Key words. Controllability, Parabolic systems, Carleman estimates, Fictitious control method.

Transcription

1 CONTROLLABILITY OF COUPLED PARABOLIC SYSTEMS WITH MULTIPLE UNDERACTUATIONS, PART : NULL CONTROLLABILITY DREW STEEVES, BAHMAN GHARESIFARD, AND ABDOL-REZA MANSOURI Abstract This paper is the second of two parts which together study the null controllability of a system of coupled parabolic PDEs Our work specializes to an important subclass of these control problems which are coupled by first and zero-order couplings and are, additionally, underactuated In the first part of our work [], we posed our control problem in a framework which divided the problem into interconnected components: the algebraic control problem, which was the focus of the first part; and the analytic control problem, whose treatment was deferred to this paper We use slightly non-classical techniques to prove null controllability of the analytic control problem by means of internal controls appearing on every equation We combine our previous results in [] with the ones derived below to establish a null controllability result for the original problem Key words Controllability, Parabolic systems, Carleman estimates, Fictitious control method AMS subject classifications 35K4, 93B5 Introduction We begin with defining some notation and conventions Notation and conventions Throughout this work, we define N := N \ {}, and similarly, R := R \ {} For n, k N, we denote the set of n k matrices with real-valued entries by M n k R, and we denote the set of n n matrices with real-valued entries by M n R We denote the set of linear maps from a vector space U to a vector space V by L U; V For X, T X a topological space and U X, we denote the closure of U by Ū We now recall the coupled parabolic system of interest A system of interest In this second part of this two-part work, the primary objective is maintained from the first part: that is, for :=, T and Σ T :=, T for some T >, we wish to study the controllability properties of the system of coupled parabolic PDEs given by t y = divd y + G y + Ay + r, in, y =, on Σ T, y, = y, in, where D := diagd,, d m, G := g pk p,k m M m R n and A := a pk p,k m M m R We associate to this system the differential operator m m m Ly = divd p y p g pk y k a pk y k e p, p= k= where g pk := g pk,, gn pk Rn, d p M n R is symmetric and e p is the p-th canonical basis vector in R m, for p {,, m} We call g pk the first-order coupling coefficients and a pk the zero-order coupling coefficients, which are constant in space and time Department of Mechanical and Aerospace Engineering, University of California, San Diego, EBU, La Jolla, CA, United States 993, dsteeves@engucsdedu Department of Mathematics and Statistics, Queen s University, Jeffery Hall, University Ave, Kingston, ON, Canada K7L3N6 bahmangharesifard@queensuca, mansouri@mastqueensuca k=

2 In this work, we assume that L satisfies the uniform ellipticity condition: that is, there exists C > such that, 3 n i,j= d ij p ξ i ξ j C ξ, ξ R n Suppose r L m, y L m For u, v H m, we define the bilinear form m n n B[u, v] := d ij p xi u p xj v p gpk i xi u k v p a pk u k v p e p dx p,k= i,j= One has the following definition Definition Suppose r L m, y L m A function i= y L, T ; H m H, T ; H m is said to be a weak solution of system provided that for every v H m and almost every t [, T ] i d y, v + B[y, v] = rt vdx, and; ii y = y, where, denotes the appropriate duality pairing It can be deduced, for example, from [6, Theorems 3 and 4, Section 7], that for any initial condition y L m and r L m, system admits a unique solution From now on, we mean by solution to a coupled parabolic system the weak solution in the sense of Definition 3 A parabolic regularity result We state a regularity result for the solution of system which is essential in the work to follow Theorem [6, Theorem 6, Subsection 73] For d N, assume y H d+ m, r L, T ; H d m H d, T ; L m, and assume that y L, T ; H m H, T ; H m is the solution of system Suppose also that the following compatibility conditions hold: g := y H m ; g := r Lg H m ; g d := dd r Lg d H d m Then y L, T ; H d+ m H d+, T ; L m and we have the estimate y L,T ;H d+ m H d+,t ;L r m L,T ;H d m H d,t ;L m 4 + y H d+ m 4 The control problem This work specializes to the case of internal or distributed control: that is, for ω nonempty and open, we study the case where r = ω Bu, for B = T Id c c m c Mm c R and c m, and henceforth,

