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1 SIAM J. CONTROL OPTIM. Vol. 48, No. 8, pp c 1 Society for Industrial and Applied Mathematics NULL CONTROLLABILITY OF A PARABOLIC SYSTEM WITH A CUBIC COUPLING TERM JEAN-MICHEL CORON, SERGIO GUERRERO, AND LIONEL ROSIER Abstract. We consider a system of two parabolic equations with a forcing term present in one equation and a cubic coupling term in the other one. We prove that the system is locally null controllable. Key words. return method null controllability, parabolic system, nonlinear coupling, Carleman estimate, AMS subject classifications. 35K4, 93B5 DOI / Introduction. The control of coupled parabolic systems is a challenging issue, which has attracted the interest of the control community in the last decade. Let Ω be a nonempty connected bounded subset of R N of class C.Letω be a nonempty open subset of Ω. In [, 1], the authors identified sharp conditions for the control of systems of the form 1.1) w t =DΔ+A)w + Bh, where w =w 1,...,w N ):Ω R N is the state to be controlled, h = ht, ) :Ω R M is the control input supported in ω, andd : R N R N, A : R N R N,and B : R M R N are linear maps. In general, the rank of B is less than N, so that the controllability of the full system depends strongly on the linear) coupling present in the system. See also [9, 11] for the Stokes system. Here, we are concerned with the control of systems of semilinear parabolic systems in which the coupling occurs through nonlinear terms only. More precisely, we study the control properties of systems of the form u t Δu = gu, v)+h1 ω in,t) Ω, 1.) v t Δv = u 3 + Rv in,t) Ω, u =, v = in,t) Ω, where g : R R R is a given function of class C vanishing at, ) R R, R is a given real number, and 1 ω is the characteristic function of ω. This is a control Received by the editors January 9, 1; accepted for publication in revised form) September 16, 1; published electronically December 16, 1. Institut Universitaire de France, F-755 Paris, France, and Université Pierre et Marie Curie - Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, F-755 Paris, France coron@ann.jussieu.fr). This author s work was partially supported by the Agence Nationale de la Recherche ANR), Project C-QUID, grant BLAN Université Pierre et Marie Curie - Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, F-755 Paris, France guerrero@ann.jussieu.fr). This author s work was partially supported by the Agence Nationale de la Recherche ANR), Project CISIFS, grant ANR-9-BLAN-13-. Institut Élie Cartan, UMR 75 UHP/CNRS/INRIA, B.P. 39, 5456 Vandœuvre-lès-Nancy Cedex, France rosier@iecn.u-nancy.fr). This author s work was partially supported by the Agence Nationale de la Recherche ANR), Project CISIFS, grant ANR-9-BLAN

2 563 J.-M. CORON, S. GUERRERO, AND L. ROSIER system where, at time t [,T], the state is ut, ),vt, )) : Ω R and the control is ht, ) :Ω R. The goal of this paper is to prove the local null controllability of system 1.). Our main result is as follows. Theorem 1. There exists δ> such that, for every u,v ) L Ω) satisfying u L Ω) + v L Ω) <δ, there exists a control h L,T) Ω) such that the solution u, v) L [,T] Ω) of the Cauchy problem u t Δu = gu, v)+h1 ω in,t) Ω, v t Δv = u 3 + Rv in,t) Ω, 1.3) u =, v = in,t) Ω, u, ) =u ), v, ) =v ) in Ω satisfies 1.4) ut, ) = and vt, ) = in Ω. Let us give a system from chemistry to which our result applies. A reactiondiffusion system describing a reversible chemical reaction see [4, 5, 1]) takes the form 1.5) 1.6) u t =Δu aku k v m ) in,t) Ω, v t =Δv + bku k v m ) in,t) Ω, together with homogeneous Neumann boundary conditions. In 1.5) 1.6), a and b denote some positive numbers, and k and m arepositiveintegers.thecorresponding reversible chemical reaction reads ka mb. Incorporating a forcing term 1 ω h in 1.5), we obtain a system of the form 1.3) when k =3andm = 1, so that Theorem 1 may be applied. In fact, Theorem 1 deals with Dirichlet homogeneous boundary conditions, but the proof we give here can easily be adapted to deal with homogeneous Neumann boundary conditions as well, and a scaling argument shows that one may assume without loss of generality that b =1/3.) Remark. In Theorem 1, it is not possible to replace u 3 by u. Indeed, by the maximum principle, for every u,v ) L Ω) and for every h L,T) Ω), the solution u, v) L [,T] Ω) of the Cauchy problem u t Δu = gu, v)+h1 ω in,t) Ω, v t Δv = u + Rv in,t) Ω, u =, v = in,t) Ω, u, ) =u ), v, ) =v ) in Ω, if it exists, satisfies vt, ) v T, ) in Ω,

3 CONTROLLABILITY PARABOLIC SYSTEM CUBIC COUPLING 5631 where v L [,T] Ω) is the solution of the linear Cauchy problem vt Δv = Rv in,t) Ω, v = in,t) Ω, v, ) =v ) in Ω. In particular, by the strong) maximum principle, if v andv,thenvt, ) > in Ω. L. Robbiano asked one of the authors whether the result in Theorem 1, with u 3 still replaced by u, could be true if we consider complex-valued functions. The following result, whose proof is sketched in the appendix, shows that this is indeed the case. Theorem 3. There exists δ > such that, for every u,v ) L Ω; C) satisfying u L Ω) + v L Ω) <δ, there exists a control h L,T) Ω; C) such that the solution u, v) L [,T] Ω; C) of the Cauchy problem u t Δu = gu, v)+h1 ω in,t) Ω, v t Δv = u + Rv in,t) Ω, 1.7) u =, v = in,t) Ω, u, ) =u ), v, ) =v ) in Ω satisfies 1.8) ut, ) = and vt, ) = in Ω. When trying to prove a local null controllability result, the first thing to do is to look at the null controllability of the linearized control system around. Here, the linearized control system reads as u t Δu = u g, )u + v g, ) + h1 ω in,t) Ω, 1.9) v t Δv = Rv in,t) Ω, u =, v = in,t) Ω. Clearly, the control h has no influence on v, andifv, ),thenvt, ). Hence the linearized control system 1.9) is not null controllable and this strategy cannot be applied to prove Theorem 1. Our proof of Theorem 1 relies on the return method, a method introduced in [6] for a stabilization problem and in [7] for the controllability of the Euler equations of incompressible fluids see [8, Chapter 6] and the references therein for other applications of this method). Applied to the control system 1.), it consists in looking for a trajectory u, v), h) of the control system 1.) such that the following hold: i) it goes from, ) to, ), i.e., u, ) =v, ) =ut, ) =vt, ) =; ii) the linearized control system around that trajectory is null controllable. With this trajectory and a suitable fixed point theory at hand, one can hope to get the null controllability stated in Theorem 1. We shall see that this is indeed the case.

