ime =T.To his end, we will need a local egula i y esul o he solu ions o linea hea equa ions wi h L∞coefficien s a and B. This will be used below o he unc ions χand wand eads as ollows: Lemma 1. Le us deno e by Y he space L2(0,T;H1(Ω)) ∩C0([0,T]; L2(Ω)).Le y∈Ybe a solu ion o he equa ion y −∆y+a(x, )y+B(x, )·∇y= , (14) whe e a∈L∞(Q),B∈L∞(Q)Nand ∈L2(Q).Le O⊂Ωbe a nonemp y open se and assume ha is L∞ in he cylinde O×(0,T).Then y∈L∞(δ, T ;W1,∞(O)) 470 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION o any δ∈(0,T)and any nonemp y open se O⊂⊂ O. Fu he mo e, he e exis s a posi i e cons an C(O) such ha he ollowing es ima e holds: yL∞(δ,T ;W1,∞(O)) ≤C(O)(T1/2+TN/2)1+δ−1+a∞+B∞N+1 yY+ L∞(O×(0,T )).(15) The p e ious egula i y also holds wi h δ=0i , besides (14), we ha e y(x, 0) = 0 in Ω. In ha case, one has an es ima e simila o (15) wi hou he e m in δ. This lemma is implied by well known pa abolic egula i y heo y. Fo comple eness, i s p oo is gi en in an Appendix, a he end o his pape . Le us now conside an open se ω0wi h ω⊂⊂ ω0⊂⊂ ωand a cu -off unc ion ξ,wi h ξ∈C2 0(ω0),ξ≡1inω and le us se w=(1−ξ)w.Thenweha e: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ w −∆w+a(x, )w+B(x, )·∇w=−η( )χ+ 1ωin Q, ∂w ∂n +β(x, )w=0 onΣ, w(x, 0) = 0,w(x, T )=0 inΩ, wi h =ηξχ+2∇ξ·∇w+∆ξw−B·∇ξw. (16) Le us ema k ha supp ⊂ω×[0,T]. The e o e, i we p o e ha ∈L∞(ω×(0,T)), we will ha e ha he unc ion y=w+ηχ sol es ( oge he wi h ) he null con ollabili y p oblem o (2). Thus, le us check ha ∈L∞(ω×(0,T)) and le us es ima e i s no m in his space: •The egula i y o he fi s e m in he igh hand side o (16) is implied by he in e io egula i y o χ no only in space bu in ime as well. F om Lemma 1 wi h O=ω, we deduce ha χ∈L∞(ω0×(δ, T )) wi h supp ξ⊂ω0⊂⊂ ω(we e en ha e χ∈L∞(δ, T ;W1,∞ loc (ω))) and χL∞(ω0×(δ,T )) ≤C(T1/2+TN/2)1+δ−1+a∞+B∞N+1 χY; ecall ha Y=L2(0,T;H1(Ω)) ∩C0([0,T]; L2(Ω)). Consequen ly aking o ins ance δ=T/8, since η≡0in(0,T/4), we ge ηξχL∞(ω×(0,T )) ≤CT−1(T1/2+TN/2)1+T−1+a∞+B∞N+1 χY. •The egula i y o he o he h ee e ms in he igh hand side o (16) is ela ed o he in e io space egula i y o w. Thus, le us in oduce ω1wi h ω0⊂⊂ ω1⊂⊂ ωand le us apply Lemma 1 wi h O=ω1 ω. This gi es w∈L∞(0,T;W1,∞(ω0 ω)) and he es ima e wL∞(0,T;W1,∞(ω0 ω)) ≤C(T1/2+TN/2)(1+a∞+B∞)N+1 (wY+ηχL∞(ω1×(0,T))), whence 2∇ξ·∇w+∆ξw−B·∇ξwL∞(ω×(0,T )) ≤C(T1/2+TN/2) ×(1 + a∞+B∞)N+2 (wY+ηχL∞(ω1×(0,T))). E. FERN´ ANDEZ-CARA ET AL. 471 Pu ing he p e ious es ima es oge he , we find ha ∈L∞(ω×(0,T)) and  L∞(ω×(0,T)) ≤C(1 + TN−1)1+T−1+a∞+B∞2N+3 (wY+χY).