NULL CONTROLLABILITY FOR LINEAR PARABOLIC CASCADE SYSTEMS WITH INTERIOR DEGENERACY

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1 Electronic Journal of Differential Equations, Vol. 16 (16), No. 5, pp. 1. ISSN: URL: or NULL CONTROLLABILITY FOR LINEAR PARABOLIC CASCADE SYSTEMS WITH INTERIOR DEGENERACY IDRISS BOUTAAYAMOU, JAWAD SALHI Abstract. We study the null controllability problem for linear degenerate parabolic systems with one control force through Carleman estimates for the associated adjoint problem. The novelty of this article is that for the first time it is considered a problem with an interior degeneracy and a control set that only requires to contain an interval lying on one side of the degeneracy points. The obtained result improves and complements a number of earlier works. As a consequence, observability inequalities are established. 1. Introduction This article is devoted to the analysis of control properties for linear degenerate parabolic systems in one space dimension, governed in the bounded domain (, 1) by means of one control force h, of the form u t ( (x)u x ) x b 11 (t, x)u = h(t, x)1 ω, (t, x), (1.1) v t (a (x)v x ) x b (t, x)v b 1 (t, x)u =, (t, x), (1.) u(t, ) = u(t, 1) = v(t, ) = v(t, 1) =, t (, T ), (1.) u(, x) = u (x), v(, x) = v (x), x (, 1). (1.4) where ω is an open subset of (, 1), T > fixed, := (, T ) (, 1), the coefficients b ij L ((, T ) (, 1)), i, j = 1,, h L ((, T ) (, 1)), and every a i, i = 1,, degenerates at an interior point x i of the spatial domain (, 1) (for the precise assumptions we refer to section ). 1 ω denotes the characteristic function of the set ω. The study of degenerate parabolic equations is the subject of numerous articles and books (see e.g., [, 1, 11, 1, 1, 14, 16,, 7]). As pointed out by several authors, many problems coming from physics (boundary layer models in [9], models of Kolmogorov type in [5], models of Grushin type in [4]), biology (Wright-Fisher models in [8] and Fleming-Viot models in []), and economics (Black-Merton- Scholes equations in [18]) are described by degenerate parabolic equations. On the other hand, the fields of applications of Carleman estimates in studying controllability and inverse problems for degenerate parabolic coupled systems are 1 Mathematics Subject Classification. 5K65, 5K4, 5B45, 9B5, 9B7. Key words and phrases. Degenerate parabolic systems; interior degeneracy; one control force; Carleman estimates, observability inequalities; null controllability. c 16 Texas State University. Submitted October 14, 16. Published November, 16. 1

2 I. BOUTAAYAMOU, J. SALHI EJDE-16/5 so wide that it is not surprising that also several papers are concerned with such a topic, see [1,, 7, 15, 6]. In most of the previous papers, the authors assume that the functions a i degenerates at the boundary of the space domain, e.g., a i (x) = x k i (1 x i) κi, x [, 1], where k i and κ i are positive constants. To our best knowledge, [] is the first paper that studies Carleman estimates for degenerate operators when the degeneracy is in the interior of the domain. Recently, in [1], the authors analyzed control properties for interior degenerate non smooth parabolic equations and studied different situations in which the degeneracy point is inside or outside the control region in order to obtain an observability inequality for a degenerate parabolic single equation. For related systems of degenerate equations we refer to [6], where lipschitz stability for the source term from measurements of the component u is also treated, but using a locally distributed observation ω (, 1), which contains the degeneracy points. For this reason, in the present paper we focus on carleman estimates (and, consequently, null controllability) for parabolic system (1.1) (1.4) with a control set ω such that, there exists a subinterval ω ω (, 1) lying on one side of the degeneracy points x i. We think that this latter situation is much more interesting. Indeed, our approach has an immediate application also in the case in which the control set ω is the union of two intervals ω i, i = 1, each of them lying on one side of the degeneracy points and thus it permits to cover more involved situations. Finally, let us emphasize the fact that the proof of Theorem.5 can be adapted to the case of a single equation in which way it unifies the results of [1, Lemma 5.1 and 5.], and one does not need to distinguish between the different situations in which the degeneracy point is inside or outside the control region ω. To study the controllability problem of linear degenerate parabolic system (1.1) (1.4), we use the developed Carleman estimates for the internal degenerate parabolic equations in [] to show a global Carleman inequality and deduce an observability estimate for the adjoint system. To prove our Carleman estimates, a crucial role is played by the following Hardy-Poincaré inequality. Theorem 1.1 ([, Proposition.1]). Assume that p is any continuous function in [, 1], with p > on [, 1] \ {x }, p(x ) = and such that there exists ϑ (1, ) so that the function x p(x) x x is non-increasing on the left of x = x ϑ and nondecreasing on the right of x = x. Then there exists a constant C HP > such that for any function w locally absolutely continuous on [, x ) (x, 1] and satisfying w() = w(1) = the following inequality holds 1 and 1 p(x) (x x ) w (x) dx HP p(x) w (x) dx <, 1 p(x) w (x) dx. The rest of this article is organized as follows. In Section we give the precise setting for the weak and the strong degenerate cases and discuss the well-posedness of the system (1.1) (1.4). The Carleman estimate is proved in Section. Finally, in Section 4, by the Hilbert Uniqueness Method (HUM, [17, 5]), we deduce null controllability result by showing that the adjoint system is observable. In appendix, we give summarized proof of a Caccioppoli inequality corresponding to our context.