3 we denote by q T the set, T ω An interesting control problem that arises in many engineering applications is underactuation, that is, when c < m Our work will further specialize to this case, where there are currently few results for first and zero-order couplings, for arbitrary m and c < m even for the case of constant coefficients We focus on a particular type of controllability property, which is defined next Definition 3 We say that is null controllable in time T if for every initial condition y L m, there exists u L c such that the solution y L, T ; H m H, T ; H m to satisfies yt = in This work s main objective that we aim to achieve by selecting appropriate control inputs is null controllability of system Next, we recall the method which we introduced in []; we employ this method to achieve our main objective 5 Fictitous control method We first described following control problem, referred to as the analytic control problem: for any ỹ L m, proving the existence of ỹ, ũ a solution of t ỹ = divd ỹ + G ỹ + Aỹ + N ω ũ, in, 5 ỹ =, on Σ T, ỹ, = ỹ, in, such that ỹt, =, where N is a differential operator that was chosen to be the identity in [], ũ acts on all equations in 5, and we denote by ω a smooth version of the indicator function this can be constructed via mollification; cf relation 4 for its exact definition We next consider a different control problem, referred to as the algebraic control problem: proving the existence of a solution ŷ, û of t ŷ = divd ŷ + G ŷ + Aŷ + Bû + N ω ũ, in, 6 ŷ =, on Σ T, ŷ, = ŷt, =, in, where û acted only on the first c equations We defined the notion of algebraic solvability of 6 in [], which is a property that enabled us to algebraically invert the differential operator associated to 6 and recover the solution to this control problem locally We recovered the following result Proposition 4 Given m, n and c in N with m + c m, if the matrix C M c m cn+ R given by C := a c+ a m gc+ gm gc+ n gm n a c+ a m gc+ gm gc+ n gm n a c+c a mc gc+c gmc gc+c n gmc n has full rank, then 6 is algebraically solvable in q T Importantly, the inverse differential operator that we recovered in [], denoted by B, was of differential order at most p + in space This differential order is of consequence in Section 4, where we require the controls in the analytic system 5 to be regular enough to withstand these p + spacial derivatives This regularity 3

4 on the controls is necessary for the solution that was constructed for the algebraic problem to be well-defined With the algebraic problem resolved, solving the analytic problem is this paper s secondary objective Achieving this secondary objective will allow us to attain this work s main objective, as will be shown in Section 4 6 Statement of contributions The first contribution is a partial generalization of [5, Theorem ] In particular, our result gives a sufficient condition for the null and approximate controllability of an underactuated system of coupled parabolic PDEs, with constant first and zero-order couplings, when more than half of the equations are actuated Importantly, this controllability condition applies to systems with multiple underactuations Furthermore, this condition, which requires the rank of a matrix containing some of the coupling coefficients as entries to be full rank, is generic The technique used to prove this result is adapted from [4] Our second contribution is Proposition 37, which is an extension of [5, Proposition ] Specifically, our Carleman estimate contains higher differential order terms on its lefthand side, which allows us to construct very regular controls in Proposition 4 Importantly, these highly regular controls are necessary when applying Theorem to problems with many underactuations Main result The main controllability theorem of this work is stated next, where we assume that more than half of the equations in system are actuated Theorem For a fixed m, suppose R n nonempty, open and bounded Furthermore, suppose is connected and of class C p+ For m + c m, if the matrix C M c m cn+ R given by C := a c+ a m gc+ gm gc+ n gm n a c+ a m gc+ gm gc+ n gm n a c+c a mc gc+c gmc gc+c n gmc n has full rank, then the system is null controllable in time T The rest of this work is devoted to proving the above result First, we will resolve the analytic control problem in Section 4; next, we will utilize the solutions to the algebraic and analytic control problems to solve our original control problem in Section 3, which is the null controllability of the underacted system 3 A Carleman estimate for the analytic problem In this section, we study the analytic system: t ỹ = divd ỹ + G ỹ + Aỹ + ω ũ, in, 3 ỹ =, on Σ T, ỹ, = ỹ, in The goal of this section is to prove that the solution ỹ, ũ to the analytic control system 3 satisfies the following so-called weighted observability inequality, which will help us deduce its null controllability To this end, we consider the adjoint system 4

5 to system 3 given by 3 t ψ = divd ψ G ψ + A ψ, in QT, ψ =, on Σ T, ψt, = ψ, in, where ψ L m We state the weighted observability inequality we aim to establish Proposition 3 For every ψ L m, the solution ψ of system 3 satisfies 33, x dx obs e sα ξ p+7 t, x dx,,t ω where C obs := CT 9 e C+3T/4+/T 5 > and denotes the Euclidean norm We call 33 a weighted observability inequality with weight ρ := e sα ξ p+7, for α and ξ defined below in 35 and 36, respectively, where s := σt 5 + T for σ > depending on and ω We utilize the Carleman estimate technique to develop an estimate which will help us establish the observability inequality stated above; the proof of Proposition 3 follows from Proposition 37 and is given in the Appendix This section builds upon the technique developed in [5, Section ]: in particular, it incorporates the higherorder terms found on the lefthand side of 3 which allow us to construct highly regular controls for system 3 Constructing a solution ỹ, ũ to system 3 with highly regular controls and satisfying ỹt, = is treated in Section 4 3 Some notation and technical results We begin by introducing some notation For the multi-index β of length l consisting of multi-indices, consider the l th -order tensor given by C := C β β, where β i has length n i, for n i N, for i {,, l} We associate to C the element-wise norm: l := n,,n l i =,,i l = C β i,,β l i l An equivalent interpretation of l is the following: given a l th -order tensor C, one vectorizes C into a vector of length l i= n i and then applies the Euclidean norm to recover l Fix a sequence ω i p+ i= of nonempty open subsets of ω such that { ω i ω i, for i {,, p + }, ω ω The next result is an adaptation of [8, Lemma ] see also [, Lemma 68] Lemma 3 Assume that is of class C p+ and connected Then there exists η C p+ such that η κ, in \ ω p+, 34 η >, in, η =, on, 5 /