4 563 J.-M. CORON, S. GUERRERO, AND L. ROSIER This paper is organized as follows. Section is devoted to the construction of the trajectory u, v), h). In section 3, using some Carleman inequality, we prove that the linearized control system around u,v), h) is null controllable subsection 3.1). Next, we deduce the local null controllability around this trajectory by using the Kakutani fixed point theorem subsection 3.). The appendix contains a sketch of the proof of Theorem 3.. Construction of the trajectory u, v), h). Let us define Q :=,T) Ω. The goal of this section is to prove the existence of u C Q), v C Q), and h C Q) such that.1).).3).4) the supports of u, v, andh are compact and included in,t) ω, u t Δu = gu, v)+h in Q, v t Δv = u 3 + Rv in Q, u. The existence of such u C Q), v C Q), and h C Q) follows from the following theorem. Theorem 4. Let ρ> and R be two constants. Then there exist two functions V :t, x) R R N V t, x) R and K :t, x) R R N Kt, x) R such that.5).6).7).8) V C R R N ), and V t, x) = t, x) R R N with max t, x ) ρ, K C R R N ), and Kt, x) = t, x) R R N with max t, x ) ρ, K, V t =ΔV + RV + K 3. Indeed, let x ω, and let ρ> be small enough so that max t T ), x x ρ t, x),t) ω. Then, it suffices to define u C Q), v C Q), and h C Q) by ut, x) :=K t T ),x x, vt, x) :=V t T ),x x t, x) Q, h := u t Δu gu, v) in Q. ProofofTheorem4. Note first that we may assume without loss of generality that R =. Indeed, setting Ṽ t, x) =e Rt V t, x), Kt, x) =e Rt/3 Kt, x),.8) is transformed into Ṽ t =ΔṼ + K 3. From now on, we assume that R =. We may also assume that ρ = 1. Indeed, if the construction has been done for ρ =1andR =, then for any ρ> the functions Ṽ t, x) =V ρ t, ρ 1 x), Kt, x) =ρ 3 Kρ t, ρ 1 x),

5 CONTROLLABILITY PARABOLIC SYSTEM CUBIC COUPLING 5633 with support in [ ρ,ρ ] { x ρ}, satisfy the equation Ṽt =ΔṼ + K 3. We assume from now on that R =andthatρ =1. Letr = x. We seek a radial function V t, x) = vt, r) fulfilling the following properties:.9).1) v C R R + ), vt, r) = for t 1orr 1, k := v t v rr N 1 ) 1 3 v r C R R + ). r The smoothness of V and K := V t ΔV ) 1 3 at the points t, ), t [ 1, 1], will follow from additional properties of v see below). As far as the construction of v is concerned, the idea is to have a precise knowledge of the place where k vanishes and a good behavior of v near the place where k vanishes to ensure that k is of class C.Forthev we are going to construct, we shall have See Figure 1. {t, r); kt, r) < } = {t, r); <λt)/ <r<λt)}, {t, r); kt, r) > } = {t, r); <r<λt)/}. Fig. 1. {t, r); kt, r) > } and {t, r); kt, r) < }. Let us introduce some notation. Let.11).1) λt) =ε1 t ), t < 1, { f t) = e 1 1 t, t < 1,, t 1, where ε> is a small) parameter chosen later. We search for v in the form 3.13) vt, r) = f i t)g i z), i=

6 5634 J.-M. CORON, S. GUERRERO, AND L. ROSIER where z := r/λt), g is defined in Lemma 5 see below), and the functions f i = f i t), 1 i 3, and g i = g i z), 1 i 3, defined during the proof, are in C R) and fulfill.14).15) supp f i [ 1, 1], [ 1 supp g i δ, 1 + δ ]. In.15), δ, 1/1) is a given number. Let us begin with the construction of g. Lemma 5. There exists a function G C [, + )) such that.16) Gz) = z 1.17) z 1 ) 3 for 1 δ<z<1 + δ, ) Gz) > for <z<1, z 1, and such that the solution g to the Cauchy problem.18).19) g z)+ N 1 g z) =Gz), z >, z g 1) = g 1) = satisfies.).1).) g z) =1 z if <z<δ, g z) =e 1 1 z if 1 δ<z<1, g z) = if z 1. Proof of Lemma 5. Note first that, by.18),.).),.3) N if <z<δ, [ Gz) = N1 z ) 8z 1 z ) 3 +4z 1 z ) 4] e 1 1 z if 1 δ<z<1, if z 1. Hence only the values of G on [δ, 1 δ] andon[1 + δ, 1 δ] remain to be defined. Let G C, + ) be any function satisfying.16) and.3), and denote by g the solution of.18).19). Clearly,.1).) are satisfied. Finally, it is clear that.) holds if and only if g + )=1andg + ) =. Note that.18) may be written as 1.4) z N 1 zn 1 g ) = G. Using.19), this gives upon integration.5) z N 1 g z) = This imposes the condition 1 z s N 1 Gs) ds..6) 1 s N 1 Gs) ds =.