3 EJDE-16/5 NULL CONTROLLABILITY OF DEGENERATE CASCADE SYSTEMS. Assumptions and well-posedness To study the well-posedness of the system (1.1) (1.4), we consider two situations, namely the weakly degenerate (WD) and the strongly degenerate (SD) cases. Towards this end, as in [], the associated weighted spaces and assumptions on diffusion coefficients are the following: Case (WD) { H i (, 1) := u L (, 1) : u is abs. cont. in [, 1], } ai u x L (, 1), u() = u(1) =, where the functions a i, i = 1,, satisfy: x i (, 1), i = 1,, s.t. a i (x i ) =, a i > in [, 1] \ {x i }, a i C 1 ([, 1] \ {x i }), K i (, 1) s.t. (x x i )a i K i a i a.e. in [, 1]. (.1) Case (SD) { H i (, 1) := u L (, 1) : u is l.a.c. in [, 1] \ {x i }, } ai u x L (, 1), u() = u(1) = and x i (, 1), i = 1,, s.t. a i (x i ) =, a i > in [, 1] \ {x i }, a i C 1 ([, 1] \ {x i }) W 1, (, 1), K i [1, ) s.t. (x x i )a i K i a i a.e. in [, 1], and if K i > 4/, a i then exists µ i (, K i ] such that is non-increasing on x x i µi the left of x i and non-decreasing on the right of x i. In both cases we consider the space with the norms H a i (, 1) := { u H 1 a i (, 1) : a i u x H 1 (, 1) } u H 1 a i := u L (,1) a i u x L (,1), u H a i := u H 1 a i (a i u x ) x L (,1). (.) We recall from [] that, for i = 1,, the operator (A i, D(A i )) defined by A i u := (a i u x ) x, with u D(A i ) = H a i (, 1) is closed self-adjoint negative with dense domain in L (, 1). In the Hilbert space H = L (, 1) L (, 1), the system (1.1) (1.4) can be transformed in the Cauchy problem (CP) X (t) = AX(t) BX(t) f(t), ( ) u X() = v