6 for some κ > Remark 33 In 47, we require η to be p + times differentiable; this is why we require spatial domain boundary regularity in Theorem For t, x we define 35 αt, x := eλ η e λ η +η x t 5 T t 5 and 36 ξt, x := eλ η +η x t 5 T t 5 Additionally, for t, T we define 37 α t := max αt, x x and 38 ξ t := min ξt, x x For s, λ > and u L, T ; H H, T ; H, let us define 39 Is, λ; u := s 3 λ 4 e sα ξ 3 u dx + sλ e sα ξ u dx In the work to follow, for u L, T ; H m H, T ; H m, we use a slight abuse of notation and define Is, λ; u as above but with replaced by, and with replaced by We now state a Carleman estimate result for the heat equation; the proof is quite technical and is omitted here Lemma 34 [7, Theorem ] Assume that d >, u L, f L and f L Σ T Then there exists a constant C := C, ω p+ > such that the solution to t u = divd u + f, in, u n = f, on Σ T, ut, = u, in, satisfies Is, λ; u s 3 λ 4 e sα ξ 3 u dx + e sα f dx,t ω p+ +sλ e sα Σ T ξ f dσ for all λ C and s CT 5 + T We can adapt the Carleman estimate in Lemma 34 to system 3 with Neumann boundary condition; this adapted Carleman estimate will be used later cf A5 6

7 Lemma 35 Assume that ψ L m and u L Σ T m Then there exists a constant C := C, ω p+ > such that the solution to 3 satisfies t ψ = divd ψ G ψ + A ψ, in QT, ψ n = u, on Σ T, ψt, = ψ, in, 3 Is, λ; ψ s 3 λ 4 e sα ξ 3 dx + sλ,t ω p+ e sα Σ T ξ u dσ for all λ C and s CT 5 + T The proof of Lemma 35 can be deduced from Lemma 34 and the definitions of ξ and α one can absorb the integral with coupling terms appearing as the integrand into Is, λ; ψ on the lefthand side We will also use the following estimate in the ensuing treatment cf A8 and A9 Lemma 36 [3, Lemma 3] Let r R There exists a C := C, ω p+, r > such that for every T > and every u L, T ; H, s r+ λ Q r+3 e sα ξ r+ u dx C s r λ r+ e sα ξ r u dx T + s r+ λ r+3,t ω p+ e sα ξ r+ u dx for every λ C and s CT 5 + T Finally, one can establish the following Carleman estimate for system 3, which is an extension of [5, Proposition ] Proposition 37 There exists a constant C := C, ω > such that for every ψ L m, the solution ψ to system 3 satisfies 3 e sα s k λ k ξ k k ψ p+5 k dx k= for every λ C and s CT 5 + T s p+7 λ p+8 e sα ξ p+7 dx,t ω Remark 38 It is worth pointing out to the fact that 3 contains spatial derivatives past order one, since ψ is assumed to be in L m, and hence ψ L, T ; H m H, T ; L m However, due to inequalities A5 and A6 and since the weight e sα absorbs the singularity of ξ at t =, one can deduce that these integrals exist 7

8 4 Proof of main theorem Recall from Section that our principal goal was to prove null controllability of system with multiple underactuations To this end, we studied an algebraic system and an analytic system both related to system In [], we developed an algebraic method which allowed us to solve the algebraic control problem under the assumption that the source term ω ũ be regular enough so that our algebraic solution B ω ũ be well-defined, where B is a differential operator of order zero in time and at most p + in space In Section 3, we established the weighted observability inequality 33 for the analytic system 3 The goal of this section is to solve the analytic control problem 5 with regular enough controls ω ũ so that the algebraic control problem 6 also be solved The treatment presented in this section is an extension of that used in [5, Section 3] In particular, since the right inverse B of L derived implicitly in [] is in general of order at most p + in space, we require higher regularity of controls in the analytic problem than in [5] 4 An optimal control result We do not use the weighted observability inequality to directly deduce null controllability of system 3 Instead, we use a method developed in [8] to construct controls with high regularity which will help us deduce controllability results; to do this, we rely on the following unconstrained optimal control result Theorem 4 [9, Section 3, Theorem ] Let y L m, u L m, B L L m ; L m, and suppose L given in satisfies the uniform ellipticity condition 3 Let N L L m ; L m such that Nu, u L m ν u L for ν > and for all u L Q m T m, and let D L H m ; H m Consider the optimal control problem associated to system with cost functional Ju : L m R + given by 4 Ju := Nu, u L m + Dy ut, z d L m, for some z d H m This problem has a unique solution, and the optimal control is characterized by the following relations: t y u = divd y u + G y u + Ay y + Bu, in, y u =, on Σ T, y u, = y, in, and t ψ u = divd ψ u G ψ u + A ψ u, in, ψ u =, on Σ T, ψ u T, = D Dy u T, z d, in, B ψ u + Nu = Hence, for this unconstrained optimal control problem, the second term in 4 has no dependence on u nor do the primal/adjoint systems 4 Null controllability of the analytic problem Recall that in [], we fixed a p large enough such that we recovered algebraic solvability of 6 In this section, we establish the following proposition 8