7 CONTROLLABILITY PARABOLIC SYSTEM CUBIC COUPLING 5635 Note that if.6) holds, then, by.5),.7) g z) = 1 z N 1 which, combined with.3), yields z s N 1 Gs) ds, g z) = z, <z<δ, and g + ) =. Integrating.7) on [z,1] and using.19),.3),.6), and an integration by parts, we obtain, for <z<δ, 1 1 y ).8) g z) = s N 1 Gs) ds dy.9) = z y N N 1 z z ygy)dy N z if N, yln y)gy)dy z ln z if N =. Then g + ) = 1, provided that 1 ygy)dy = N if N,.3) 1 yln y)gy)dy =1 ifn =. It is then an easy exercise to extend G on [δ, 1 δ] [ 1 + δ, 1 δ] in such a way that G is smooth and.17),.6), and.3) are satisfied. This concludes the proof of Lemma 5. Let us turn now to the definition of the functions f i and g i for 1 i 3. Let t, z) range over 1, 1), 1), so that t, r) ranges over the domain O := {t, r); 1 <t<1, <r<λt)}. Differentiating in.13), we obtain 3 v t = f i g i λ ) λ f izg 1) i, i= v rr + N 1 v r = r 3 i= λ f i g ) i + N 1 z ) g 1) i, where f i := df i /dt and g j) i := d j g i /dz j. Let us introduce the function V = Vt, z) defined by [ V := λ v rr + N 1 ] v r v t r 3 = [f i g ) i + N 1 ) ].31) g 1) i + zλ λf i g 1) i λ f i g i. z i=

8 5636 J.-M. CORON, S. GUERRERO, AND L. ROSIER We aim to define f i and g i so that V = V z = V zz = for 1 <t<1, z= 1, V zzz cf for 1 <t<1, z= 1, for some constant c>. By.16),.18), and.31), V, 1 ) = 1 ) ) λ λf g 1) 1 λ f 1 g 3 ) + [f i g ) 1 i +N 1)g 1) i i=1 )]. λ f i g i 1 )) λ λf i g 1) i ) 1 We impose the condition ) {.3) g j) 1 1 if i =1andj =, i = for 1 i 3, j. otherwise It follows that V, 1 ) = 1 ) ) λ λf g 1) 1 λ f 1 g + f 1. The function f 1 is then defined by.33) f 1 := 1 λ λf g 1) ) ) 1 + λ f 1 g, so that.34) V, 1 ) = on 1, 1). Differentiating with respect to z in.31) yields, once more using.18), V z = f G 1) + zλ λf g ) +λ λf λ f )g 1) 3 + [f i g 3) i + N 1 g ) i N 1 ).35) z z g 1) i i=1 We infer from.16),.3), and.35) that V z, 1 ) = 1 λ λf g ) f i g 3) 1 i We impose the condition.36) g 3) i i=1 + zλ λf i g ) i ) +λ λf λ f )g 1) ] +λ λf i λ f i )g 1) i. ) 1 ) ) + f 1 N 1) + 1 λ λ ) { 1 1 if i =, = if i {1, 3} ).

9 and define f as.37) f := CONTROLLABILITY PARABOLIC SYSTEM CUBIC COUPLING 5637 [f 1 N 1) + 1 ) λ λ + 1 λ λf g ) ) 1 +λ λf λ f )g 1) It follows that.38) V z, 1 ) = on 1, 1). Differentiating.35) with respect to z, weget V zz = f G ) + zλ λf g 3) +λ λf λ f )g ) 3 + [f i g 4) i + N 1 g 3) i N 1 z z g ) i + N 1 ) z 3 g 1) i i=1 ].39) +λ λf i λ f i )g ) i, which, together with.16),.3), and.36), leads to.4) V zz, 1 ) = 1 ) λ λf g 3) 1 +λ λf λ f )g ) 3 + i=1 f i g 4) i 1 We impose the condition.41) g 4) i ) 1 )] 1. + zλ λf i g 3) i ) ) +N 1)f 8N 1)f λ λf +λ λf 1 λ f 1 ). ) { 1 1 if i =3, = if i {1, } and define f 3 as [ f 3 := N 1) + 1 ) λ λ f +λ λ 8N 1))f 1 λ f ) )] λ λf g 3) 1 +λ λf λ f )g ) 1.4). This gives.43) V zz, 1 ) = on 1, 1). By.16) and.39), we have, for t, z) 1, 1) 1 δ, 1 + δ), V zzz =6f + R, where R := zλ λf g 4) +3λ λf λ f )g 3) 3 + [f i g 5) i + N 1 g 4) i 3 N 1 z z g 3) i +6 N 1 z 3 g ) i 6 N 1 ) z 4 g 1) i i=1 ].44) + zλ λf i g 4) i +3λ λf i λ f i )g 3) i.

10 5638 J.-M. CORON, S. GUERRERO, AND L. ROSIER Let C denote various constants independent of ε, t, andz, which may vary from line to line. We claim that 1.45) R Cε f for t, z) 1, 1) δ, 1 ) + δ. First, we have that ) R C λ λf + λ f 3 + f i + λ λf i + λ f i ). Since.46) λ λf = 4ε t1 t ) 3 f, λ f = ε t1 t ) f, onemaywriteforeachi {1,...,3} i=1 f i t) =ε p i t, ε)f t), where p i R[t, ε]. Therefore, there exists some constant C> such that λ λf + λ f + 3 f i + λ f i ) Cε f, i=1 and.45) follows. We infer that for ε small enough V zzz 6 Cε )f f for t, z) [ 1, 1] 1 δ, 1 ) + δ. In view of the definitions of V and of f i for 1 i 3, we may write.47) Vt, z) =f t) p P j t)k j z) =:f t)at, z), j=1 where p 1, P j R[t], and k j C [, + )). Since A, 1 ) = A z, 1 ) = A zz, 1 ) = while 1 A zzz t, z) 1 for t, z) [ 1, 1] δ, 1 ) + δ, we conclude that we can write.48) At, z) = z 1 ) 3 ϕt, z) for t [ 1, 1], z where ϕ C [ 1, 1] t 1 δ, 1 + δ) z)and ϕt, z) > for t [ 1, 1], z 1 δ, 1 ) + δ. 1 δ, 1 ) + δ,

