Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient

Insensi izing con ols o a hea equa ion wi h a nonlinea e m in ol ing he s a e and he g adien ? O. Boda aM. Gonz´alez-Bu gos bR. P´e ez-Ga c´ıa b,∗ aUni e si ´e Blaise-Pascal, Labo a oi e de Ma h´ema iques Appliqu´ees, UMR CNRS 6620, Cle mon -Fe and 2, 63177 Aubi`e e, F ance bUni e sidad de Se illa, Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Ap do. 1160, 41080 Se illa, Spain Abs ac In his pape we p esen wo esul s on he exis ence o insensi izing con ols o a hea equa ion in a bounded domain o IRN. We i s conside a semilinea hea equa ion in ol ing g adien e ms wi h homogeneous Di ichle bounda y condi- ions. Then a hea equa ion wi h a nonlinea e m F(y) and linea bounda y con- di ions o Fou ie ype is conside ed. The nonlinea i ies a e assumed o be globally Lipschi z-con inuous. In bo h cases, we p o e he exis ence o con ols insensi izing he L2−no m o he obse a ion o he solu ion in an open subse Oo he domain, unde sui able assump ions on he da a. Each p oblem boils down o a special ype o null con ollabili y p oblem. Gene al obse abili y inequali ies a e p o ed o lin- ea sys ems simila o he linea ized p oblem. The p oo s o he main esul s in his pape in ol e such inequali ies and ely on he s udy o hese linea p oblems and app op ia e ixed poin a gumen s. Key wo ds: con ollabili y, nonlinea PDE o pa abolic ype, nonlinea g adien e ms 1991 MSC: 93B05, 35K55, 35K05 ?This wo k has been pa ially inanced by D.G.E.S. (Spain), G an PB98–1134. ∗Co esponding au ho . Email add esses: [email p o ec ed]le mon . (O. Boda ), [email p o ec ed] (M. Gonz´alez-Bu gos), [email p o ec ed] (R. P´e ez-Ga c´ıa). P ep in submi ed o Nonlinea Analysis 20 Feb ua y 2004 1 Se ing he p oblems and main esul s Le Ω ⊂IRN,N≥1, be a bounded connec ed open se wi h bounda y ∂Ω∈ C2. Fo T > 0, we deno e Q= Ω ×(0, T ) and Σ = ∂Ω×(0, T ). Le ωand O be nonemp y open subse s o Ω. We i s conside he nonlinea hea equa ion:      ∂ y−∆y+ (y, ∇y) = ξ+ 1ωin Q, y= 0 on Σ, y(x, 0) = y0(x) + τˆy0(x) in Ω, (1) whe e is a C1globally Lipschi z-con inuous unc ion de ined on IR ×IRN, ξ∈L2(Q) and y0∈L2(Ω) a e gi en, ˆy0∈L2(Ω) is unknown wi h |ˆy0|L2(Ω) = 1, τis a small unknown eal numbe , and ∈L2(Q) is a con ol unc ion o be de e mined. He e, ∂ deno es he ime de i a i e and 1ωis he cha ac e is ic unc ion o he se ω. Le us de ine Φ(y(·,·;τ, )) = 1 2ZZO×(0,T )|y(x, ;τ, )|2dx d , (2) whe e y(·,·;τ, ) is he solu ion o (1) associa ed o τand . A con ol unc ion is said o insensi ize he unc ional Φ i ∂Φ(y(·,·;τ, )) ∂τ τ=0 = 0,∀ˆy0∈L2(Ω) wi h |ˆy0|L2(Ω) = 1.(3) This p oblem, o iginally add essed by J.-L. Lions in [1], has been s udied in he semilinea case o globally Lipschi z-con inuous nonlinea i ies = (y). In [2], he au ho s weakened he unde lying p oblem, de ining app oxima ely insensi izing con ols. They p o ed he exis ence o such con ols o unknown da a in bo h he ini ial and bounda y condi ions. In [3] wo mains esul s a e gi en. On one hand, he au ho p o es ha one canno expec he exis ence o insensi izing con ols o e e y y0∈L2(Ω) when Ω ω6=∅, e en i ≡0. On he o he hand, o y0= 0 and sui able assump ions on ξ, L. de Te esa p o es he exis ence o con ols such ha (3) holds (see Theo em 1 in [3]). This esul is gene alized in [4] and [5] o nonlinea i ies wi h ce ain supe linea g ow h a in ini y. One o he pu poses o his pape is o ex end Theo em 1 in [3] o he case o a semilinea hea equa ion whe e he nonlinea i y is allowed o depend on bo h he s a e yand i s g adien . Then, an insensi i i y esul o a semilinea hea equa ion wi h a nonlinea e m F(y) and linea bounda y condi ions o Fou ie ype is gi en. The i s insensi i i y esul we p esen in his pape is he ollowing one: Theo em 1.1 Assume ha ω∩O 6=∅and y0= 0. Le : IR ×IRN→IR be aC1globally Lipschi z–con inuous unc ion such ha (0,0) = 0. Then, he e 2 exis s a posi i e cons an Mdepending on Ω,ω,O,T, and such ha o any ξ∈L2(Q) e i ying ZZQexp M |ξ|2dx d < ∞,(4) one can ind a con ol unc ion ∈L2(Q)insensi izing he unc ional Φgi en by (2). Adap ing he compu a ions in [1] and [2] o he p esen case, one ge s ha he exis ence o a con ol such ha (3) holds is equi alen o he exis ence o a con ol such ha he solu ion (y, q) o      ∂ y−∆y+ (y, ∇y) = ξ+ 1ωin Q, y= 0 on Σ, y(x, 0) = y0(x) in Ω, (5)     −∂ q−∆q+∂s (y, ∇y)q−∇·(∂p (y, ∇y)q) = y1Oin Q, q= 0 on Σ, q(x, T) = 0 in Ω, (6) e i ies q(x, 0) = 0 in Ω.