9 Proposition 4 Consider θ C p+ such that Suppθ ω, 4 θ =, in ω, θ, in Then there exists v L m such that ỹ, θv L, T ; H m H, T ; H m L m is a solution to the analytic control problem 5 satisfying ỹt, = in Moreover, for every K,, we have 43 e Ksα v L, T ; H p+ H m H, T ; L m, and e Ksα v L,T ;H p+ H m H,T ;L m ỹ L m Proof Let ỹ L m, ρ := e sα ξ p+7 and C := C, ω, T > Let k N and denote by L, ρ / m the space of functions which, when multiplied by ρ /, are L -integrable ie, for u L, ρ / m, we require ρ u dx < Consider the following optimal control problem: { minimize J k v := Q 44 T ρ v dx + k ỹt, dx, subject to v L, ρ / m, where ỹ L, T ; H m H, T ; H m The functional J k is differentiable, coercive and strictly convex on L, ρ / m By Theorem 4 for D = k, N = ρ and z d = in, there exists a unique solution to this problem, and the optimal control is characterized by the solution ỹ k to the analytic system t ỹ k = divd ỹ k + G ỹ k + Aỹ k + θv k, in, 45 ỹ k =, on Σ T, ỹ k, = ỹ, in, the solution ψ k to its adjoint system 46 and the relation t ψk = divd ψ k G ψ k + A ψk, in, ψ k =, on Σ T, ψ k T, = kỹt,, in, 47 { vk = ρθ ψ k, in, v k L, ρ / m 9

10 From 45 and 46, we calculate 48 and = d ỹ k, t ψk L m + tỹ k, ψ k L m ỹ k, ψ k L m = ỹ k T,, kỹ k T, L m ỹ, ψ k, L m, ỹ k, t ψk L m + tỹ k, ψ k L m = ỹ k, divd ψ k + G ψ k A ψk L m 49 + divd ỹ k + G ỹ k + Aỹ k + θv k, ψ k L m = θv k, ψ k L m It follows from 47, 48 and 49 that J k v k = 4 = θ ψ k, v k L m + ỹ kt,, ψ k T, L m ψ k, θv k L m + + y, ψ k, L m= y, ψ k, L m ỹ k, t ψk L m + tỹ k, ψ k L m Moreover, employing the weighted observability inequality 33 along with 4, 47, 44, 4 and the Cauchy-Schwarz inequality successively, we have ψ k, L m obs ρθ k dx,t ω obs ρθ k dx from which we deduce = C obs ρ v k dx obs J k v k obs ψ k, L m y L m, 4 ψ k, L m obs y L m Furthermore, by 4, 4 and the Cauchy-Schwarz inequality, we obtain 4 J k v k obs y L m

11 One can deduce from parabolic regularity, 4 and 4 that 43 ỹ k L,T ;H m H,T ;H m θv k L m + ỹ L m v k L m + ỹ L m + C obs ỹ L m, since for our choice of s which depends on p; see A33 and by 35 and 36, ρ in Owing to the well-known result that in Hilbert spaces, bounded sequences have weakly convergent subsequences see, for example, [], along with 44 4, and 43, one can extract subsequences of v k k and ỹ k k which we still denote by v k and ỹ k such that v k v in L, ρ / m, ỹ k ỹ in L, T ; H m H, T ; H m, ỹ k T, in L m Hence, ỹ, θv is the solution to the analytic control problem 5 with θv L, ρ / Furthermore, we deduce from 44 by taking k that ỹt, = in the sense of Definition In addition, by 4 and since ρ in for our choice of s, v L C obs y L m, as claimed It is left to show that 43 is verified Note that for every K,, there exists a C K := C K such that 44 e Ksα K ξ p 7 e sα, for all t, x Hence, utilizing 44, 44 and then 4, we obtain e Ksα v k L m K ρ v k dx 45 For a >, one has see A9 K ỹ L m 46 t ξ a e sα T ξ a+6/5 e sα Furthermore, for r = {,, p + } one has 47 r ξ a e sα r ξ a+r e sα Indeed, ξ a e sα = aξ a λ η ξe sα s ξ a e sα λ η ξ a = λ η ξ + s ξ a+ e sα, and since C := C, ω, T, 47 is verified for r = The same reasoning can be used for the r-th derivative, where we have fixed η C p+ Hence, by 47, the