11 CONTROLLABILITY PARABOLIC SYSTEM CUBIC COUPLING 5639 From.15),.18),.31), and.47) we have that, for t [ 1, 1] and z 1 δ, At, z) =G + zλ λg 1) λ f g, f which, combined with.46) and.17), yields At, z) G Cε Gz). It follows that for ε> small enough.49) At, z) > 1 [ Gz) > for t [ 1, 1], z, 1 δ ] [ 1 + δ ), 1. Gathering.48).49) we obtain that λ V = λ f t)at, z) =Bt, z) 3, t 1, 1), z [, 1), for some B C 1, 1) [, 1)). Now define V t, x) by V t, x) :=vt, x ) = 3 i= ) x f i t)g i. λt) From.15) and.) we have that ) V t, x) =f t) 1 x λ for x <δ λt). t) Combined with.11),.1), and.1), this yields.5) V C R R N ). On the other hand, it follows from.15),.3),.31),.47),.48), and.49) that λ V = λ f z 1 ) 3 1 z ) 4 e 1 1 z ψt, z) for t, z) 1, 1) [, 1) for some function ψ C [ 1, 1] [, 1]) with ψt, z) η>on[ 1, 1] [, 1]. We observe that the function.51) λ V) 1 3 =λ f ) 1 3 r λ 1 )1 r λ ) e r λ ψ 1 3 t, r ) λ when extended by for t 1orr λt) isofclassc on R t [, + ) r. Therefore ) ΔV V t ) 1 3 t, x) =λ V) 1 x 3 t, λt) is of class C on R R N \ [ 1, 1] {}) and on a neighborhood of 1, 1) {} by.5) and the fact that ΔV V t )t, ) = Nε 1 t ) 4 f t)+t1 t ) f t) < for ε> small enough. The smoothness of v t Δv) 1/3 near ±1, ) follows from.1) and.51). The proof of Theorem 4 is complete.

12 564 J.-M. CORON, S. GUERRERO, AND L. ROSIER 3. Local null controllability around the trajectory u, v), h). We consider the trajectory u,v), h) of the control system 1.) constructed in section. Let u, v),h):q R R, and let ζ 1,ζ ), h) :=u u, v v),h h). Then u, v),h) is a trajectory of 1.) if and only if ζ 1,ζ ), h) fulfills ζ 1,t Δζ 1 = G 11 ζ 1,ζ )ζ 1 + G 1 ζ 1,ζ )ζ + h1 ω in,t) Ω, 3.1) ζ,t Δζ = G 1 ζ 1,ζ )ζ 1 + G ζ 1,ζ )ζ in,t) Ω, ζ 1 =, ζ = in,t) Ω, where G 11 ζ 1,ζ )t, x) := = 1 g u λζ 1t, x)+ut, x),ζ t, x)+vt, x)) dλ gζ 1 + u, ζ + v) g j u, ζ + v) t, x) if ζ 1 t, x), ζ 1 u gut, x),ζ t, x)+vt, x)) if ζ 1 t, x) =, G 1 ζ 1,ζ )t, x) := = 1 g v ut, x),λζ t, x)+vt, x)) dλ gu, ζ + v) gu, v) t, x) if ζ t, x), ζ v gut, x), vt, x)) if ζ t, x) =, and G 1 ζ 1,ζ ):=3u +3uζ 1 + ζ 1 ), G ζ 1,ζ ):=R. Note that 3.) G 1, ) = 3u. By.4) and 3.), there exist t 1,T), t,t), a nonempty open subset ω of Ω, and M> such that 3.3) 3.4) ω ω, t 1 <t, G 1, )t, x) M t, x) t 1,t ) ω. Increasing M> if necessary, we may also assume that 3.5) G ij, ) L Q) M i, j) {1, }. It is therefore natural to study the null controllability of the linear systems ζ 1,t Δζ 1 = a 11 t, x)ζ 1 + a 1 t, x)ζ + h1 ω in,t) Ω, 3.6) ζ,t Δζ = a 1 t, x)ζ 1 + a t, x)ζ in,t) Ω, ζ 1 =, ζ = in,t) Ω

13 CONTROLLABILITY PARABOLIC SYSTEM CUBIC COUPLING 5641 under the following assumptions: 3.7) 3.8) a 1 t, x) 1 M t, x) t 1,t ) ω, a ij L Q) M i, j) {1, }. We will do this study in subsection 3.1. In subsection 3., we deduce from this study the local null controllability around the trajectory u,v), h) and therefore get Theorem Null controllability of a family of linear control systems. Let E be the set of a 11,a 1,a 1,a ) L Q) 4 such that 3.7) and 3.8) hold. The goal of this subsection is to prove the following lemma. Lemma 6. There exists C>such that, for every a 11,a 1,a 1,a ) E and for every α 1,α ) L Ω), there exists a control h L Q) satisfying 3.9) h L Q) C α 1 L Ω) + α L Ω)) such that the solution to the Cauchy problem ζ 1,t Δζ 1 = a 11 t, x)ζ 1 + a 1 t, x)ζ + h1 ω in,t) Ω, ζ,t Δζ = a 1 t, x)ζ 1 + a t, x)ζ in,t) Ω, 3.1) ζ 1 =, ζ = in,t) Ω, ζ 1, ) =α 1, ζ, ) =α in Ω satisfies 3.11) ζ 1 T, ) = and ζ T, ) =. Note that the coefficients a jk are in L Q), so that we have existence and uniqueness of solutions of 3.1) in C [,T]; L Ω) ) L,T; H 1 Ω) ). In order to prove Lemma 6, we take ht, ) = for every t,t 1 ). Note that there then exists C> such that, for every a 11,a 1,a 1,a ) E and for every α 1,α ) L Ω), the solution to the Cauchy problem 3.1) satisfies with α 1,α ) L Ω) C α 1,α ) L Ω), α 1,α ):=ζ 1 t 1, ),ζ t 1, )). Then, our goal will be to find h :t 1,t ) Ω R satisfying 3.1) h L t 1,t ) Ω) C ζ 1 t 1, ) L Ω) + ζ t 1, ) L Ω)) such that the solution ζ 1,ζ ) of the Cauchy problem ζ 1,t Δζ 1 = a 11 t, x)ζ 1 + a 1 t, x)ζ + h1 ω in t 1,t ) Ω, ζ,t Δζ = a 1 t, x)ζ 1 + a t, x)ζ in t 1,t ) Ω, 3.13) ζ 1 =, ζ = in t 1,t ) Ω, ζ 1 t 1, ) =α 1, ζ t 1, ) =α in Ω