(7) He e we no ed (s, p)7→ (s, p), s∈IR, p∈IRN,∂s he de i a i e o wi h espec o s, and ∂p he g adien o wi h espec o p. Thus, so as o p o e Theo em 1.1, we will es ic ou a en ion o sol e he nons anda d null con ollabili y p oblem (5)–(7) o y0= 0. Le us now conside a semilinea hea equa ion wi h linea bounda y condi- ions o Fou ie ype and pa ially known ini ial da a:      ∂ y−∆y+F(y) = ξ+ 1ωin Q, ∂ny+hy = 0 on Σ, y(x, 0) = y0(x) + τˆy0(x) in Ω, (8) whe e F: IR →IR is a C1globally Lipschi z-con inuous unc ion, h∈L∞(Σ) (a leas ), ξ,y0,τ, and ˆy0a e as in (1), and ∈L2(Q) is again a con ol unc ion o be de e mined. He e, ∂ndeno es he de i a ion wi h espec o he uni ou wa d no mal o ∂Ω and he no m in L∞(Σ) will be deno ed by k·k∞;Σ. The nex aim in his pape is o p o e he exis ence o con ols insensi izing he L2–no m o he obse a ion o he solu ion o (8) in he open se O. Theo em 1.2 Assume ha ω∩O 6=∅and y0= 0. Le F∈C1(IR) be a glob- ally Lipschi z–con inuous unc ion (wi h Lipschi z cons an L>0) sa is ying F(0) = 0 and le h∈L∞(Σ) be such ha ∂ h∈L∞(Σ). Then, he e exis s a posi i e cons an N(depending on Ω,ω,O,T,L,khk∞;Σ, and k∂ hk∞;Σ) 3 such ha , o any ξ∈L2(Q) e i ying ZZQexp N |ξ|2dx d < ∞,(9) one can ind a con ol unc ion ∈L2(Q)insensi izing he unc ional de ined in (2),y(·,·;τ, )being he solu ion o (8) associa ed o τand . In his case, he e exis s a con ol unc ion such ha (3) holds i and only i he e exis s a con ol such ha he solu ion (y, q) o      ∂ y−∆y+F(y) = ξ+ 1ωin Q, ∂ny+hy = 0 on Σ, y(x, 0) = y0(x) in Ω, (10)     −∂ q−∆q+F0(y)q=y1Oin Q, ∂nq+hq = 0 on Σ, q(x, T) = 0 in Ω, (11) e i ies (7). To p o e Theo em 1.2, i will hen su ice o ind an L2–con ol sol ing his new null con ollabili y p oblem o y0= 0. As in [2] and [5], one can expec o choose a con ol unc ion such ha he associa ed solu ion (y, q) o (5), (6) (wi h y0= 0), in addi ion o insensi ize he unc ional Φ, i also e i ies y(x, T ) = 0 in Ω. This can be done wi h an ex a assump ion on ξ: Theo em 1.3 Assume ha ω∩O 6=∅and y0= 0. Le be as in Theo em 1.1. Then, he e exis s M>0(depending on Ω,ω,O,T, and ) such ha o any ξ∈L2(Q) e i ying ZZQexp M (T− )!|ξ|2dx d < ∞, one can ind a con ol unc ion ∈L2(Q)insensi izing he unc ional Φgi en by (2) and such ha he solu ion y(·,·;τ, )|τ=0 o (1) (wi h y0= 0) sa is ies y(x, T;τ, )|τ=0 = 0 in Ω. We will no gi e he p oo o his esul , since i is simila o he one o Theo em 1.1. The es o his pape is o ganized as ollows: in sec ion 2, we i s p o e an obse abili y inequali y ha gene alizes he one in [3]. This esul is indeed one o he main esul s in his wo k and we will use i in o he o hcoming pape s (c . [4], [5]). We also gi e an obse abili y inequali y o he case o linea Fou ie bounda y condi ions, which will also be used in [7]. In sec ion 3, we p o e Theo ems 1.1 and 1.2. We end wi h commen s and conclusions. 4 2 The Obse abili y Inequali ies In his sec ion we i s p o e an obse abili y inequali y ha is a gene aliza ion o he one gi en in [3] o he case o linea sys ems wi h i s o de e ms. This inequali y will be he main ool in he p oo o Theo em 1.1. We also gi e an obse abili y inequali y o linea sys ems wi h linea bounda y condi ions o Fou ie ype, which will be essen ial o p o e Theo em 1.2. Le us conside ϕand ψsol ing he ollowing sys ems:      ∂ ϕ−∆ϕ+cϕ +D·∇ϕ= 0 in Q, ϕ= 0 on Σ, ϕ(x, 0) = ϕ0(x) in Ω, (12)     −∂ ψ−∆ψ+aψ −∇·(Bψ) = ϕ1Oin Q, ψ= 0 on Σ, ψ(x, T ) = 0 in Ω, (13) wi h a, c ∈L∞(Q), B, D ∈L∞(Q)N, and ϕ0∈L2(Ω). In he sequel, k · k∞ will deno e he no m in bo h L∞(Q) and L∞(Q)N. I is known (c . [8], p. 356) ha ϕ, ψ ∈L2(0, T ;H1 0(Ω)) ∩C([0, T]; L2(Ω)), ∂ ϕ, ∂ ψ∈L2(0, T;H−1(Ω)). The main esul in his sec ion is he ollowing one: Theo em 2.1 Assume ha ω∩ O 6=∅. Then, he e exis posi i e cons an s Mand Hsuch ha , o e e y ϕ0∈L2(Ω), he co esponding solu ion (ϕ, ψ) o (12) and (13) sa is ies ZZQexp −M |ψ|2dx d ≤HZZω×(0,T )|ψ|2dx d . Mo e p ecisely, M=C1 + TM0and H= exp"C M0+1 T+T1 + kak∞+kck∞+kBk2 ∞+kDk2 ∞!