12 triangle inequality and then 47 for a = p + 7, we obtain 48 e Ksα v k L m = e Ksα v k dx Q T = ξ p+7 e sα θ ψ k dx e Ksα e Ksα e Ksα ξ p+7 e sα k + ξ p+7 e sα ψ k dx 4s α ξ 4p+6 ψk + ξ 4 ψ k dx, and similarly, for r {,, p + }, we obtain 49 r e Ksα r v k L 4s α ξ 4+l r l ψk m r l+ dx e Ksα l= By 46 and since ψ k satisfies system 46, we obtain 4 4 t e Ksα v k L m e Ksα 4s α e Ksα 4s α ξ p+8/5 k ξ p+8/5 k + ξ ψ k 3 + ψ k + k dx + ξ t ψk dx Note that for every a, b > and K,, there exists C a,b,k := C a,b,k > such that 4 ξ a e Ksα 4s α a,b,k ξ b e sα From 45, 48, 49, 4 and utilizing 4 for appropriate a and b, e Ksα v k L,T ;H p+ H m H,T ;L m max,k ξ k k ψk p+5 kdx, e sα k= where C max,k := max{max a,b {C a,b,k }, C K } Owing to 4, Proposition 37 and 47, we deduce e Ksα v k L,T ;H p+ H m H,T ;L m max,k C obs e sα ξ p+7 θ ψ k dx = C max,k C obs v k L m

13 Lastly, for C K := C K, ω, T, 4 yields the inequality e Ksα v k L,T ;H p+ H m H,T ;L m C K ỹ L m, from which 43 is verified by taking a convergent subsequence and k With algebraic solvability of the algebraic control problem 6 and null controllability of the analytic control problem 5 established for highly regular controls, we can now prove null controllability of the system with internal controls û L q T c, where c < m In Proposition 4, we showed the existence of ỹ, θv L, T ; H m H, T ; H m L m satisfying t ỹ = divd ỹ + G ỹ + Aỹ + θv, in, 43 ỹ =, on Σ T, ỹ, = y, in, such that ỹt, = in Furthermore, we established the following higher regularity for v: 44 e Ksα v L, T ; H p+ H m H, T ; L m, for all k, Notice that 44 implies that v is exponentially decaying as t and t T For the linear partial differential operator B of order zero in time and at most p + in space constructed implicitly in [], let us define ŷ := B θv, û which is well-defined by 44 By virtue of B being a linear partial differential operator of the stated orders with constant coefficients, we conclude that 45 ŷ, û L q T L q T c ; we then extend ŷ, û by zero to Since v decays exponentially as t and t T, ŷ, = ŷt, = in Furthermore, it follows from the discussions in Subsection 5 that ŷ, û is the solution to t ŷ = divd ŷ + G ŷ + Aŷ + Bû + θv, in, 46 ŷ =, on Σ T, ŷ, = ŷt, =, in, where, by 45 and by parabolic regularity, ŷ, û satisfies Definition Defining y, u := ỹ ŷ, û, it is immediate that y, u is the solution to with yt, = in This finishes the proof of Theorem 5 Conclusion Using the powerful fictitious control technique, which has allowed us to pose our controllability problem as two interconnected problems, we have derived a sufficient condition for the null controllability of a system of coupled parabolic PDEs, where the couplings were constant in space and time and of first and zero-order and more than half of the equations in the system were actuated This controllability condition is generic 3

14 REFERENCES [] JB Conway, A Course in Functional Analysis, vol 96 of Graduate Texts in Mathematics, Springer-Verlag, New York, second ed, 99 [] J-M Coron, Control and Nonlinearity, vol 36 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 7 [3] J-M Coron and S Guerrero, Null controllability of the N-dimensional Stokes system with N scalar controls, J Differential Equations, 46 9, pp 98 9 [4] J-M Coron and P Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent Math, 98 4, pp [5] M Duprez and P Lissy, Indirect controllability of some linear parabolic systems of m equations with m controls involving coupling terms of zero or first order, J Math Pures Appl 9, 6 6, pp [6] LC Evans, Partial Differential Equations, vol 9 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second ed, [7] E Fernández-Cara, M González-Burgos, S Guerrero, and J-P Puel, Null controllability of the heat equation with boundary Fourier conditions: the linear case, ESAIM Control Optim Calc Var, 6, pp [8] AV Fursikov and OY Imanuvilov, Controllability of Evolution Equations, vol 34 of Lecture Notes Series, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 996 [9] J-L Lions, Contrôle Optimal de Systemes Gouvernés par des Équations aux Dérivées Partielles, Dunod; Gauthier-Villars, Collier-Macmillan Ltd, 968 [] L Nirenberg, On elliptic partial differential equations, Ann Scuola Norm Sup Pisa 3, 3 959, pp 5 6 [] D Steeves, B Gharesifard, and A-R Mansouri, Controllability of coupled parabolic systems with multiple underactuations, part : algebraic solvability, 7, pp 3 bahman/ds-bg-arm-partpdf Appendix In a proof to follow, we rely on the so-called Gagliardo-Nirenberg interpolation inequality, which is stated next Theorem A [] For R n open, for q, r R such that q, r and for m N, let u : R such that u L q W m,r For j m, we have A where p satisfies u W j,p u α W m,r u α L q, p = j n + α r m + α n q j for all α in the interval m α, where C := Cn, m, j, q, r, α, with the following exceptional assumptions: i if j =, rm < n, q =, then we require u at infinity, and; ii if < r < and m j n r a nonnegative integer, then A only holds for α satisfying j m α < Proof Proof of Proposition 37: We denote by C various positive constants which depend on and ω We define the operator A L := divd + G A By density of H k m H m in L m for k N this follows from the inclusion Cc m H k m H m L m and since Cc m dense in L m, we assume without loss of generality that ψ H p+5 m and L k ψ p+ H 4 k=