14 564 J.-M. CORON, S. GUERRERO, AND L. ROSIER satisfies 3.14) ζ 1 t, ) = and ζ t, ) =. Finally, we take ht, ) = for every t t,t). Of course, this construction provides a control h L,T) Ω) driving the solution of 3.1) to, ) at time t = T.In subsection we prove the existence of h :t 1,t ) Ω R satisfying the required property but with an L -bound instead of 3.1). In subsection 3.1. we deal with the condition 3.1). For the sake of simplicity, in these two subsections, we write,t) instead of t 1,t )andqinstead of t 1,t ) Ω Controls in L. The goal of this subsection is to prove a null controllability result for the linear control systems 3.6) with L controls. Lemma 7. There exists C>such that, for every a 11,a 1,a 1,a ) E and for every α 1,α ) L Ω), there exists a control h L Q) satisfying h L Q) C α 1 L Ω) + α L Ω)) such that the solution to the Cauchy problem 3.1) satisfies 3.11). In order to prove Lemma 7, we consider the associated adjoint system ϕ 1,t Δϕ 1 = a 11 t, x)ϕ 1 + a 1 t, x)ϕ in,t) Ω, ϕ,t Δϕ = a 1 t, x)ϕ 1 + a t, x)ϕ in,t) Ω, 3.15) ϕ 1 =, ϕ = in,t) Ω, ϕ 1 T, ) =ϕ 1,T ), ϕ T, ) =ϕ,t ) in Ω and set ϕ := ϕ 1,ϕ ). For this system, we intend to prove the following observability inequality: 3.16) ϕ,x) dx C ϕ 1 dx dt. Ω,T ) ω From estimate 3.16), with C>independent of a 11,a 1,a 1,a ) E, it is classical to deduce Lemma 7. Let us recall the following Carleman inequality, proved in [1, Chapter 1], for the heat equation with Dirichlet boundary conditions. Lemma 8. Let ηt) =t 1 T t) 1. Let ω 1 be a nonempty open set included in Ω. There exist a constant C>andafunctionρ C Ω;, + )) such that, for every z H 1,T; L Ω)) L,T; H Ω)) and for every s C, 3.17) e sρx)ηt) sη) 3 z + sη z +sη) 1 Δz + z t )) dx dt,t ) Ω ) C e sρx)ηt) z t +Δz dx dt + e sρx)ηt) sη) 3 z dx dt.,t ) Ω,T ) ω 1 For ω 1 we take a nonempty open set of R N whose closure in R N ) is included in ω. Unless otherwise specified, we denote by C various positive constants varying from line to line which may depend on Ω, ω, ω 1, T, ρ, M, and other variables which will be specified later. However, they are independent of a 11,a 1,a 1,a ) E,of ϕ 1,T,ϕ,T ), and of other variables which will be specified later.

15 CONTROLLABILITY PARABOLIC SYSTEM CUBIC COUPLING 5643 We start by applying 3.17) to ϕ 1 and ϕ the solution of 3.15)): e sρx)ηt) sη) 3 ϕ + sη ϕ +sη) 1 Δϕ + ϕ t )) dx dt,t ) Ω C e sρx)ηt) a 11 t, x)ϕ 1 + a 1 t, x)ϕ + a 1 t, x)ϕ 1,T ) Ω ) + a t, x)ϕ )dx dt + e sρx)ηt) sη) 3 ϕ dx dt,t ) ω 1 for every s C. Using 3.8) and taking s large enough, we have e sρx)ηt) sη) 3 ϕ + sη ϕ +sη) 1 Δϕ + ϕ t )) dx dt,t ) Ω 3.18) C e sρx)ηt) sη) 3 ϕ dx dt,t ) ω 1 for every s C. Finally, we estimate the local integral of ϕ. We multiply the first equation in 3.15) by χx)e sρx)ηt) sη) 3 ϕ,where 3.19) χ C Ω; [, + )), the support of χ is included in ω and χ =1inω 1. Integrating in,t) ω,thisgives a 1 t, x)χx)e sρx)ηt) sη) 3 ϕ dx dt,t ) ω 3.) = χx)e sρx)ηt) sη) 3 ϕ ϕ 1,t Δϕ 1 a 11 t, x)ϕ 1 ) dx dt.,t ) ω Thanks to 3.7) and 3.19), the integral in the left-hand side of 3.) is bounded from below by C 1,T ) ω 1 e sρx)ηt) sη) 3 ϕ dx dt. Let us now estimate the integral in the right-hand side of 3.). Let ε, 1). From now on the constant C>maydepend on ε, 1). Using 3.8) for i, j) =1, 1)), we have that 3.1) χx)e sρx)ηt) sη) 3 ϕ a 11 t, x)ϕ 1 dx dt,t ) ω ɛ e sρx)ηt) sη) 3 ϕ dx dt + C e sρx)ηt) sη) 3 ϕ 1 dx dt.,t ) Ω,T ) ω Next, for the time derivative term, we integrate by parts with respect to t. Weget χx)e sρx)ηt) sη) 3 ϕ ϕ 1,t dx dt,t ) ω 3.) = χx)e sρx)ηt) sη) 3 ϕ,t ϕ 1 dx dt,t ) ω + χx)e sρx)ηt) sη) 3 ) t ϕ ϕ 1 dx dt.,t ) ω

16 5644 J.-M. CORON, S. GUERRERO, AND L. ROSIER Using that e sρx)ηt) sη) 3 ) t Cs 4 e sρx)ηt) ηt) 5 and Cauchy Schwarz s inequality, we can estimate this term in the following way: χx)e sρx)ηt) sη) 3 ϕ ϕ 1,t dx dt,t ) ω 3.3) ɛ e sρx)ηt) sη) 1 ϕ,t +sη) 4 ϕ ) dx dt,t ) Ω + C e sρx)ηt) sη) 7 ϕ 1 dx dt,t ) ω for s C. Finally, for the integral term with Δϕ 1, we integrate by parts twice with respect to x to get χx)e sρx)ηt) sη) 3 ϕ Δϕ 1 dx dt,t ) ω = Δχx)e sρx)ηt) sη) 3 ϕ ) ϕ 1 dx dt.,t ) ω Using that Δχx)e sρx)ηt) sη) 3 ϕ ) Ce sρx)ηt) sη) 3 Δϕ + sη ϕ +sη) ϕ ), t, x),t) ω in the previous identity, together with Cauchy Schwarz s inequality, we deduce that χx)e sρx)ηt) sη) 3 ϕ Δϕ 1 dx dt,t ) ω 3.4) ɛ e sρx)ηt) sη) 1 Δϕ +sη) ϕ +sη) 4 ϕ ) dx dt,t ) Ω + C e sρx)ηt) sη) 7 ϕ 1 dx dt.,t ) ω Combining inequalities 3.1), 3.3), and 3.4) with 3.18) and 3.) and taking ε, 1) small enough, we obtain e sρx)ηt) sη) 3 ϕ + sη ϕ +sη) 1 Δϕ + ϕ t )) dx dt 3.5),T ) Ω C e sρx)ηt) sη) 7 ϕ 1 dx dt,t ) ω for every s C. From this estimate and taking into account the dissipation in time) of the heat system 3.15), one gets for s large enough 3.6) ϕ, ) L Ω) C,T ) ω e sρx)ηt) sη) 7 ϕ 1 dx dt, which gives 3.16). Note that the constant C in 3.6) may depend on s at this time. The proof of Lemma 7 is finished.