4 4 I. BOUTAAYAMOU, J. SALHI EJDE-16/5 where X = ( ) ( ) u(t) A1, A =, D(A) = D(A v(t) A 1 ) D(A ), ( ) ( ) b11 h(t, )1ω B =, f(t) = b 1 b As the operator A is diagonal and since B is a bounded perturbation, the following well-posedness and regularity results hold. Proposition.1. (i) The operator A generates a contraction strongly continuous semigroup (T (t)) t. (ii) for all h L () and u, v L (, 1), there exists a unique weak solution (u, v) C([, T ]; H) L (, T ; H 1 (, 1) H (, 1)) of (1.1) (1.4) and T ( sup (u, v)(t) H ) u x L (,1) a v x L (,1) dt t [,T ] (.) T ( (u, v ) H h L () ), for a positive constant C T. (iii) Moreover, if (u, v ) H 1 H 1 a, then (u, v) is in the space H 1 (, T ; H) L (, T ; H (, 1) H a (, 1)) C([, T ]; H 1 (, 1) H 1 a (, 1)), (.4) and there exists a positive constant C such that ( ) sup (u, v)(t) H 1 a (,1) H 1 1 a (,1) t [,T ] T ( (u t, v t ) ( ) ) (a H 1 u x ) x, (a v x ) x H ( (u, v ) H 1 (,1) H 1 a (,1) h L (). Carleman estimates for the adjoint cascade system dt ). (.5) The goal of this section is to establish a Carleman estimate for the homogeneous adjoint system of (1.1) (1.4). Thus, let us consider the problem U t ( (x)u x ) x b 11 (t, x)u b 1 (t, x)v =, (t, x), (.1) V t (a (x)v x ) x b (t, x)v =, (t, x), (.) U(t, 1) = U(t, ) = V (t, 1) = V (t, ) =, t (, T ), (.) U(, x) = U (x), V (, x) = V (x), x (, 1). (.4) Towards this end, we define the following time and space weight functions. For x [, 1], ϕ i (t, x) = θ(t)ψ i (x), (.5) where For x [A, B]: θ(t) := 1 [t(t t)] 4, ψ i(x) := c i [ x x i y x ] i a i (y) dy d i. Φ i (t, x) = θ(t)ψ i (x), Ψ i (x) = e ρi e riζi(x), (.6)

5 EJDE-16/5 NULL CONTROLLABILITY OF DEGENERATE CASCADE SYSTEMS 5 where ζ i (x) = B x dy ai (y), ρ i = r i ζ i (A). Here the functions a i, i = 1,, satisfy (.1) or (.) and the positive constants c i, d i, r i and ρ i are chosen such that where d 1 > d 1, d > 16d, ρ > ln(), (.7) e ρ 1 d d e ρ1 e ρ1 e ρ 1, (.8) c < 4 d (e ρ e ρ ), (.9) and c 1 max { e ρ1 1 d 1 d, 1 c d d 1 d 1 x d y x i i := sup [,1] x i a i (y) dy. Remark.1. The interval [ e ρ 1 e d d, 4(eρ ρ ) ) d is not empty. In fact, from ρ > ln, and d > 16d, we have d d < < 1 4d d e ρ < 1 4d d eρ 1 e ρ e ρ < 4(d d ) d eρ 1 d d From (.7)-(.1), we have the following results. Lemma.. (i) For (t, x) [, T ] [, 1], (ii) For (t, x) [, T ] [, 1], Proof. (i) < 4 d (e ρ e ρ ). }, (.1) ϕ 1 ϕ, Φ 1 Φ, ϕ i Φ i. (.11) 4Φ (t, x) ϕ (t, x) >. (.1) (1) ϕ 1 ϕ : since θ it is sufficient to prove ψ 1 ψ. By the choice of c 1 we have c 1 cd d 1 d. 1 Then, max{ψ 1 (), ψ 1 (1)} c d. Hence, ψ 1 (x) ψ (x). () Φ 1 Φ : since Ψ i is increasing, it is sufficient to prove that min Ψ 1 (x) max Ψ (x). Indeed Ψ 1 () = e ρ1 e r1ζ1() e ρ 1 = Ψ (1). () ϕ i Φ i : since c i eρ i 1 d i d, it follows that max{ψ i (), ψ i (1)} Ψ i (1) i and the conclusion follows immediately.