15 Hence by Theorem, the solution ψ to system 3 is an element of A3 L, T ; H p+6 m H, T ; L m We apply the differential operator p+ to system 3 and, for β a multi-index with β = p +, we denote β ψ by φβ so that φ β satisfies t φ β = divd φ β G φ β + A φ β, in, φ A4 β n = φ β n, on Σ T, φ β T, = ψ β, in Indeed, since D, G and A are constant, p+ commutes with all the terms in system 3 We define the p + 3-th order tensor φ := φ β β,,β p+ n; applying Lemma 35 to system A4, we have a Carleman inequality for φ: Is, λ; φ A5 s 3 λ 4 e sα ξ 3 φ dx + sλ,t ω p+ e sα Σ T ξ φ n dσ for every λ C and s CT 5 + T The rest of this proof follows three steps: i We will estimate the boundary term on the righthand side of A5 with a global interior term involving ψ, which will be absorbed into the lefthand side; ii we will relate Is, λ; φ with the lefthand side of 3; iii we will estimate the local term on the righthand side of A5 with a local term of zero differential order as appearing in 3 and some other local terms which will be absorbed into the lefthand side Step i: Consider a function θ C such that θ n = θ = in, where n is the outward pointing normal of With this construction, θ = n Indeed, for any q and for any parametrized curve γ : R passing through point q at time, we have d θγt t= = θ dγt q =, t= since θ = in Hence, since θ is orthogonal to the tangent of any curve passing through any arbitrary point q at t =, it must be equal to n Let β and γ be 5

16 multi-indices of length n; we integrate the boundary term by parts to obtain sλ e sα ξ φ n dσ Σ T = sλ ξ β ψ θ β ψ n dσ = β = β = γ = + sλ sλ e sα Σ T e sα e sα ξ γ ψ β ψ θ dx ξ β ψ θ β ψdx Next, we employ Cauchy-Schwarz and Young s inequalities to obtain sλ ξ φ n dσ Σ T e sα A6 T λ e sα sξ k ψ H m + e sα sξ k ψ H m, for k, to be chosen later We define ˆ ψ := ρ ψ, with ρ C [, T ] defined by ρ := sξ a e sα for some a R to be chosen later Note that ˆ ψt, = in, d since ρ decays exponentially to zero as t T Similarly, i ρ =, for all i N i Furthermore, ˆ ψ is the solution to t ˆ ψ = divd ˆ ψ G ˆ ψ + A d ˆ ψ ρ ψ, in, A7 ˆ ψ =, on Σ T, ˆ ψt, =, in Hence, by A3, one can utilize Theorem to get the estimate A8 ˆ ψ L,T ;H d+ m H d+,t ;L m d ρ ψ L,T ;H d m H d,t ;L m for d {,, p + } Owing to 35 and 36, we have the bound A9 d ρ T sξ a+6/5 e sα 6

17 Indeed, for c := min x {e λ η +η x } and and c := max x {e η e λ η +η x }, we have d ρ = assξ a e d sα ξ ssξ a e d sα α = e sα ssξ a 5t T t 6 T t 6 a c sξ c = sξ a t 5T e sα a sξ c tt t c = sξ a+6/5 e sα t 5T c 6/5 at 5 T t 5 s 6/5 c s /5 and since s CT 5 + T, one can obtain A9 Similarly, we have A d r r ρ T r sξ a+6r/5 e sα, for r N We apply A8 to ˆ ψ for a = k and d = p+ to obtain, A e sα sξ k ψ H m d e sα sξ k ψ + p+ r= d r r H p+ m d e sα sξ k ψ L m We now apply A8 to ˆ ψ = d ρ ψ which satisfies a system similar to A7 and verifies the compatibility conditions in Theorem for a = k and d = p+ to obtain d e sα sξ k ψ A + p+ + r= d p+ r= d r r H p+ m d e sα sξ k ψ e sα sξ k ψ d r r H p+ L m m d e sα sξ k ψ L m 7

18 Repeating this way p+ more times and utilizing A yields the inequality A3 e sα sξ k ψ H m d p+ + e sα sξ k ψ p+ + p+ T + L m e sα sξ k+ 5 p+ + ψ L m We can get very similar estimates A and A for a = 3k, d = p+, and by using A, we obtain A4 e sα sξ 6k ψ H m d p+ + e sα sξ 3k ψ p+ + p+ T + L m e sα sξ 6k + 5 p+ + ψ L m Suppose for the moment that p is odd By applying Theorem A to the appropriate spatial derivative of ψ with j =, m = q = p = r = and α = /, and then employing the Cauchy-Schwarz inequality, we obtain e sα sξ k ψ H m e sα sξ 3k ψ H e sα sξ k ψ m H m e sα sξ 6k ψ H m e sα sξ k ψ H m Choosing k = + 3 k + 5 p+ p+ verifies p + + = 6k + 5 p + +, and hence by utilizing A3 and A4, we obtain A5 e sα sξ k ψ H m T p+ + p+ + e sα sξ p p+ ψ L m 8