(17) A his poin , no ice ha o any ∈L2(Q)andanyy0∈L2(Ω) he solu ion y o he linea sys em ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ y −∆y+a(x, )y+B(x, )·∇y= in Q, ∂y ∂n +β(x, )y=0 onΣ, y(x, 0) = y0(x)inΩ (18) sa isfies yY≤eCT(1+a∞+B2 ∞+β2 ∞)( L2(Q)+y0L2(Ω)). Fo a de ailed p oo , see o example p oposi ion 1 in [10]. Thiscanbeused oes ima ewYand χYin e ms o  L2(ω×(0,T)) and y0L2(Ω). In iew o (17), we see ha  L∞(ω×(0,T)) ≤L( L2(ω ×(0,T )) +y0L2(Ω)),(19) whe e L=CT−1(1 + TN−1)1+T−1+a∞+B∞2N+3 exp{CT(1 + a∞+B2 ∞+β2 ∞)}. Combining his es ima e and (13), we finally ob ain ha  L∞(ω×(0,T)) ≤eCK(T,a∞,B∞,β∞)y0L2(Ω),(20) whe e Kis gi en by (12). This ends he p oo o P oposi ion 1. 3. Con ollabili y o he nonlinea sys em In his sec ion we will p o e Theo em 1. The ollowing auxilia y esul will be needed: P oposi ion 2. Le us assume ha , in (18), we ha e ∈L∞(Q)and y0∈L∞(Ω). Le us also assume ha he coefficien s a,Band βsa is y (3).Theny∈L∞(Q)and y∞≤eCT(1+a∞+B2 ∞+β2 ∞)y0∞+ ∞.(21) o some C=C(Ω). P oo . We will conside wo diffe en si ua ions: Case 1. We will fi s assume ha a≥1andβ≥0 and we will es ablish (21) in his case. In ac , we will show ha , unde hese assump ions, y∞≤y0∞+ ∞.(22) To his end, le us in oduce he sys em ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ z −∆z+a(x, )z+B(x, )·∇z=hin Q, ∂z ∂n +β(x, )z=kon Σ, z(x, 0) = z0(x)inΩ, 472 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION whe e h∈L∞(Q), k∈L∞(Σ) and z0∈L∞(Ω) and le us show ha , i h,z0and ka e nonnega i e, hen his is also he case o z. Indeed, by mul iplying he equa ion sa isfied by zby z−(·, ) ( he nega i e pa o z(·, )) o each ∈(0,T) and in eg a ing in Ω, a e se e al simplifica ions, we find: 1 2 d d Ω |z−(x, )|2dx+Ω |∇z−(x, )|2dx +∂Ω β(x, )|z−(x, )|2dσ+∂Ω k(x, )z−(x, )dσ+Ω a(x, )|z−(x, )|2dx= −Ω h(x, )z−(x, )dx−Ω B(x, )·∇z−(x, )z−(x, )dx. F om his iden i y, in iew o he posi i eness o a,h,βand k, we easily deduce ha d d Ω |z−(x, )|2dx≤B2 ∞Ω |z−(x, )|2dx, whence z≥0inQ. Now, le M>0 be a la ge cons an ( o be chosen below). The unc ion z=M−ysa isfies ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ z −∆z+a(x, )z+B(x, )·∇z=a(x, )M− in Q, ∂z ∂n +β(x, )z=β(x, )Mon Σ, z(x, 0) = M−y0(x)inΩ. The e o e, i we ake M≥max{ L∞(Q),y0L∞(Ω)}, we can apply he p e ious a gumen and deduce ha y≤M. In