6 6 I. BOUTAAYAMOU, J. SALHI EJDE-16/5 (ii) 4Φ (t, x) ϕ (t, x) > : This follows easily by the assumption c d < 4Ψ (). We show now an intermediate Carleman-type estimate which could be used to show the null controllability for parabolic systems with two control forces. As a first step, consider the adjoint problem U t ( (x)u x ) x b 11 (t, x)u b 1 (t, x)v =, (t, x), V t (a (x)v x ) x b (t, x)v =, (t, x), U(t, 1) = U(t, ) = V (t, 1) = V (t, ) =, t (, T ), (.1) U(, x) = U (x) D(A 1), V (, x) = V (x) D(A ), where D(A i {u ) = D(A i ) } A i u D(A i ), for i = 1,. Observe that for i = 1,, D(A i ) is densely defined in D(A i) for the graph norm (see, e.g., [8, Lemma 7.]) and hence in L (, 1). As in [1] or [], letting (U, V ) vary in ( D(A 1), D(A ) ), we define the class of functions W := { (U, V ) is a solution of (.1) }. Obviously (see, e.g., [8, Theorem 7.5]) W C 1( [, T ]; D(A) ) V U, where V := L (, T ; D(A) ) H 1 (, T ; H 1 (, 1) H 1 a (, 1)), U := C ( [, T ]; H ) L (, T ; H 1 (, 1) H 1 a (, 1)). To prove the forthcoming theorems we use the following Carleman estimate proved in [, Corollary 5.1]. Theorem.. Let w L (, T ; H a(, 1) ) H 1(, T ; H 1 a(, 1) ) solution of w t (aw x ) x cw = H, (t, x), w(t, ) = w(t, 1) =, t (, T ), w(, x) = w (x), x (, 1), where c L () and a satisfies Hypothesis.1 or. and H L (). Then, there exist two positive constants C and s, such that, for all s s, (sθaw x s θ (x x ) w ) e sϕ dx dt a ( T 1 T ) H e sϕ dx dt sc 1 [aθe sϕ (x x )wxdt] x=1 x=. Remark.4. In this article, we are interested in the case where the control subdomain ω is such that, it contains a subinterval lying on one side of the degeneracy points x i, more precisely: ω = (α, β) ω (, 1), such that < x 1 < x < α < β < 1. Now we are ready to state Carleman estimates related to (.1). Theorem.5. Let T > be given. There exist two positive constants C and s such that, every solution (U, V ) W of (.1) satisfies [sθ(t) (x)ux(t, x) s θ (t) (x x 1) U (t, x)]e sϕ1(t,x) dx dt (x)

7 EJDE-16/5 NULL CONTROLLABILITY OF DEGENERATE CASCADE SYSTEMS 7 [sθ(t)a (x)vx (t, x) s θ (t) (x x ) V (t, x)]e sϕ(t,x) dx dt a (x) T s θ [U (t, x) V (t, x)]e sφ(t,x) dx dt ω for all s s. For the proof of the above theorem, we shall use the following classical Carleman estimate in suitable interval (A, B) []. Proposition.6. Let z be the solution of z t (az x ) x cz = h, x (A, B), t (, T ), z(t, A) = z(t, B) =, t (, T ), where a C 1 ([A, B]) is a strictly positive function and c L. Then there exist two positive constants r and s, such that for any s > s T B A ( T c sθe rζ z xe sφ dx dt B A h e sφ dx dt T B A T s θ e rζ z e sφ dx dt [σ(t, )z x(t, )e sφ(t, )] x=b x=a ) dt, (.14) for some positive constant c. Here the functions θ, Φ and ζ are defined, as in (.5) (.6), with σ(t, x) := rsθ(t)e rζ(x), for r, s >. Proof of Theorem.5. Let us suppose that < x 1 < x < α < β < 1 (the proof is analogous wehn we assume that < α < β < x 1 < x < 1 with obvious adaptation). Also set λ := αβ and γ := αβ, so that α < λ < γ < β. Now, we consider a smooth function η : [, 1] [, 1] such that η(x) = 1, x [γ, 1] η(x) =, x [, λ]. Then, define ˆp = ηu and ˆq = ηv, where (U, V ) is the solution of (.1). Hence, ˆp and ˆq satisfy the system ˆp t ( ˆp x ) x b 11 ˆp = b 1ˆq ( η x U) x η x U x, (t, x), ˆq t (a ˆq x ) x b ˆq = (a η x V ) x η x a V x, (t, x), ˆp(t, α) = ˆp(t, 1) = ˆq(t, α) = ˆq(t, 1) =, t (, T ). Applying the classical Carleman estimate stated in Proposition.6, with A = α and B = 1, one has C α α ˆq e sφ1 dx dt C (sθe r1ζ1 ˆp x s θ e r1ζ1 ˆp )e sφ1 dx dt T (U Ux)e sφ1 dx dt, ˆω for all s s with ˆω = [λ, γ]. Let us remark that the boundary term in x = 1 is nonpositive, while the one in x = α is, so that they can be neglected in the classical Carleman estimate. Analogously, one can prove that ˆq satisfies α (sθe rζ ˆq x s θ e rζ ˆq )e sφ dx dt