#, whe e C=C(Ω,ω,O)and M0is gi en by M0= 1 + kak2/3 ∞+kck2/3 ∞+ka−ck1/2 ∞+kBk∞+kB−Dk∞+kBk2 ∞+kDk2 ∞. The basic ool o p o e his heo em is a global Ca leman inequali y o linea 5 sys ems o he o m      ∂ z−∆z=Fin Q, z= 0 on Σ, z(x, 0) = z0(x) in Ω, (14) wi h z0∈L2(Ω) and Fin L2(Q) o in L2(0, T ;H−1(Ω)). Fo his we need o in oduce an auxilia y unc ion whose exis ence is gua an eed by he ollowing esul (see Lemma 1.1. in [9]): Lemma 2.2 Le B ⊂⊂ Ωbe a nonemp y open subse . Then he e exis s a unc ion η0∈C2(Ω) such ha η0>0in Ω,η0= 0 on ∂Ωand |∇η0|>0in Ω B. Fo a ixed nonemp y open subse B ⊂⊂ Ω, le us se α0(x) = e2C∗kη0k∞−eC∗η0(x), x ∈Ω (15) and e α0(x) = e2C∗kη0k∞−e−C∗η0(x), x ∈Ω,(16) C∗being an app op ia e posi i e cons an depending on Ω and B. Using esul s in [9] and [10], one can p o e he ollowing Lemma 2.3 Le zbe he solu ion o (14) associa ed o z0∈L2(Ω). Le Bbe an open subse o Ω. The e exis posi i e cons an s C0,σ0, and σ0(depending only on Ωand B) such ha : (1) I F∈L2(Q), o e e y s≥s0=σ0(Ω,B) (T+T2)one has 1 sZZQe−2sα (T− )|∂ z|2+|∆z|2+sZZQe−2sα −1(T− )−1|∇z|2 +s3ZZQe−2sα −3(T− )−3|z|2≤C0 s3ZZB×(0,T )e−2sα −3(T− )−3|z|2 +ZZQe−2sα|F|2 , wi h αde ined by α(x, ) = α0(x) (T− ), x ∈Ω, ∈(0, T ), and α0gi en by (15). 6 (2) I F= 0+ N X i=1 ∂ i ∂xi ,wi h i∈L2(Q),i= 0,1,...,N, hen sZZQe−2sα −1(T− )−1|∇z|2+s3ZZQe−2sα −3(T− )−3|z|2 ≤C0 s3ZZB×(0,T )e−2sα −3(T− )−3|z|2+ZZQe−2sα| 0|2 +s2 N X i=1 ZZQe−2sα −2(T− )−2| i|2 , o s≥s0=σ0(Ω,B) (T+T2),αbeing as abo e. The explici dependence o s0on Thas been analyzed in [6]. A guing in a simila way, we can ob ain he p ecise way s0depends on T(also see [11]). We will also need he ollowing echnical lemma, which p oo will be gi en u he o he sake o cla i y. Lemma 2.4 Le α0and αbe gi en as in Lemma 2.3, m0= minΩα0, and M0= maxΩα0. (1) One has s4e−2sα −7(T− )−7≤22e−77 m04 T−6, o e e y s≥7T2 23m0 and (x, )∈Q. (2) Fo s≥3T2 2M0 , one has e−2sα −3(T− )−3≥Asexp (−Ms/ ), o (x, )∈ Ω×(0, T/2), wi h As= 26T−6exp −4M0s/T2, Ms= 2M0s/T. (17) (3) Fo e e y s≥0, one has e2sα 3(T− )3≤2−6T6exp 25M0s 3T2!,(x, )∈Ω×(T/4,3T/4) . P oo o Theo em 2.1: The s uc u e o he p oo is simila o ha o P oposi ion 2 in [3]. In he i s place, using app op ia e Ca leman inequal- i ies, we p o e an inequali y in ol ing he unc ions ϕand ψwhich sol e (12) and (13). This inequali y allows us o bound he unc ion ϕin e ms o ψ(see (32)). Combining i wi h ene gy es ima es yields he esul . He e we adap he me hod exhibi ed in [3] o he lack o egula i y in he e m ∇·(Bψ) in equa ion (13). Mo eo e , he cons an s in he inequali ies a e explici . Le us conside wo open se s B1and B2such ha B1⊂⊂ B2⊂ω∩ O. Applying Lemma 2.3 o he solu ion ϕo (12) wi h F=−cϕ −D·∇ϕand B=B1, he e exis posi i e cons an s C1=C1(Ω, B1) and σ1=σ1(Ω, B1) 7 such ha sZZQe−2sα −1(T− )−1|∇ϕ|2+s3ZZQe−2sα −3(T− )−3|ϕ|2 ≤C1s3ZZB1×(0,T )e−2sα −3(T− )−3|ϕ|2, (18) o e e y s≥s1, wi h s1=σ1(Ω, B1)T+T2+T2kck2/3 ∞+T2kDk2 ∞.(19) Then applying Lemma 2.3 o he solu ion ψo (13) wi h B=B1⊂B2and F=−aψ +∇ · (Bψ) + ϕ1O, he e exis posi i e cons an s C2=C2(Ω, B1) and s2=σ2(Ω, B1) (T+T2+T2kak2/3 ∞+T2kBk2 ∞) such ha , o s≥s2, one has sZZQe−2sα −1(T− )−1|∇ψ|2+s3ZZQe−2sα −3(T− )−3|ψ|2 ≤C2 s3ZZB2×(0,T )e−2sα −3(T− )−3|ψ|2+ZZO×(0,T )e−2sα|ϕ|2!. (20) In a i s s ep, we p o e an inequali y which bounds ϕwi h espec o ψ. Conside a unc ion ξ1∈C∞ 0(Ω) such ha 0≤ξ1≤1 in Ω, ξ1= 1 in B1,supp ξ1⊂B2⊂ω∩O,(21) ∆ξ1/ξ1/2 1∈L∞(Ω),and ∇ξ1/ξ1/2 1∈L∞(Ω)N.