19 Identical steps can be followed for the case when p is even to obtain A6 e sα sξ k ψ H m T p+ + p+ + e sα sξ p p+ ψ L m It follows from A6, A3 and A5 that sλ ξ φ n dσ Σ T e sα p+ λ T + + T p+ + p+ + e sα sξ p p+ ψ L m, for p odd, and it follows from A6, A4 and A6 sλ ξ φ n dσ Σ T e sα p+ λ T + + T p+ + p+ + e sα sξ p p+ ψ L m, for p even In what follows, we choose p even without loss of generality the exact same technique can be used for p odd, and since p+ T + + T p+ + p+ + s p 3 5 p+ 9 5 p , for s CT 5 + T, we use 37 and 38 to obtain sλ ξ φ n dσ Σ T e sα s 4/5 λ s 4/5 λ Denoting by lp the exponent 7 to conclude that Is, λ; φ e sα ξ p p+ ψ L m e sα ξ p p+ p dx p+, we arrive at the end of Step i A7 s 3 λ 4 e sα ξ 3 φ dx + s 4/5 λ e sα ξ lp dx,t ω p+ for every λ C and s CT 5 + T Step ii: In this step, we relate Is, λ; φ to the lefthand side of 3 We apply 9

20 Lemma 36 to ψ for r = p + 5 to obtain s p+7 λ p+8 e sα ξ p+7 dx s p+5 λ p+6 e sα ξ p+5 ψ dx +s p+7 λ p+8 e sα ξ p+7 A8 dx,,t ω p+ for every λ C and s CT 5 + T Similarly, for k {,, p}, we apply Lemma 36 to p+ k ψ for r = k + 3 to obtain s k+5 λ k+6 e sα ξ k+5 p+ k ψ p+ k dx A9 s k+3 λ k+4 e sα ξ k+3 p+ k ψ dx k +s k+5 λ k+6,t ω p+ e sα ξ k+5 p+ k ψ p+ k dx, for every λ C and s CT 5 + T One can upper bound the first term in the righthand side of A8 by A9 for k = p and continue this way by backwards iteration on k The global terms on the righthand side of A9 can be absorbed in the exact same way Hence, a combination of A7, A8 and A9 gives e sα s k λ k ξ k k ψ p+5 k dx k= e sα,t ω p+ s k λ k ξ k k ψ p+5 k dx k= + s 3 λ Q 4 e sα ξ 3 p+ ψ dx + s 4/5 λ e sα ξ lp dx, T for every λ C and s CT 5 + T By utilizing A7 once more, we arrive at the inequality A e sα s k λ k ξ k k ψ p+5 k dx k= e sα,t ω p+ s k λ k ξ k k ψ p+5 k dx k= + s 4/5 λ e sα ξ lp dx, which is verified for every λ C and s CT 5 + T Step iii: In this final step, we absorb the higher-order local terms in the righthand

21 side of A Consider the function θ p+ C satisfying Suppθ p+ ω p+, A θ p+ =, in ω p+, θ p+ in Let β be a multi-index of length n Since ω p+ ω p+, where ω p+ is an open subset of, we integrate the rightmost term in A by parts and employ the the Cauchy-Schwarz inequality to obtain s 3 λ 4 e sα ξ 3 p+ ψ dx,t ω p+ s 3 λ 4 θ p+ e sα ξ 3 p+ ψ dx,t ω p+ = s 3 λ 4 s 3 λ 4 n,t ω p+ i= β =p+,t ω p+ i θ p+ e sα ξ 3 i β ψ + θp+ e sα ξ 3 i β ψ β ψ dx θp+e sα ξ 3 A +θ p+ e sα ξ 3 p+ ψ dx p+ p+ p+ ψ p+ By 35 and 36, we have that A3 θp+ e sα ξ 3 sλe sα ξ 4 Indeed, θp+ e sα ξ 3 = e sα ξ 3 θ p+ + sλθ p+ ξ η + 3λθ p+ η = sλe sα ξ 4 θ p+ + θ p+ η + 3θ p+ η sλξ sξ, and since s CT 5 + T, A3 is verified Hence, by A, A3 and using Young s inequality with ɛ >, we have s 3 λ 4 e sα ξ 3 p+ ψ dx,t ω p+ s 3 λ 4 sλe sα ξ 4 p+ p+ ψ A4,T ω p+ p+ +e sα ξ 3 p+ ψ dx p+ e sα ɛs 3 λ 4 ξ 3 p+ ψ + ɛsλ ξ,t ω p+ + ɛ s5 λ 6 ξ 5 p+ dx p+