8 8 I. BOUTAAYAMOU, J. SALHI EJDE-16/5 T (V Vx )e sφ dx dt. ˆω Thus combining the last two inequalities, it follows that α C (sθe r1ζ1 ˆp x s θ e r1ζ1 ˆp )e sφ1 dx dt α C α T (sθe rζ ˆq x s θ e rζ ˆq )e sφ dx dt ˆq e sφ1 dx dt C T (V Vx )e sφ dx dt. ˆω (U Ux)e sφ1 dx dt ˆω Taking s such that C 1 s θ e rζ, using Φ 1 Φ and Caccioppoli s inequality (5.1), we obtain α α T (sθe r1ζ1 ˆp x s θ e r1ζ1 ˆp )e sφ1 dx dt (sθe rζ ˆq x s θ e rζ ˆq )e sφ dx dt ω s θ [U V ]e sφ dx dt. Then, by Lemma. one can prove that there exists a positive constant C such that for every (t, x) [, T ] [α, 1] a i (x)e sϕi(t,x) e riζi e sφi, Consequently, α α T By the definition of ˆp and ˆq, we obtain (x x i ) e sϕi(t,x) e riζi e sφi. (.15) a i (x) (sθ ˆp x s θ (x x 1) ˆp ) e sϕ1 dx dt γ γ T (sθa ˆq x s θ (x x ) ˆq ) e sϕ dx dt a ω s θ [U V ]e sφ dx dt. (sθ Ux s θ (x x 1) U ) e sϕ1 dx dt (sθa Vx s θ (x x ) V ) e sϕ dx dt a ω s θ [U V ]e sφ dx dt, for a positive constant C and for s large enough. (.16)

9 EJDE-16/5 NULL CONTROLLABILITY OF DEGENERATE CASCADE SYSTEMS 9 On the other hand, by the properties of the weight functions, calculations show that s θ (x x i) a i e sϕi s θ e sφ, (t, x) (, T ) (λ, γ) (.17) for a positive constant C. In addition, arguing as in the proof of Caccioppoli s inequality 5.1, one can easily show that T γ T sθ[ux Vx ]e sφ dx dt s θ [U V ]e sφ dx dt, (.18) ω λ for some constant C >. By (.17) and (.18) we can find a positive constant C such that T γ λ T γ λ T Thus (.16) and (.19) imply (sθ Ux s θ (x x 1) U ) e sϕ1 dx dt (sθa Vx s θ (x x ) V ) e sϕ dx dt a ω s θ [U V ]e sφ dx dt. (.19) λ λ T (sθ Ux s θ (x x 1) U ) e sϕ1 dx dt (sθa Vx s θ (x x ) V ) e sϕ dx dt a ω s θ [U V ]e sφ dx dt, (.) for a positive constant C and for s large enough. To complete the proof, it is sufficient to prove a similar inequality on the interval [, λ]. To this aim, we follow a reflection procedure. Consider the functions W (t, x) := { U(t, x), x [, 1], U(t, x), x [ 1, ], Z(t, x) := { V (t, x), x [, 1], V (t, x), x [ 1, ], where (U, V ) solves (.1), and { ψi (x), x [, 1], ψ i (x) := c i [ x yx i x i ã dy d i(y) i], x [ 1, ], { { a i (x), x [, 1], b ij (x), x [, 1], ã i (x) = bij (x) := a i ( x), x [ 1, ], b ij ( x), x [ 1, ]. Therefore, (W, Z) solves the system W t (ã 1 W x ) x b 11 W = b 1 Z, x ( 1, 1), t (, T ), Z t (ã Z x ) x b Z =, x ( 1, 1), t (, T ), W (t, 1) = W (t, 1) = Z(t, 1) = Z(t, 1) =, t (, T ). (.1)