a simila way, one can deduce ha y≥−M and, consequen ly, |y|≤M. This p o es ha whene e a≥1andβ≥0, he es ima e (22) holds. Case 2. We will now p o e (21) o gene al L∞coefficien s aand β. Le γ∈C2(Ω) be a unc ion sa is ying γ≥0inΩ,∂γ ∂n ≤−β∞on ∂Ω,γ∞≤1, ∇γ∞≤Cβ∞,D2γ∞≤Cβ2 ∞. (23) We gi e he e a ske ch o he p oo o he exis ence o such a unc ion γ. To his end, le δ>0beapa ame e (depending on Ω) such ha x∈Ωδ→ dis (x, ∂Ω) is C2,wi hΩ δ={x∈Ω:dis (x, ∂Ω) <δ}. We dis inguish wo cases. Le usfi s assume ha β∞≥1/δ.Thenwe akeγ(x)≡1inΩ Ωδ,γ(x)=β∞dis (x, ∂Ω) in Ωεwi h ε=1/(2β∞) and a egula iza ion o γin Ωδ Ωε. This gi es he desi ed p ope ies o γ. On he o he hand, i β∞<1/δ,we akeγ(x)=δβ∞in Ω Ωδ,γ(x)=β∞dis (x, ∂Ω) in Ωδ/2and a egula iza ion in Ωδ Ωδ/2. This also p o ides a desi ed unc ion in his case. E. FERN´ ANDEZ-CARA ET AL. 473 Le us now se y=e γ(x)y.Thenysa isfies ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩y −∆y+a(x, )y+ B(x, )·∇y=e γ(x) in Q, ∂y ∂n + β(x, )y=0 onΣ, y(x, 0) = eγ(x)y0(x)inΩ, (24) whe e a=a+∆γ−|∇γ|2−B·∇γ,  B=B+2∇γ,  β=β−∂γ ∂n ≥0onΣ. No ice ha , om he inequali ies (23) sa isfied by γ, we know ha |a+∆γ−|∇γ|2−B·∇γ|≤C1(a∞+B2 ∞+β2 ∞)inQ o some C1>0. Now, le us se y=e −(C1(a∞+B2 ∞+β2 ∞)+1) y. Then ysa isfies ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩y −∆y+a(x, )y+(B(x, )+2∇γ(x)) ·∇y= in Q, ∂y ∂n + β(x, )y=0 onΣ, y(x, 0) = eγ(x)y0(x)inΩ, (25) whe e a=a+∆γ−|∇γ|2−B·∇γ+C1(a∞+B2 ∞+β2 ∞)+1,  =e −(C1(a∞+B2 ∞+β2 ∞)+1) +γ(x) and  β= β. Since a≥1and β≥0, we can apply case 1 o y. This p o ides he es ima es y∞≤y∞≤eCT(1+a∞+B2 ∞+β2 ∞)(y0∞+ ∞), whence we deduce (21).  Le us now s a wi h he p oo o Theo em 1. Le y0∈L∞(Ω) and ybe gi en and assume ha ysa isfies (6) and (4) in he weak sense. Le us conside he nonlinea sys em ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ w −∆w+F1(w,∇w;x, )w+F2(∇w;x, )·∇w= 1ωin Q, ∂w ∂n +F3(w;x, )w=0 onΣ, w(x, 0) = y0(x)−y(x, 0) in Ω, (26) whe eweha eused heno a ion F1(s, p;x, )=F(y(x, )+s, ∇y(x, )+p)−F(y(x, ),∇y(x, )+p) s,(27) F2=(F21,...,F 2N),F 2j(p;x, )=1 0 ∂F ∂pj (y(x, ),∇y(x, )+λp)dλ(28) 474 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION and F3(s;x, )= (y(x, )+s)− (y(x, )) s(29) o s∈Rand p∈RN. We will p o e ha he e exis a con ol ∈L∞(ω×(0,T)) and an associa ed solu ion o (26) such ha w(x, T )=0 in Ω.