(22) This is achie ed by se ing ξ1=ζ4, wi h ζ∈C∞ 0(Ω) e i ying (21). To simpli y no a ions, we se u=e−2sαs3 −3(T− )−3.(23) Le s≥s1,s1gi en by (19). Mul iplying (13) by ϕξ1u, in eg a ing o e Q, and aking in o accoun ha u(0) anishes in Ω, we ha e ZZO×(0,T )e−2sαs3 −3(T− )−3|ϕ|2ξ1=ZZQ(a−c)ϕψξ1u +ZZQ(B−D)·∇ϕ ψξ1u+ϕψξ1∂ u−∆(ξ1u) + B·∇(ξ1u) −2ZZQ∇(ξ1u)·∇ϕ ψ := I1+I2+I3+I4+I5+I6. (24) Le us es ima e each Ii,1≤i≤6. In he sequel, Cwill deno e a posi i e cons an depending only on Ω and B1( hus on B2) which may change om one line o ano he . In he i s place, using H¨olde and Young inequali ies, we ha e I1=ZZQ(a−c)ϕψξ1u≤δ1ZZQξ1u|ϕ|2+1 4δ1ka−ck2 ∞ZZQξ1u|ψ|2,(25) 8 o any δ1>0. Then, I2=ZZQ(B−D)·∇ϕ ψξ1u≤γ1ZZQe−2sαs −1(T− )−1|∇ϕ|2ξ1 +1 4γ1kB−Dk2 ∞ZZQe−2sαs5 −5(T− )−5|ψ|2ξ1, (26) o any γ1>0. Le us now obse e ha |∂ u| ≤ T s3e−2sα −5(T− )−5Cs + 3T2/4≤CTs4e−2sα −5(T− )−5, since s≥σ1(Ω, B1)T2. Thus, we can es ima e I3≤ZZQ|ϕ||ψ|ξ1|∂ u| ≤ ZZQCTe−2sαs4 −5(T− )−5|ϕ||ψ|ξ1 ≤δ2ZZQe−2sαs3 −3(T− )−3|ϕ|2ξ1+CT2 δ2ZZQe−2sαs5 −7(T− )−7|ψ|2ξ1 ≤δ2ZZQξ1u|ϕ|2+C δ2ZZQe−2sαs7 −7(T− )−7|ψ|2ξ1, (27) o δ2>0, since s≥σ1(Ω, B1)T. In o de o es ima e I4=−ZZQϕψ∆(ξ1u), le us obse e ha ∆(ξ1u) = s3 −3(T− )−3(∆ξ1)e−2sα + 2∇ξ1·∇(e−2sα) + ξ1∆(e−2sα), wi h |∇(e−2sα)|= 2se−2sα −1(T− )−1|∇α0| ≤ Cse−2sα −1(T− )−1, |∆(e−2sα)| ≤ 2se−2sα −2(T− )−2(2s|∇α0|2+ (T− )|∆α0|) ≤Cse−2sα −2(T− )−2(s+T2)≤Cs2e−2sα −2(T− )−2. Taking hese conside a ions and (22) in o accoun , we ha e I4≤CZZQe−2sαs3 −3(T− )−3|ϕ||ψ|ξ1/2 1 +ZZQe−2sαs4 −4(T− )−4|ϕ||ψ|ξ1/2 1+ZZQe−2sαs5 −5(T− )−5|ϕ||ψ|ξ1. We now use H¨olde and Young inequali ies and (21) o ge I4≤δ3ZZQξ1u|ϕ|2+C δ3ZZQe−2sαs3 −3(T− )−3|ψ|21B2 +C δ3ZZQe−2sαs5 −5(T− )−5|ψ|21B2+C δ3ZZQe−2sαs7 −7(T− )−7|ψ|21B2, 9 o δ > 0. We can also ob ain he co esponding es ima es o Ii+e Ii,i= 3,4,6, simila o (27), (29) and (31), espec i ely, and alid o any s≥ max {s1, C(T+T2)},wi h C > 0 depending only on Ω and B1. Taking such es ima es o (46) and using (44) (and (21)), we can es ima e s3ZZQρ −3(T− )−3|ϕ|2≤Cka−ck2 ∞ZZB2×(0,T )ρs3 −3(T− )−3|ψ|2 +CZZB2×(0,T )ρs7 −7(T− )−7|ψ|2, o s≥max {s1, C(T+T2))}. Then, i s≥s3= max ns1, C T+T2+T2ka−ck1/2 ∞o, he ollowing es ima e o ϕholds ZZQρ −3(T− )−3|ϕ|2≤C3ZZB2×(0,T )ρs4 −7(T− )−7|ψ|2,(47) wi h C3>0 depending on Ω, B1,T,kkk∞;Σ, and k∂ kk∞;Σ. Now, using (45) and (47), a new es ima e o ψanalogous o (34) is ob ained. Mo e p ecisely, he e exis s C4=C4(Ω, B1,T,khk∞;Σ,k∂ hk∞;Σ,kkk∞;Σ,k∂ kk∞;Σ)>0 such ha ZZQρ −3(T− )−3|ψ|2≤C4ZZB2×(0,T )ρs4 −7(T− )−7|ψ|2,(48) o any s≥s4, wi h s4= max ns1, s2, C T+T2+T2ka−ck1/2 ∞o.(49) On he o he hand, mul iplying he equa ion in (41) by ϕand in eg a ing o e Ω, we ge 1 2 d d ZΩ|ϕ( )|2+ZΩ|∇ϕ( )|2≤ kkk∞;Σ Z∂Ω|ϕ( )|2dσ +kck∞ZΩ|ϕ( )|2,(50) o a.e. in (0, T ). We claim ha d d |ϕ( )|2 L2(Ω) ≤K1|ϕ( )|2 L2(Ω),a.e. in (0, T ),(51) K1being a posi i e cons an depending on kck∞and kkk∞;Σ. Indeed, in iew o he chain o embeddings H1(Ω) ⇒Hγ(Ω) ,→L2(Ω), γ < 1, he i s one being compac , o any ε > 0 he e exis s C(ε)>0 such ha kuk2 Hγ(Ω) ≤εZΩ|∇u|2dx +C(ε)|u|2 L2(Ω),∀u∈H1(Ω). 16 Taking also in o accoun he con inuous embedding o Hγ(Ω) in o L2(∂Ω), o γ > 1/2, he e exis s C(kkk∞;Σ)>0 such ha kkk∞;Σ Z∂Ω|ϕ( )|2dσ ≤1 2ZΩ|∇ϕ( )|2+C(kkk∞;Σ)|ϕ( )|2 L2(Ω), o 1/2< γ < 1. Combining his es ima e wi h (50), yields (51), wi h K1gi en by K1= 2(C(kkk∞;Σ) + kck∞). Then |ϕ( +T/4)|2 L2(Ω) ≤exp (K1T/4) |ϕ( )|2 L2(Ω),∀ ∈(T/4,3T/4) , and hence ZZΩ×(T/2,T )|ϕ|2≤exp (K1T/4) ZZΩ×(T/4,3T/4) |ϕ|2.(52) Now, mul iply he equa ion in (42) by ψand in eg a e o e Ω. Using again a compac ness–uniqueness a gumen , we ob ain −d d |ψ( )|2 L2(Ω) ≤K2|ψ( )|2 L2(Ω) +|ϕ( )|2 L2(O),a.e. in (0, T ), wi h K2=K2(kak∞,khk∞;Σ)>0. Then |ψ( )|2 L2(Ω) ≤ZT exp (K2(s− )) |ϕ(s)|2 L2(O)ds, ∀ ∈(0, T), whence ZZΩ×(T/2,T )|ψ|2≤exp (K2T)ZZO×(T/2,T )|ϕ|2.