17 CONTROLLABILITY PARABOLIC SYSTEM CUBIC COUPLING Controls in L. Let us remark that the proof in this subsection follows ideas of [3]. Let ε, 1). In this subsection the constants C>donotdepend on ε, 1). We choose s> large enough so that 3.5) and therefore also 3.6)) holds. Let α 1,α ) L Ω). Let us consider, for each ε>, the extremal problem 1 3.7) inf h L,T ) ω ),T ) ω e sρx)ηt) sη) 7 h dx dt + 1 ε ζt, ) L Ω), where ζ := ζ 1,ζ ) is the solution of 3.6) satisfying the initial condition 3.8) ζ 1, ) =α 1, ζ, ) =α. Clearly, there exists a unique) solution of 3.7) h ε with e sρx)ηt) sη) 7 ) 1/ h ε belonging to L,T) ω ). We extend h ε to all of Q by letting h ε := on Q \,T) ω. Let us call ζ ε := ζ1 ε,ζε ) the solution of 3.6) associated with hε with, again, the initial condition 3.8). The necessary condition of the minimum yields 3.9) e sρx)ηt) sη) 7 h ε hdxdt+ 1 ζ ε T ) ζt ) dx = h L,T) ω ),,T ) ω ε Ω where ζ := ζ 1,ζ ) is the solution of ζ 1,t Δζ 1 = a 11 t, x)ζ 1 + a 1 t, x)ζ + h1 ω in,t) Ω, ζ,t Δζ = a 1 t, x)ζ 1 + a t, x)ζ in,t) Ω, 3.3) ζ 1 =, ζ = on,t) Ω, ζ 1, ) =ζ, ) = inω. Let us now introduce ϕ ε 1,ϕ ε ) as the solution of the following homogeneous adjoint system: ϕ ε 1,t Δϕ ε 1 = a 11 t, x)ϕ ε 1 + a 1 t, x)ϕ ε in,t) Ω, 3.31) ϕ ε,t Δϕε = a 1t, x)ϕ ε 1 + a t, x)ϕ ε in,t) Ω, ϕ ε 1 = ϕ ε = on,t) Ω, ϕ ε T, ) = 1 ε ζε T, ) in Ω. Then, the duality properties between ϕ ε and ζ provide 1 ζ ε T ) ζt ) dx = hϕ ε 1 dx dt, ε Ω,T ) ω which, combined with 3.9), yields hϕ ε 1 dx dt =,T ) ω e sρx)ηt) sη) 7 h ε hdxdt,t ) ω h L,T) ω ). Consequently, we can identify h ε : 3.3) h ε = e sρx)ηt) sη) 7 ϕ ε 1 1 ω.

18 5646 J.-M. CORON, S. GUERRERO, AND L. ROSIER From the systems fulfilled by ζ ε and ϕ ε, we find, using 3.3), 1 ε ζε T, ) L Ω),T = e sρx)ηt) sη) 7 ϕ ε 1 dx dt + ϕ ε, ) αdx, ) ω Ω with α := α 1,α ). Inequality 3.6) used for ϕ ε tells us that ϕ ε, ) L Ω) C,T ) ω e sρx)ηt) sη) 7 ϕ ε 1 dx dt, so, once more using 3.3), 3.33) 1 ε ζε T, ) L Ω) + esρx)ηt) sη) 7 ) 1/ h ε L,T ) ω ) C α L Ω). Consequently, we deduce the existence of a control h such that e sρx)ηt) sη) 7 ) 1/ h L,T) ω ) whose corresponding solution we denote by ζ) such that ζt, ) = and 3.34) e sρx)ηt) sη) 7 ) 1/ h L,T ) ω ) C α L Ω). Let us finally bound the L -norm of the control h ε. For this, we now develop a bootstrap argument. Let ψ ε, := e sρx)ηt)/ sη) 1/ ϕ ε, ψ ε,1 := e sρx)ηt)/ sη) 5/ ϕ ε = 1 sη) ψε,. Let LR ; R ) be the vector space of linear maps from R into itself. Using 3.31), one easily checks that ψ ε,1 fulfills a backward heat system with a homogeneous Dirichlet boundary condition and a final null condition of the following form: ψ ε,1 t Δψ ε,1 = d 1 in,t) Ω, 3.35) ψ ε,1 = on,t) Ω, ψ ε,1 T, ) = inω, with 3.36) d 1 t, x) =A 1 t, x)ψ ε, +sη) 1 ψ ε, ρ, where A 1 L Q; LR ; R )) satisfies see, in particular, 3.8)) 3.37) A 1 L Q;LR ;R )) C. In 3.36) and in the following, we use the notation θ ρ)t, x) := θ 1 t, x) ρx), θ t, x) ρx)) for θ =θ 1,θ ):Q R. Thanks to 3.5), 3.3), 3.33), 3.36), and 3.37), 3.38) d 1 L Q) and d 1 L Q) C α L Ω).