10 1 I. BOUTAAYAMOU, J. SALHI EJDE-16/5 Now, we consider a smooth function τ : [ 1, 1] [, 1] such that { 1, x [ x 1 /, λ], τ(x) =, x [ 1, x 1 /] [γ, 1], and define the functions p = τw and q = τz, where (W, Z) is the solution of (.1). Then ( p, q) satisfies p t (ã 1 p x ) x b 11 p = b 1 q (ã 1 τ x W ) x τ x ã 1 W x, x ( 1, 1), t (, T ), q t (ã q x ) x b q = (ã τ x Z) x τ x ã Z x, x ( 1, 1), t (, T ), p(t, x 1 ) = p(t, 1) = q(t, x 1 ) = q(t, 1) =, t (, T ). Now, define ϕ i := θ(t) ψ i (x), where ψ i is defined as above. Using the analogue of Theorem. for the first component p on ( x1, 1) in place of (, 1) and with ϕ i replaced by ϕ i, by the equalities p x (t, x1 ) = p x(t, 1) = and the definition of W, we obtain x 1/ C C (sθã 1 p x s θ (x x 1) p )e s ϕ1 dx dt x 1/ T x 1 x 1 x 1/ T x 1 ã 1 b 1 q e s ϕ1 dx dt (W W x )e s ϕ1 dx dt C b 1 q e s ϕ1 dx dt (U U x 1 x)e sϕ1 dx dt C } {{} J T γ T γ λ λ (W W x )e sϕ1 dx dt (U U x)e sϕ1 dx dt. To absorb J, let ɛ 1 > be small enough. Since inf t [,T ] θ(t) >, inf x x [ 1, x 1 ] (x) > (x x and inf x 1) x [ 1, x 1 ] (x) >, for s large enough, it follows that Therefore, T x 1 x 1 ɛ 1 T ɛ 1 T x 1/ (U U x)e sϕ1 dx dt x 1 x 1 λ (sθ Ux s (x x1) θ U )e sϕ1 dx dt (sθ Ux s θ (x x 1) U )e sϕ1 dx dt. (sθã 1 p x s θ (x x 1) p )e s ϕ1 dx dt x 1/ ã 1 b 1 q e s ϕ1 dx dt C T γ λ (U U x)e sϕ1 dx dt

11 EJDE-16/5 NULL CONTROLLABILITY OF DEGENERATE CASCADE SYSTEMS 11 ɛ 1 T λ (sθ Ux s θ (x x 1) U )e sϕ1 dx dt. Using similar arguments, for the second component q, we obtain x 1/ T γ ɛ T (sθã q x s θ (x x ) q )e s ϕ dx dt λ λ ã (V V x )e sϕ dx dt (sθa Vx s θ (x x ) V )e sϕ dx dt, where ɛ > is taken small. Combining the last two inequalities and using Lemma., by Caccioppoli s inequality 5.1, we obtain a x 1/ x 1/ T ɛ 1 T ɛ T (sθã 1 p x s θ (x x 1) p )e s ϕ1 dx dt ã 1 (sθã q x s θ (x x ) q )e s ϕ dx dt ã ω s θ (U V )e sφ dx dt C λ λ x 1/ (sθ Ux s θ (x x 1) U )e sϕ1 dx dt (sθa Vx s θ (x x ) V )e sϕ dx dt. On the other hand, using that ϕ 1 < ϕ, we have x 1/ b 1 = b 1 b 1 4 b 1 4 b 1 q e s ϕ1 dx dt x 1/ x 1/ x 1/ ( qe s ϕ ) dx dt ( x x x 1/ ã (x) a b 1 q e s ϕ1 dx dt q e s ϕ ) 1/4 ( ã1/ (x) x x / q e s ϕ ) /4 dx dt x x q e s ϕ dx dt ã (x) ã 1/ (x) x x / q e s ϕ dx dt. (.) (.) Now, define w(t, x) := e s ϕ(t,x) q(t, x), one has x 1/ ã 1/ (x) x x / q e s ϕ dx dt = x 1/ ã 1/ (x) x x / w dx dt.