(53) The o m o he weigh unc ion ρde ined in Lemma 2.5 allows one o p o e es ima es simila o hose in Lemma 2.4, wi h e−2sα eplaced by ρ, alid o s≥CT2. Le us ix s= max {s4, CT 2}, wi h s4gi en by (49). We can hus bound bo h sides o (48) and deduce ZZΩ×(0,T/2) exp −N |ψ|2≤C5ZZB2×(0,T )|ψ|2,(54) wi h N > 0 and C5>0 depending on Ω, B1,T,kak∞,kck∞,khk∞;Σ,kkk∞;Σ, k∂ hk∞;Σ, and k∂ kk∞;Σ. In addi ion, due o (53), (52), and (47), we can also es ima e (see a simila p oo in page 12) ZZΩ×(T/2,T )exp −N |ψ|2≤exp K1 T 4+K2T+C6ZZB2×(0,T )|ψ|2,(55) wi h C6>0 depending on Ω, T,kak∞,kck∞,khk∞;Σ,kkk∞;Σ,k∂ hk∞;Σ, k∂ kk∞;Σ, and B1, hus on ωand O. Finally, ga he ing (54) and (55) yields he desi ed obse abili y inequali y, since B2⊂ω, wi h Nas in (54) and K=C5+ exp (K1T/4 + K2T+C6).  17 3 P oo o Theo ems 1.1 and 1.2 We de o e his sec ion o p o e Theo ems 1.1 and 1.2. Bo h p oo s, which a e inspi ed in hose o o he known con ollabili y esul s o nonlinea sys ems (see [12], [13], [3], [14],...), ely on con ollabili y esul s o linea p oblems simila o he linea ized sys em and app op ia e ixed poin a gumen s. The p oo o Theo em 1.2 is simila o he one o Theo em 1.1 and i will be omi ed he e. P oo o Theo em 1.1: We s a wi h he exis ence o app oxima ely insensi izing con ols o a linea ized e sion o (5), (6) o y0= 0. Fo gi en a, c ∈L∞(Q), B, D ∈L∞(Q)Nand ξ∈L2(Q), we conside he linea sys ems      ∂ y−∆y+ay +B·∇y=ξ+ 1ωin Q, y= 0 on Σ, y(x, 0) = 0 in Ω, (56)     −∂ q−∆q+cq −∇·(Dq) = y1Oin Q, q= 0 on Σ, q(x, T ) = 0 in Ω, (57) and he co esponding adjoin sys ems (12) and (13). The ollowing esul holds: P oposi ion 3.1 Assume ha ω∩ O 6=∅. Le Mand Hbe he posi i e cons an s p o ided by Theo em 2.1. Fo any ε > 0, he e exis s a con ol unc ion ε∈L2(ω×(0, T)) such ha he associa ed solu ion (yε, qε)o (56), (57) sa is ies |qε(0)|L2(Ω) ≤ε. (58) In addi ion, i ξ∈L2(Q)sa is ies ZZQexp M |ξ|2dx d < ∞,(59) hen he con ols { ε}ε>0a e uni o mly bounded in L2(ω×(0, T )). Mo e p e- cisely, k εkL2(ω×(0,T )) ≤√HZZQexp M |ξ|2dx d 1/2 ,∀ε > 0.(60) P oo : The s uc u e o he p oo being iden ical o he one in [2] and [3], we will no go in o de ails. Fo ixed ε > 0, we in oduce he unc ional de ined on L2(Ω) J(ϕ0;a, c, B, D) = 1 2ZZω×(0,T )|ψ|2+ε|ϕ0|L2(Ω) +ZZQξψ, (61) 18 whe e ψsol es (13), ϕbeing he solu ion o (12) wi h ini ial da a ϕ0∈L2(Ω). In iew o a unique con inua ion p ope y o he adjoin sys ems (which ollows, o ins ance, om Theo em 2.1), he con inuous and con ex unc ional J(·;a, c, B, D) is s ic ly con ex and sa is ies lim in |ϕ0|L2(Ω)→+∞ J(ϕ0;a, c, B, D) |ϕ0|L2(Ω) ≥ε. (62) Thus, J(·;a, c, B, D) is coe ci e and he e o e i eaches i s minimum a a unique ϕ0 ε∈L2(Ω). Se ε=ψε1ω,(63) (ϕε, ψε) sol ing (12), (13) wi h ini ial da a ϕ0 ε. Then, he solu ion (yε, qε) o (56), (57) associa ed o εsa is ies (58). Indeed, εis he unique con ol o minimal L2–no m sol ing (56)–(58). Now, assume ha ξsa is ies (59). The op imali y condi ion o ϕ0 εand Theo- em 2.1 gi e ZZω×(0,T )|ψε|2+ε|ϕ0 ε|L2(Ω) =−ZZQξ ψε ≤ HZZω×(0,T )|ψε|2!1/2ZZQexp M |ξ|21/2 , which yields, oge he wi h (63), he uni o m es ima e (60).  Rema k 1 In iew o (60), o any ξ∈L2(Q) e i ying (59), one can p o e he exis ence o a con ol ∈L2(ω×(0, T )) such ha he associa ed solu ion (y, q)o (56),(57) sa is ies (7). Mo eo e , his con ol sa is ies he es ima e k kL2(ω×(0,T )) ≤√HZZQexp M |ξ|2dx d 1/2 , wi h Mand Has abo e. Tha is o say, an insensi i i y esul in he linea case can also be p o ed. We now apply a ixed poin a gumen o p o e an app oxima e insensi i i y esul in he nonlinea case. P oposi ion 3.2 Fo ixed ε > 0, unde he assump ions in Theo em 1.1, he e exis a posi i e cons an M(depending on Ω,ω,O,T, and ) such ha o any ξ∈L2(Q)sa is ying (4), one can ind a con ol ε∈L2(ω×(0, T)) so ha he associa ed solu ion (yε, qε)o (5),(6) sa is ies (58). Fu he mo e, k εkL2(ω×(0,T )) ≤ HZZQexp M |ξ|2dx d 1/2 ,∀ε > 0,(64) Hbeing a new posi i e cons an depending on Ω,ω,O,T, and . 