22 Observe that the first two terms in the righthand side of A4 can be bounded above by employing A and A4 recursively: indeed, by positivity of the integrand in and by A, we obtain ɛ ɛ,t ω p+ e sα s 3 λ 4 ξ 3 p+ ψ + sλ ξ e sα,t ω p+ dx s k λ k ξ k k ψ p+5 k dx k= +s 4/5 λ e sα ξ lp dx = Cɛ s k λ k ξ k k ψ p+5 k dx,t ω p+ e sα k=3 A5 +s 3 λ 4 e sα ξ 3 p+ ψ dx + s 4/5 λ e sα ξ lp dx,,t ω p+ for λ C and s CT 5 + T Combining A5 and A4 yields ɛ s 3 λ 4 ξ 3 p+ ψ + sλ ξ A6,T ω p+ e sα ɛ e sα,t ω p+ dx s k λ k ξ k k ψ p+5 k dx k=3 + e sα ɛ s 3 λ 4 ξ 3 p+ ψ + sλ ξ,t ω p+ + e sα s 5 λ 6 ξ 5 p+ ψ dx,t ω p+ p+ + ɛs 4/5 λ e sα ξ lp dx, for λ C and s CT 5 + T Using the same treatment by adapting A4, one can bound the terms with ɛ in A6; after r of these recursions, ɛ s 3 λ 4 ξ 3 p+ ψ + sλ ξ dx,t ω p+ e sα r ɛ j j= + ɛ r+ e sα,t ω p+,t ω p+ e sα +js 5 λ 6 ξ 5 p+ p+ s k λ k ξ k k ψ p+5 k dx k=3 s 3 λ 4 ξ 3 p+ ψ + sλ ξ dx + ɛ j s 4/5 λ e sα ξ lp dx,

23 for λ C and s CT 5 + T Taking ɛ sufficiently small and using A4, s 3 λ 4 e sα ξ 3 p+ ψ dx,t ω p+ s k λ k ξ k k ψ p+5 k dx A7,T ω p+ e sα k=3 + s 4/5 λ e sα ξ lp dx, for λ C and s CT 5 + T, since by A, if ψ Hence from A7, we obtain A8 e sα,t ω p+ p+ ψ s k λ k ξ k k ψ p+5 k dx k= e sα,t ω p+ =, then so does s k λ k ξ k k ψ p+5 k dx, k=3 for λ C and s CT 5 + T For r {,, p + }, consider the functions θ r C satisfying Suppθ p+ r ω p+ r, θ p+ r =, in ω p+ r, θ p+ k, in Using the exact same approach as was used for r =, one obtains the estimate s r+3 λ r+4 e sα ξ r+3 p+ r ψ dx,t ω p+ r r s k λ k ξ k k ψ p+5 k dx,t ω p+ e sα k=3+r + s 4/5 λ e sα ξ lp dx, for λ C and s CT 5 + T Hence, it follows that A9 e sα s k λ k ξ k k ψ p+5 k dx k= s p+7 λ p+8 e sα ξ p+7 dx,t ω + s 4/5 λ dx, e sα ξ lp 3

24 for λ C and s CT 5 + T Finally, by 36 we have the estimate s 4/5 λ e sα ξ lp dx Csp+7 λ p+8 e sα ξ p+7 dx, for λ C and s CT 5 + T large enough; from now on, we denote this choice of s by s Hence, one can absorb the global term in the righthand side of A9 into its lefthand side, and thus 3 is verified Proof Proof of Proposition 3: We denote by C various positive constant depending on and ω From 3, we deduce A3 e sα ξ p+7 dx e sα ξ p+7 dx,,t ω for λ C and s s Note that for t [ T 4, ] 3T 4, we have A3 min t [ T 4, 3T 4 ] {e sα ξ p+7 } = e sα ξ p+7 T = e s , = e sα ξ p+7 3T 4, e λ η e λ η +η x T 4 e p+7λ η +η x 3 5 T We can choose s sufficiently large such that A3 for all t [ T 4, ] 3T 4 Indeed, choosing { 3 5 p + 7λ A33 s s := max s, T 4, 3T T e s T e sα ξ p+7, 4 { max x η + η x e λ η e λ η +η x in A3 will ensure that A3 is verified Note that we can write s as s = σ T 5 + T, where σ > depends only on and ω Fixing s = s from now on, we deduce from A3 and A3 that dx T e C+/T 5 e sα ξ 7 dx,t ω for every λ C and s s We claim that A34 T/4, dx T ect/ T 4, 3T 4 dx }} and A35, dx e CT/4 T/4, dx, 4

25 from which we can deduce 33 Indeed, we can multiply system 3 by ψ, integrate the resulting equation by parts over and use the Cauchy-Schwarz and Young s inequalities to obtain d dx + D ψ dx = = t ψ ψdx + divd ψ ψdx G ψ ψdx + A ψ ψdx G ψ dx + + A Hence, since satisfies the uniform ellipticity condition see 3, we obtain d ψ dx + ψ dx dx, from which we deduce A36 d e Ct dx = e Ct C dx + d dx e Ct for all t > We integrate A36 over [ T 4, t] to obtain dx e CT/4 t A37 T/4, dx e CT/ dx ψ dx, T/4, dx, for every t [ T 4, ] [ 3T 4 Integrating A37 once more over T 4, 3T 4 Finally, to show A35, we integrate A36 over t [ ], T 4 ] now yields A34 5