19 CONTROLLABILITY PARABOLIC SYSTEM CUBIC COUPLING 5647 For r [1, + ], let X r := L r,t; W,r Ω) ) W 1,r,T; L r Ω) ). We denote by Xr its usual norm. Let 3.39) p 1 :=. From 3.35), 3.38), 3.39), and a standard parabolic regularity theorem, we have 3.4) ψ ε,1 X p1, ψ ε,1 Xp1 C α L Ω). For k N \{}, let ψ ε,k := e sρx)ηt)/ sη) 1/ k ϕ ε = 1 sη) k ψε,. Let us define, by induction on k, a sequence p k ) k N\{} of elements of [, + ] by N +)p k 1 if p k 1 <N+, N + p p k := k 1 p k 1 if p k 1 = N +, + if p k 1 >N+. One easily checks the following: IfN =l, with l N \{}, one has N + p k = if k<l+, N k ) p l+ =N +), p k =+ k l +3. IfN =l + 1, with l N, one has N + p k = if k l +, N k ) p k =+ k>l+. In particular, 3.41) p k =+ k N +3. We now use an induction argument on k. We assume that 3.4) ψ ε,k 1 X pk 1 and ψ ε,k 1 Xpk 1 C α L Ω) now C is allowed to depend on k) andthatψ ε,k 1 fulfills a heat system of the following form: ψ ε,k 1 t Δψ ε,k 1 = d k 1 in,t) Ω, 3.43) ψ ε,k 1 = on,t) Ω, ψ ε,k 1 T, ) = inω, with 3.44) d k 1 t, x) =A k 1 t, x)ψ ε,k +sη) 1 ψ ε,k ρ,

20 5648 J.-M. CORON, S. GUERRERO, AND L. ROSIER where A k 1 L Q; LR ; R )) satisfies 3.45) A k 1 L Q;LR ;R )) C. Note that we have just proved above that this induction assumption holds for k =. Using 3.43) and 3.44), one gets that ψ ε,k fulfills the following backward heat system with a homogeneous Dirichlet boundary condition and a final null condition: ψ ε,k t Δψ ε,k = d k in,t) Ω, 3.46) ψ ε,k = on,t) Ω, ψ ε,k T, ) = inω, with 3.47) d k t, x) =A k t, x)ψ ε,k 1 +sη) 1 ψ ε,k 1 ρ, where A k : Q LR ; R ) is defined by 3.48) A k := A k 1 + η t s η 3 Id, Id denoting the identity map of R. From 3.45) and 3.48), one gets that 3.49) A k L Q; LR ; R )) and A k L Q;LR ;R )) C. Let us recall the following embeddings between Sobolev spaces see, e.g., [13, Lemma 3.3, page 8]). Lemma 9. Let p 1, + ). i) If p<n+,let N +)p r := N + p. Then X p is continuously embedded in L r,t; W 1,r Ω) ). ii) If p = N +, for every r [1, + ), X p is continuously embedded in L r,t; W 1,r Ω) ). iii) If p>n+, X p is continuously embedded in L,T; W 1, Ω) ). Applying Lemma 9 with p = p k 1 and using 3.4), we get that 3.5) ψ ε,k 1 L p k,t; W 1,p k Ω) ) and ψ ε,k 1 L p k,t ;W 1,p k Ω) ) C α L Ω). From 3.47), 3.49), and 3.5), we have 3.51) d k L p k Q) and d k L p k Q) C α L Ω). Using 3.46), 3.51), and a classical parabolic regularity theorem see, e.g., [13, Theorem 9.1, pages ]), we have 3.5) ψ ε,k X pk and ψ ε,k Xpk C α L Ω). Hence 3.5) holds for every positive integer k. Let us choose an integer k such that k N/) + 3. Then using 3.41) and 3.5) we get that 3.53) ψ ε,k L Q) and ψ ε,k L Q) C α L Ω). This shows that the control defined in 3.3) is bounded in L Q) independently of ε by C α L Ω). This concludes the proof of Lemma 6.

21 CONTROLLABILITY PARABOLIC SYSTEM CUBIC COUPLING Local null controllability around the trajectory u, v), h). Let ν> be small enough so that, for every z =z 1,z ) L Q), 3.54) z L Q) ν) G 11 z 1,z ),G 1 z 1,z ),G 1 z 1,z ),G z 1,z )) E). The existence of such a ν>follows from 3.4) and 3.5).) Let Z be the set of z =z 1,z ) L Q) such that z L Q) ν. By 3.54) and Lemma 6, there exists C > such that, for every z =z 1,z ) Zand for every α 1,α ) L Ω), there exists a control h L Q) satisfying 3.55) h L Q) C α 1 L Ω) + α L Ω)) such that the solution ζ 1,ζ ) to the Cauchy problem ζ 1,t Δζ 1 = G 11 z 1,z )ζ 1 + G 1 z 1,z )ζ + h1 ω in,t) Ω, ζ,t Δζ = G 1 z 1,z )ζ 1 + G z 1,z )ζ in,t) Ω, 3.56) ζ 1 =, ζ = on,t) Ω, ζ 1, ) =α 1, ζ, ) =α in Ω satisfies 3.11). We now define a set-valued mapping B : Z L Q) in the following way. Fix first any α 1,α ) L Ω). For any z =z 1,z ) Z, Bz) isthesetof ζ 1,ζ ) L Q) such that, for some h L Q) fulfilling 3.55), ζ 1,ζ )isthe solution of 3.56) and this solution satisfies 3.11). As we have just pointed out, Bz) is never empty. Theorem 1 will be proved if one can check that the set-valued mapping z Bz) has a fixed point i.e., a point z such that z Bz)) with the additional hypothesis α 1,α ) L Ω) being small enough. To get the existence of this fixed point, we apply Kakutani s fixed point theorem see, e.g., [14, Theorem 9.B, page 45]): i) for every z Z, Bz) is a nonempty closed convex subset of L Q) ; ii) there exists a convex compact set K Z such that 3.57) Bz) K z Z; iii) B is upper semicontinuous in L Q) ; i.e., for every closed subset A of Z, B 1 A) :={z Z; Bz) A } is closed see, e.g., [14, Definition 9.3, page 45]). If i) iii) hold, then there exists z Z such that z Bz). Clearly i) holds. Let us prove that ii) holds. By standard estimates, there exists C 1 > such that 3.58) ζ L Q) C 1 α 1 L Ω) + α L Ω)) z Z, ζ Bz). From now on we assume that α 1,α ) L Ω) satisfies 3.59) α 1 L Ω) + α L Ω) ν C 1. From 3.58) and 3.59), one has 3.6) Bz) Z z Z.