19 P oo : Fo a gi en unc ion as in Theo em 1.1, we can w i e (s, p) = g(s, p)s+G(s, p)·p o all (s, p)∈IR ×IRN, whe e g: IR ×IRN→IR and G: IR ×IRN→IRNa e he bounded con inuous unc ions de ined by g(s, p) = Z1 0∂s (σs, σp)dσ, G(s, p) = Z1 0∂p (σs, σp)dσ. (65) Since i is a ixed pa ame e , he dependence on εwill be omi ed in his p oo . Fo any z∈L2(0, T;H1 0(Ω)), we conside he linea sys ems (56) and (57), wi h a=az=g(z, ∇z), c =cz=∂s (z, ∇z)∈L∞(Q) and B=Bz= G(z, ∇z), D =Dz=∂p (z, ∇z)∈L∞(Q)N. Indeed, he hypo hesis on gi es kazk∞,kczk∞,kBzk∞,kDzk∞≤L, ∀z∈L2(0, T;H1 0(Ω)),(66) whe e L > 0 is a bound o ∂s and ∂p in IR×IRN. In iew o P oposi ion 3.1, he e exis s a con ol z∈L2(ω×(0, T)) such ha he co esponding solu ion (yz, qz) o hese sys ems sa is ies |qz(0)|L2(Ω) ≤ε. (67) Le Mzand Hzbe he posi i e cons an s p o ided by Theo em 2.1 o a=az, c=cz,B=Bz, and D=Dz. Recalling he exp essions o Mzand Hz, and using (66), he e exis posi i e cons an s Mand Ho he o m      M=C(Ω,ω,O) (1 + T(1 + L2)) , H= exp C(Ω,ω,O)1 + 1 T+T+ (1 + T)L2,(68) such ha , o all z∈L2(0, T ;H1 0(Ω)), Mz≤ M and √Hz≤ H. Then, i ξ sa is ies (4), using (60) we ha e he ollowing es ima e (uni o m wi h espec o zand ε) k zkL2(ω×(0,T )) ≤ HZZQexp M |ξ|21/2 ,∀z∈L2(0, T;H1 0(Ω)).(69) We now conside he mapping Λε:L2(0, T;H1 0(Ω)) →L2(0, T;H1 0(Ω)) de ined by Λε(z) = yz,yzbeing he solu ion o (56) associa ed o he po en ials a=az and B=Bzand he con ol zp o ided by P oposi ion 3.1. We will apply he Schaude ixed poin heo em o p o e ha Λεpossesses a leas one ixed poin . Fi s , by classical egula i y esul s on he hea equa ion, yzlies in he space Y={u:u∈L2(0, T;H2(Ω) ∩H1 0(Ω)), ∂ u∈L2(Q)}, wi h kyzkY≤exp hC1 + T+T1/2kazk∞+TkBzk2 ∞ikξ+ z1ωkL2(Q) 20 (he e kyzkY=kyzkL2(H2∩H1 0)+k∂ yzkL2(Q)and k·kL2(H2∩H1 0)deno es he no m in L2(0, T;H2(Ω) ∩H1 0(Ω))). Taking in o accoun (66) and (69), one deduces ha Λεmaps L2(0, T;H1 0(Ω)) in o a bounded se o Y. This space being com- pac ly embedded in L2(0, T;H1 0(Ω)), he e exis s a ixed compac se Kin L2(0, T;H1 0(Ω)) such ha Λε(L2(0, T;H1 0(Ω))) ⊂K. (70) Thus, Λεis a compac mapping. Now, le {zj} ⊂ L2(0, T ;H1 0(Ω)) be such ha zj→zin L2(0, T;H1 0(Ω)). F om (66) and he egula i y assump ions on , one has azj=g(zj,∇zj)* az, czj=∂s (zj,∇zj)* czweak-?in L∞(Q), Bzj=G(zj,∇zj)* Bz, Dzj=∂p (zj,∇zj)* Dzweak-?in L∞(Q)N. (71) Le ˆϕ0( esp. ˆϕ0 j,j≥1) be he unique minimize in L2(Ω) o he unc ional J de ined by (61) wi h a=az,c=cz,B=Bzand D=Dz( esp. wi h a=azj, c=czj,B=Bzjand D=Dzj). Reasoning as in [12] and [15], he coe ci i y p ope y (62) is p o ed o be hold uni o mly on po en ials a,c,Band D uni o mly bounded. Then, one can see ha he sequence {ˆϕ0 j}is bounded in L2(Ω) and, inally, one p o es ha ˆϕ0 j→ˆϕ0in L2(Ω).(72) Le now ( ˆϕ, ˆ ψ) ( esp. ( ˆϕj,ˆ ψj), j≥1) be he solu ion o (12), (13) wi h a=az, c=cz,B=Bz,D=Dz( esp. a=azj,c=czj,B=Bzj,D=Dzj) and he ini ial condi ion ˆϕ0( esp. ˆϕ0 j). F om (71) and (72), we ha e ˆϕj→ˆϕ, ˆ ψj→ˆ ψin L2(Q).(73) By de ini ion o Λε, Λε(z) ( esp. Λε(zj), j≥1) is he solu ion o (56) associa ed o he con ol ˆ =ˆ ψ1ω( esp. ˆ j=ˆ ψj1ω) wi h a=az,c=cz,B=Bzand D=Dz( esp. a=azj,c=czj,B=Bzj, and D=Dzj). F om (73) one has ˆ j→ˆ in L2(Q),so ha om (71) one ge s ha Λε(zj)→Λε(z) in L2(Q) and also in L2(0, T;H1 0(Ω)), due o (70). This p o es he con inui y o Λε. All he assump ions o he Schaude heo em being ul illed, Λεpossesses a leas one ixed poin yε∈L2(0, T ;H1 0(Ω)). Then, he con ol ε= yεis such ha yεsol es      ∂ yε−∆yε+g(yε,∇yε)yε+G(yε,∇yε)·∇yε=ξ+ ε1ωin Q, yε= 0 on Σ, yε(x, 0) = 0 in Ω, (74) 21 and he solu ion qεo     −∂ qε−∆qε+∂s (yε,∇yε)qε−∇·(∂p (yε,∇yε)qε) = yε1Oin Q, qε= 0 on Σ, qε(x, T ) = 0 in Ω, (75) sa is ies (58). In o he wo ds, we ha e ound a con ol unc ion ε∈L2(ω× (0, T)) such ha he associa ed solu ion o (5), (6) (wi h y0= 0) e i ies (58). Finally, es ima e (64) ollows eadily om (69), which ends he p oo o P oposi ion 3.2.  