(30) Wi h his con ol and he s a e y=w+y, we will ha e sol ed he exac con ollabili y p oblem o (1) and we will ha e hus p o ed Theo em 1. We will fi s assume ha he unc ions Fand a e con inuously diffe en iable. Then, by a densi y a gumen , we will be able o p o e he esul in he gene al case. 3.1. The case in which Fand a e C1 The idea o he p oo is well known: we in oduce an app op ia e (se - alued) fixed poin mapping and we check ha i possesses a leas one fixed poin ; his will be a solu ion o he null con ollabili y p oblem associa ed o (26). Le R>0 be gi en and le us in oduce he ollowing unc ion: MR(s)=⎧ ⎪ ⎨ ⎪ ⎩ −Ri s<−R, si −R≤s≤R, Ri s>R. Le us deno e by Z he Hilbe space Z=L2(0,T;H1(Ω)) and le us se o each R>0andeachz∈Z aR,z(x, )=F1(MR(z(x, )),∇z(x, ); x, ), Bz(x, )=F2(∇z(x, ); x, ) and βR,z(x, )=F3(MR(z(x, )); x, ). Conside he linea null con ollabili y p oblem ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ w −∆w+aR,z(x, )w+Bz(x, )·∇w= 1ωin Q, ∂w ∂n +βR,z(x, )w=0 onΣ, w(x, 0) = y0(x)−y(x, 0) in Ω, (31) oge he wi h (30). Le us in oduce he unc ion w0,wi hw0(x)=y0(x)−y(x, 0) o all x∈Ω. F om (6), (8) and he ac ha ∈C1(R), we ha e aR,z ∈L∞(Q),B z∈L∞(Q)N,β R,z ∈L∞(Σ). Consequen ly, in iew o P oposi ion 1, (30)–(31) can be sol ed wi h con ols in L∞(ω×(0,T)). We a e now going o selec a pa icula solu ion o (30)–(31) cons uc ed as in [11]. To do his, we fi s se TR=min{T,a−1/3 R}>0, whe e aR=sup |s|≤R, p∈RN ess sup (x, )∈Q |F1(s, p;x, )|. E. FERN´ ANDEZ-CARA ET AL. 481 We ha e 0∈L2(Q). Consequen ly, ∆y0∈L2(Q), y0∈C0([0,T]; H1 0(Ω)) and app op ia e es ima es a e sa isfied. Indeed, by mul iplying he equa ion sa isfied by y0by −∆y0and in eg a ing wi h espec o xin Ω, we find 1 2 d d ∇y0(·, )2 L2+Ω |∆y0(x, )|2dx=−Ω 0(x, )∆y0(x, )dx. (45) Since  0L2≤C L2+(1+δ−1+a∞+B∞)yY, we easily ob ain om (45) ha y0C0([0,T];H1 0(Ω)) +∆y0L2(Q)≤C L2+1+δ−1+a∞+B∞yY.(46) Clea ly, he same es ima e holds o yL∞(δ/(N+1),T ;H1(O0)) +∆yL2(O0×(δ/(N+1),T )). Le us now y o imp o e he local space egula i y p ope ies o y. To his end, we will use he ollowing lemma: Lemma 2. Le us se p0=2,le pibe defined by 1 pi =1 pi−1 −1 2N o 1≤i≤N−1and le us se pN=+∞. Le us deno e by Xi he space Xi=L∞((i+1)δ/(N+1),T;W1,pi(Oi)) o 0≤i≤Nand suppose ha y∈Xj−1 o some j. Then we also ha e y∈Xjand yXj≤C(O)(T1/2 ∞+D(T,δ,a∞,B∞)yXj−1), whe e D(T,δ,a∞,B∞)=(T1/4+T1/2)(1 + δ−1+a∞+B∞).