22 565 J.-M. CORON, S. GUERRERO, AND L. ROSIER Let λ 1,λ ) L Q) be the solution to the following Cauchy problem: λ 1,t Δλ 1 = in,t) Ω, λ,t Δλ = in,t) Ω, 3.61) λ 1 =, λ = on,t) Ω, λ 1, ) =α 1, λ, ) =α in Ω. Let ζ1 := ζ 1 λ 1 and ζ := ζ λ. Then ζ1,ζ ) is the solution to the following Cauchy problem: ζ1,t Δζ 1 = D 1 in,t) Ω, ζ,t 3.6) Δζ = D in,t) Ω, ζ1 =, ζ = on,t) Ω, ζ1, ) =, ζ, ) = inω, with 3.63) 3.64) D 1 := G 11 z 1,z )ζ 1 + G 1 z 1,z )ζ + h1 ω, D := G 1 z 1,z )ζ 1 + G z 1,z )ζ. Note that there exists C > such that 3.65) D 1 L Q) + D L Q) C z Z, ζ Bz). From 3.6), 3.65), and a classical parabolic regularity theorem see, e.g., [13, Lemma 3.3, page 8 and Theorem 9.1, pages ]), ζ := ζ 1,ζ ) C Q) and there exists C 3 > such that, for every z Z and for every ζ Bz), 3.66) ζ t, x) ζ t,x ) C 3 t t 1/ + x x ) t, x) Q, t,x ) Q. Let K be the set of ζ =ζ 1,ζ ) C Q) such that 3.66) holds. Then λ 1,λ )+K is a compact convex subset of L Q) and 3.67) Bz) λ 1,λ )+K z Z. Then K := λ 1,λ )+K ) Z is a convex compact subset of Z such that 3.57) holds. Let us finally prove the upper semicontinuity of B. Let A be a closed subset of Z. Letz k ) k N be a sequence of elements in Z, letζ k ) k N be a sequence of elements in L Q), and let z Z be such that 3.68) 3.69) 3.7) z k z in L Q) ask +, ζ k A k N, ζ k Bz k ) k N. By 3.7), for every k N there exists a control h k L Q) satisfying 3.71) h k L Q) C α 1 L Ω) + α L Ω))

23 CONTROLLABILITY PARABOLIC SYSTEM CUBIC COUPLING 5651 such that ζ k =ζ1 k,ζk ) is the solution of the Cauchy problem ζ1,t k Δζ1 k = G 11 z1 k,z k )ζ1 k + G 1 z1 k,z k )ζ k + h k 1 ω in,t) Ω, ζ,t k Δζ k = G 1 z1 k,z k )ζ1 k + G z1 k,z k )ζ k in,t) Ω, 3.7) ζ1 k =, ζk = on,t) Ω, ζ1 k, ) =α 1, ζ k, ) =α in Ω, and this solution satisfies 3.73) ζ1 k T, ) =,ζk T, ) =. From ii) and 3.71), there exist a strictly increasing sequence k l ) l N of integers, h L Q), and ζ Z such that 3.74) 3.75) h k l h for the weak-* topology on L Q) asl +, ζ k l ζ in L Q) as l +. Note that, since A is closed, 3.69) and 3.75) imply that ζ A. Hence, in order to prove iii), it suffices to check that 3.76) ζ Bz). Letting l + in 3.7) and 3.73) and using 3.68), 3.74), and 3.75), we get 3.11) and 3.56). The two equalities in 3.11) and the last two equalities of 3.56) have to be understood as equalities in L Ω) for ζ k l ζ in X.) Letting l + in 3.71) and using 3.74), we get 3.55). Hence 3.76) holds. This concludes the proof of iii) and of Theorem Appendix: Sketch of the proof of Theorem 3. As the proof is very similar to that of Theorem 1, we limit ourselves to pointing out only the differences. First, Lemma 5 should be replaced by the following lemma. Lemma 1. There exists a function G C [, + ); C) such that 4.1) Gz) = z 1 ) for 1 δ<z<1 + δ, 4.) Im Gz) < for <z< 1 δ, 4.3) Re Gz) > Im Gz) > for 1 + δ<z<1 δ and such that the solution g to the Cauchy problem 4.4) 4.5) g z)+ N 1 g z) =Gz), z >, z g 1) = g 1) = satisfies 4.6) 4.7) 4.8) g z) =1 z if <z<δ, g z) =e 1 1 z if 1 δ<z<1, g z) = if z 1.

24 565 J.-M. CORON, S. GUERRERO, AND L. ROSIER The proof is carried out in the same way as for Lemma 5. Note that the conditions.6) and.3) are easily satisfied thanks to the change of sign of Re G and Im G. Theorem 4 is still true for some functions V :t, x) R R N V t, x) C and K :t, x) R R N Kt, x) C when.8) is replaced by V t =ΔV + RV + V. In the proof, we consider the same functions λt) andf t) andsearchforvt, r) in the form vt, r) = f i t)g i z). f 1,f,g 1, and g are defined in the same way as for Theorem 4, so that V = V x = for 1 <t<1, z= 1, V xx =f + R for 1 <t<1, z 1 <δ, with R Cε f. Letting At, z) =f t) 1 Vt, z), we have that At, z) = z ) 1 1 ϕt, z) for t [ 1, 1], z δ, 1 ) + δ, where ϕ C [ 1, 1] t 1 δ, 1 + δ)) and 1 Re ϕt, x) >f t) for t [ 1, 1], z δ, 1 ) + δ. On the other hand, i= At, z) G Cε Gz) for 1 t 1and z 1 δ/. Using 4.), 4.3), it is then clear that for ε small enough we have At, z) ir + for 1 t 1, z, 1) \{ 1 }. Defining the square root as an analytic function on the complement of ir +,weseethat λ V = λ f t)at, z) =Bt, z) for some B C 1, 1) [, 1)). The end of the construction of V,K) is as in the proof of Theorem 4. In the study of the local null controllability around the trajectory ū, v), h), the functions G ij are defined in the same way, except G 1 ζ 1,ζ )=ū + ζ 1. Therefore, 3.4) and 3.7) have to be changed to Im G 1, )t, x) M t, x) t 1,t ) ω, Im a 1 t, x) 1 M t, x) t 1,t ) ω, respectively. Note that the functions in the control systems are complex-valued, so that we have to conjugate the coefficients in the right-hand side of the adjoint system 3.15). To estimate the local integral of ϕ, we multiply the first equation in 3.15) by χx)e sρx)ηt) sη) 3 ϕ where ϕ stands for the conjugate of ϕ ) and take the absolute value of the imaginary part of the left-hand side of 3.). The remaining part of the proof is the same as for Theorem 1.

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