We will end he p oo o Theo em 1.1 by passing o he limi in (74), (75), and (58). Since he con ols εp o ided by P oposi ion 3.2 a e uni o mly bounded in L2(ω×(0, T)) and (66) holds, due o he egula izing e ec o he hea equa ion, {(yε, qε)}lies in a bounded se o Y×W(0, T ) (Yde ined in page 20 and W(0, T ) := {u:u∈L2(0, T ;H1 0(Ω)), ∂ u∈L2(0, T;H−1(Ω))}) and acco dingly, in a compac se o L2(0, T ;H1 0(Ω)) ×L2(Q). Then, up o a subsequence, one has ε* weakly in L2(ω×(0, T)), (yε, qε)→(y, q) in L2(0, T ;H1 0(Ω)) ×L2(Q), qε(0) →q(0) in L2(Ω), o some ∈L2(ω×(0, T )), y∈Y,q∈W(0, T). Due o he con inui y o g and G, one can pass o he limi in (74) and (75), deducing ha (y, q) sol es (5), (6) wi h con ol e m and ini ial da um y0= 0. Mo eo e , om (58), he unc ion qsa is ies (7). Thus, he unc ion is an insensi izing con ol o he unc ional Φ gi en by (2). Finally, (64) and he con e gences abo e allow one o es ima e k kL2(ω×(0,T )) ≤ HZZQexp M |ξ|2dx d 1/2 ,(76) wi h Mand Hgi en by (68) and he p oo is comple e.  Rema k 2 The me hod used in Theo em 1.1 o ob ain such a con ol p o- ides an uppe bound o he cos o insensi izing he unc ional Φ. Indeed, in he p oo o he heo em i is shown ha he con ol unc ion can be chosen sa is ying es ima e (76), wi h Mand Hgi en by (68). Inspi ed in [6], deno e by Uad he nonemp y se Uad ={ ∈L2(ω×(0, T)) : (y, q) sa is ies (5)–(7) wi h y0= 0}. Thus, he quan i y Cins = in {k kL2(ω×(0,T )) : ∈ Uad},which measu es he cos o insensi izing he unc ional Φ, can be es ima ed as ollows Cins ≤ HZZQexp M |ξ|2dx d 1/2 . 22 4 Commen s and conclusions Bounda y Fou ie condi ions. P o ing an insensi i i y esul o he sys- em:      ∂ y−∆y+ (y, ∇y) = ξ+ 1ωin Q, ∂ny+hy = 0 on Σ, y(x, 0) = τˆy0(x) in Ω, wi h a C1globally Lipschi z-con inuous unc ion is a much mo e di icul p oblem. Le us obse e ha such an insensi i i y esul is equi alen o he ollowing null con ollabili y p oblem:      ∂ y−∆y+ (y, ∇y) = ξ+ 1ωin Q, ∂ny+hy = 0 on Σ, y(x, 0) = 0 in Ω,     −∂ q−∆q−∇·(∂p (y, ∇y)q) + ∂s (y, ∇y)q=y1Oin Q, ∂nq+hq + (∂p (y, ∇y)·n)q= 0 on Σ, q(x, T ) = 0 in Ω, q(x, 0) = 0 in Ω. This leads us o analyze he null con ollabili y p oblem o he cascade linea sys em      ∂ y−∆y+ay +B·∇y=ξ+ 1ωin Q, ∂ny+hy = 0 on Σ, y(x, 0) = 0 in Ω, (77)     −∂ q−∆q−∇·(Dq) + cq =y1Oin Q, ∂nq+ (h+D·n)q= 0 on Σ, q(x, T ) = 0 in Ω, (78) unde he hypo hesis a, c ∈L∞(Q) and B, D ∈L∞(Q)N(which a e he na u al assump ions on hese po en ials o he gi en unc ion ). Fo his, an obse abili y inequali y o he co esponding adjoin p oblem should be p o ed. This adjoin p oblem is:      ∂ ϕ−∆ϕ+cϕ +D·∇y= 0 in Q, ∂nϕ+hϕ = 0 on Σ, ϕ(x, 0) = ϕ0in Ω,     −∂ ψ−∆ψ−∇·(Bψ) + aψ =ϕ1Oin Q, ∂nψ+ (h+B·n)ψ= 0 on Σ, ψ(x, T) = 0 in Ω. In o de o ob ain such an obse abili y inequali y, we need a Ca leman inequali y o hese adjoin p oblems. The p esence o he e m (B·n)ψ in he bounda y condi ion o ψand he unique hypo hesis B∈L∞(Q)N makes i qui e di icul (e en in he case o a null con ollabili y p oblem o 23 a unique linea hea equa ion wi h he same kind o bounda y condi ions) and his is ou o he scope o his pape . Supe linea nonlinea i ies. The obse abili y esul s p o ed in his pape a e o wide use han he scope o his a icle. Fi s , Theo em 2.1 is used in [4] and [5] o a semilinea hea equa ion wi h a supe linea nonlinea - i y (y). Theo em 2.6 is also used in [7] o he case o nonlinea Fou ie bounda y condi ions. I is o in e es o no ice ha in Theo em 2.6, he dependency o he cons an s wi h espec o he bounda y da a hand k is no explici . This comes om he p oo o Lemma 2.5 (see Lemma 1.2 in [9]). In iew o known null con ollabili y esul s, i is na u al o hink o ex ending Theo em 1.1 o C1locally Lipschi z-con inuous unc ions such ha (0,0) = 0 and lim |(s,p)|→∞ |g(s, p)| log3/2(1 + |s|+|p|)= 0,lim |(s,p)|→∞ |G(s, p)| log1/2(1 + |s|+|p|)= 0, wi h gand G he unc ions gi en by (65) (see Theo em 1.1 in [11]). Obse e ha such nonlinea i ies may lead o blow-up phenomena. Howe e , he idea in [11] o aking sho con ol imes o a oid blow-up o occu ails he e (e en i G≡0), since he ini ial and inal imes