(47) P oo o Lemma 2. Le us in oduce ξj∈C2 c(Oj−1)andηj∈C1([0,T]), wi h ξj(x)=1inOj,η j( )=1in[(j+1)δ/(N+1),T], ηj( )=0in[0,jδ/(N+1)],|ηj, ( )|≤C δin (0,T) and le us pu yj=ηjξjy.Thenyjsa isfies he ollowing: ⎧ ⎪ ⎨ ⎪ ⎩ yj, −∆yj= jin Q, yj=0 onΣ, yj(x, 0) = 0 in Ω (48) wi h j= j,1+ j,2+ j,3, whe e j,1=ηjξj , j,2=ηj, ξjy−ηj∆ξjy−aη jξjy−ηj(B·∇ξj)y, j,3=−2ηj∇ξj·∇y−ηjξjB·∇y. 482 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION F om he ac ha he sys em (48) is linea , we see ha yjcan be w i en as he sum o h ee solu ions o simila sys ems wi h igh hand sides j,1, j,2and j,3. Le us espec i ely deno e hem by yj,1,yj,2and yj,3. We a e now going o deduce es ima es o yj,k in Xj o 1 ≤k≤3. To his end, we will use he usual ep esen a ion o yj,k p o ided by he semig oup S( )associa ed o he hea equa ion wi h homogeneous Di ichle condi ions, say yj,k(·, )= 0 S( −s) j,k(·,s)ds o all ∈(0,T). Since ∈L∞(Q), we can w i e yj,1(·, )W1,pj(Ω) ≤C 0 ( −s)−1/2 j,1(·,s)Lpj(Ω) ds. The e o e, om Young’s inequali y we find ha yj,1∈L∞(0,T;W1,pj(Ω)) and yj,1L∞(0,T ;W1,pj(Ω)) ≤CT1/2 j,1L∞(0,T ;Lpj(Ω)) ≤C(O)T1/2 L∞(O×(0,T)) . Taking in o accoun ha j,2∈L∞(0,T;Lp∗ j−1(Ω)) wi h p∗ j−1=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ∞i j>N−1, p(a bi a y in (1,+∞)) i j=N−1, 2N N−j−1i j<N−1, we see ha j,2is no wo se han j,1and, again, yj,2(·, )W1,pj(Ω) ≤C 0 ( −s)−1/2 j,2(·,s)Lpj(Ω) ds o all . F om Young’s inequali y and he assump ion y∈Xj−1,wealsoge yj,2∈L∞(0,T;W1,pj(Ω)) and yj,2L∞(0,T ;W1,pj(Ω)) ≤CT1/2 j,2L∞(0,T ;Lp∗ j−1(Ω)) ≤C(O)T1/2(1 + δ−1+a∞)yXj−1. In he defini ion o j,3, we find ∇y. Consequen ly, we can only ensu e ha j,3∈L∞(0,T;Lpj−1(Ω)). Since −N 21 pj−1 −1 pj−1 2=−3 4, we ha e yj,3(·, )W1,pj(Ω) ≤C 0 ( −s)−3/4 j,3(·,s)Lpj−1(Ω) ds and now Young’s inequali y gi es yj,3∈L∞(0,T;W1,pj(Ω)) and yj,3L∞(0,T ;W1,pj(Ω)) ≤CT1/4 j,3L∞(0,T ;Lpj−1(Ω)) ≤C(O)T1/4(1 + B∞)yXj−1. E. FERN´ ANDEZ-CARA ET AL. 483 Pu ing he es ima es o yj,kL∞(0,T;W1,pj(Ω)) oge he and aking in o accoun he defini ions o ηjand ξj, we ob ain he desi ed inequali y o yXj. This concludes he p oo o Lemma 2.  Since we al eady had y∈X0, we deduce om Lemma 2 ha y∈XNand yXN≤C(O)(T1/2 ∞+D(T,δ,a∞,B∞)yXN−1), whe e, Dis gi en by (47). We can apply Lemma 2 subsequen ly o j=N,N −1,...,1. The es ima es we find yield yXN≤C(T1/2(1 + DN−1) L∞(O×(0,T )) +DNyX0). This, oge he wi h (46), yields yXN≤C(T1/2+TN/2)D(T,δ,a∞,B∞)N+1( L∞(O×(0,T )) +yY), which is exac ly (15). Re e ences [1] H. Amann, Pa abolic e olu ion equa ions and nonlinea bounda y condi ions. J. Diff. Equ. 72 (1988) 201–269. [2] J. A ie a, A. Ca alho and A. Rod ´ıguez-Be nal, Pa abolic p oblems wi h nonlinea bounda y condi ions and c i ical nonlin- ea i ies. J. Diff. Equ. 156 (1999) 376–406. [3] J.P. Aubin, L’analyse non lin´eai e e ses mo i a ions ´economiques. Masson, Pa is (1984). [4] O. Boda , M. Gonz´alez-Bu gos and R. P´ ez-Ga c´ıa, Insensi izing con ols o a semilinea hea equa ion wi h a supe linea nonlinea i y. C. R. Ma h. Acad. Sci. Pa is 335 (2002) 677–682. [5] A.Doubo a,E.Fe n´andez-Ca a and M. Gonz´alez-Bu gos, On he con ollabili y o he hea equa ion wi h nonlinea bounda y Fou ie condi ions. J. Diff. Equ. 196 (2004) 385–417. [6]A.Doubo a,E.Fe n´andez-Ca a, M. Gonz´alez-Bu gos and E. Zuazua, On he con ollabili y o pa abolic sys ems wi h a nonlinea e m in ol ing he s a e and he g adien . SIAM J. Con ol Op im. 41 (2002) 798–819. [7] L. E ans, Regula i y p ope ies o he hea equa ion subjec o nonlinea bounda y cons ain s. Nonlinea Anal. 1(1997) 593–602. [8] C. Fab e, J.P. Puel and E. Zuazua, App oxima e con ollabili y o he semilinea hea equa ion. P oc. Roy. Soc. Edinbu gh 125A (1995) 31–61. [9] L.A. Fe n´andez and E. Zuazua, App oxima e con ollabili y o he semi-linea hea equa ion in ol ing g adien e ms. J. Op im. Theo y Appl. 101 (1999) 307–328. [10] E. Fe n´andez-Ca a, M. Gonz´alez-Bu gos, S. Gue e o and J.P. Puel, Null con ollabili y o he hea equa ion wi h bounda y Fou ie condi ions: The linea case. ESAIM: COCV 12 442–465. [11] E. Fe n´andez-Ca a and E. Zuazua, Null and app oxima e con ollabili y o weakly blowing up semilinea hea equa ions. Ann. Ins . H. Poinca ´e, Anal. non Lin´eai e 17 (2000) 583–616. [12] A. Fu siko and O.Yu. Imanu ilo , Con ollabili y o E olu ion Equa ions. Lec u e No es #34, Seoul Na ional Uni e si y, Ko ea (1996). [13] I. Lasiecka and R. T iggiani, Exac con ollabili y o semilinea abs ac sys ems wi h applica ions o wa es and pla es bounda y con ol. Appl. Ma h. Op im. 23 (1991) 109–154. [14] I. Lasiecka and R. T iggiani, Con ol Theo y o Pa ial Diffe en ial Equa ions: Con inuous and App oxima ion Theo ies. Camb idge Uni e si y P ess, Camb idge (2000). [15] E. Zuazua, Exac bounda y con ollabili y o he semilinea wa e equa ion, in Nonlinea Pa ial Diffe en ial Equa ions and hei Applica ions, Vol. X, H. B ezis and J.L. Lions Eds. Pi man (1991) 357–391. [16] E. Zuazua, Exac con ollabili y o he semilinea wa e equa ion in one space dimension. Ann. I.H.P., Analyse non Lin´eai e 10 (1993) 109–129.