a e ixed in insensi i i y p oblems. In [4] and [5] he au ho s in oduce a new echnique and p o e an insen- si i i y esul o nonlinea i ies = (y) wi h ce ain supe linea g ow h a in ini y, e.g. o such as | (s)|=|p1(s)|logα(1+|p2(s)|) o all |s| ≥ s0>0, wi h α∈[0,1), p1and p2being i s o de eal polynomial unc ions. The c ucial poin in hese wo ks is he cons uc ion, in he linea case, o egula con ols s a ing om insensi izing con ols in L2. The idea is as ollows. Le us conside wo open se s B0and Bsuch ha B0⊂⊂ B ⊂ ω∩O. Le ˆ be an L2–con ol, wi h supp ˆ ⊂ B0×[0, T ], such ha he co espond- ing solu ion (ˆy, ˆq) o (56), (57) o B=D= 0 sa is ies (7). Then, se ing q= (1 −θ)ˆq, y = (1 −θ) ˆy+ 2∇θ·∇ˆq+ (∆θ)ˆq, wi h θ∈ D(B) such ha θ≡1 in a neighbo hood o B0, i is possible o u nish a egula insensi izing con ol suppo ed on B × [0, T ]. This cons uc ion uses local egula iza- ion p ope ies o he hea equa ion. This echnique does no apply o he case in ol ing g adien e ms because o he lack o egula i y in oduced by he e m −∇ · (Dq) in (57). Indeed, in his case, he exp ession o he egula con ol we wish o build con ains some e ms, which a e no egula enough o make he s a e ylie in a sui able space o apply a ixed poin a gumen . This is why in Theo em 1.1 we canno conside nonlinea i ies o highe o de . In [7], a local esul on he exis ence o insensi izing con ols o a semi- linea hea equa ion wi h nonlinea bounda y condi ions o Fou ie ype is p o ed. Such bounda y condi ions lead o seek a ixed poin , hus also con- ol unc ions, in ce ain H¨olde spaces. A cons uc ion simila o ha used in [4] and [5], allows one o build, in he linea case, con ols wi h h¨olde ian 24 egula i y s a ing om L2–con ols. Again, his is one o he essen ial poin s in he e e enced wo k. Re e ences [1] Lions, J.-L., Quelques no ions dans l’analyse e le con ˆole de sys `emes `a donn´ees incompl`e es, P oceedings o he XI h Cong ess on Di e en ial Equa ions and Applica ions/Fi s Cong ess on Applied Ma hema ics, Uni e si y o M´alaga, M´alaga, 1990, 43–54. [2] Boda , O., Fab e, C., Con ols insensi izing he no m o he solu ion o a semilinea hea equa ion, J. Ma h. Anal. Appl. 195 (3), 1995, 658–683. [3] De Te esa, L., Insensi izing con ols o a semilinea hea equa ion, Comm. Pa ial Di e en ial Equa ions 25 (1&2), 2000, 39–72. [4] Boda , O., Gonz´alez-Bu gos, M., P´e ez-Ga c´ıa, R., Insensi izing con ols o a semilinea hea equa ion wi h a supe linea nonlinea i y, C. R. Acad. Sci. Pa is, Se . I 335 (8), 2002, 677–682. [5] Boda , O., Gonz´alez-Bu gos, M., P´e ez-Ga c´ıa, R., Exis ence o insensi izing con ols o a semilinea hea equa ion wi h a supe linea nonlinea i y, submi ed o Comm. Pa ial Di e en ial Equa ions. [6] Fe n´andez-Ca a, E., Zuazua, E., The cos o app oxima e con ollabili y o hea equa ions: he linea case, Ad . Di e en ial Equa ions 5 (4–6), 2000, 465–514. [7] Boda , O., Gonz´alez-Bu gos, M., P´e ez-Ga c´ıa, R., A local esul on insensi izing con ols o a semilinea hea equa ion wi h nonlinea bounda y Fou ie condi ions, o appea in SIAM J. Con ol Op im. [8] E ans, L.C., Pa ial Di e en ial Equa ions, G adua e S udies in Ma hema ics 19, Ame ican Ma hema ical Socie y, P o idence, RI, 1998. [9] Fu siko , A., Imanu ilo , O.Yu., Con ollabili y o E olu ion Equa ions, Lec u e No es Se ies #34, Seoul Na ional Uni e si y (Seoul, 1996). [10] Imanu ilo , O.Yu., Yamamo o, M., On Ca leman inequali ies o pa abolic equa ions in Sobole spaces o nega i e o de and exac con ollabili y o semilinea pa abolic equa ions, UTMS 98-46. [11] Doubo a, A., Fe n´andez-Ca a, E., Gonz´alez-Bu gos, M., Zuazua, E., 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 (3), 2002, 798–819. [12] Fab e, C., Puel, J.-P., Zuazua, E., App oxima e con ollabili y o he semilinea hea equa ion, P oc. Royal Soc. Edinbu gh, 125 A, 1995, 31–61. [13] Fe n´andez-Ca a, E., Zuazua, E., 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 (5), 2000, 583–616. 25