12 1 I. BOUTAAYAMOU, J. SALHI EJDE-16/5 First observe that, since w( x 1 /) = w() =, by the classical Poincaré inequality, one has T x 1/ max x [ x 1 P C P ã 1/ x x / w dx dt 1/ { ã (x) },] x x / max x [ x 1 T 1/ { ã (x) },] x x / max x [ x 1 1/ { ã (x) },] x x / x 1/ T x 1/ T w dx dt x 1/ w x dx dt ã w x dx dt, (.4) where C P is the Poincaré constant. Now, to obtain an estimate on (, 1), we distinguish between two cases. First, if K 4, we consider the function p (x) = x x 4. Obviously, there exists q ( ) 1, 4 such that the function p (x) x x is non-increasing on the left of x = x q and nondecreasing on the right of x = x. Then, we can apply the Theorem 1.1, obtaining / x x / w dx dt max x [,1] a1/ (x) = max x [,1] a1/ (x) T max x [,1] a1/ (x)c HP 1 x x / w dx dt p (x) x x w dx dt 1 1 T = max x [,1] a1/ (x)c HP T max x [,1] a1/ (x)c HP C 1 1 where C HP is the Hardy-Poincaré constant and C 1 = max ( x p (x)w x dx dt x x 4 a wx dx dt a a w x dx dt, 4 a, (1 x) 4 () a (1) ). In the previous inequality we have used the property that the map x x x µ a (x) is nonincreasing on the left of x = x and nondecreasing on the right of x = x for all µ K, see [, Lemma.1]. If K > 4/, we can consider the function p (x) = (a (x) x x 4 ) 1/. Then ( (x x ) ) / p (x) = a (x) 1 a (x), a (x) where { ( x C 1 := max a () ) /, ((1 x ) a (1) ) / }, / x x / = p (x) (x x ). Moreover, using Hypothesis., one has that the function p(x) x x, where q := q 4µ (1, ), is non-increasing on the left of x = x and nondecreasing on the right

13 EJDE-16/5 NULL CONTROLLABILITY OF DEGENERATE CASCADE SYSTEMS 1 of x = x. The Hardy-Poincaré inequality implies / x x / w dx dt = HP T p (x) (x x ) w dx dt 1 HP C 1 T 1 p (x)w x dx dt a w x dx dt, where C HP and C 1 are the Hardy-Poincaré constant and the constant introduced before, respectively. Thus, in every case, / x x / w dx dt a w x dx dt, (.5) for a positive constant C. Combining (.4) and (.5), we obtain x 1/ T = C P T x 1/ ã 1/ x x / w dx dt max x [ x 1 1 x 1/ x 1/ ã 1/ x x / w dx dt { ã 1/ (x),] x x } / ã w x dx dt T ã q xe s ϕ dx dt C From (.) and (.6), it results that x 1/ / x x / w dx dt ã w x dx dt C x 1/ a w x dx dt s θ x x q e s ϕ dx dt. ã (x) (.6) x 1/ x 1/ b 1 q e s ϕ1 dx dt (ã q x s θ x x q ) e s ϕ dx dt, ã (x) (.7) for a positive constant C.

14 14 I. BOUTAAYAMOU, J. SALHI EJDE-16/5 Taking s large enough, by (.7), one can estimate (.) in the following way x 1/ x 1/ T (sθã 1 p x s θ (x x 1) p )e s ϕ1 dx dt ã 1 (sθã q x s θ (x x ) q )e s ϕ dx dt s θ (U V )e sφ dx dt ω T λ ɛ 1 (sθ Ux s θ (x x 1) U )e sϕ1 dx dt ɛ T λ ã (sθa Vx s θ (x x ) V )e sϕ dx dt. Hence, by (.8), the definition of W, Z, p and q, we obtain T λ T λ = = T λ a (sθ Ux s θ (x x 1) U )e sϕ1 dx dt T λ T λ x 1 T λ x 1 T λ x 1 T λ x 1 (sθa Vx s θ (x x ) V )e sϕ dx dt x 1/ a (sθ Wx s θ (x x 1) W )e sϕ1 dx dt (sθa Zx s θ (x x ) Z )e sϕ dx dt a (sθã 1 Wx s (x x1) θ W )e s ϕ1 dx dt ã 1 (sθã Zx s (x x) θ Z )e s ϕ dx dt ã (sθã 1 p x s (x x1) θ p )e s ϕ1 dx dt ã 1 x 1/ T (sθã q x s (x x) θ q )e s ϕ dx dt ã (sθã 1 p x s θ (x x 1) p )e s ϕ1 dx dt ã 1 (sθã q x s θ (x x ) q )e s ϕ dx dt s θ (U V )e sφ dx dt ω T λ ɛ 1 (sθ Ux s θ (x x 1) U )e sϕ1 dx dt ɛ T λ ã (sθa Vx s θ (x x ) V )e sϕ dx dt. Adding up (.) and (.9), we finally obtain Theorem.5. a (.8) (.9)

15 EJDE-16/5 NULL CONTROLLABILITY OF DEGENERATE CASCADE SYSTEMS 15 To study the null-controllability of the parabolic system (1.1) (1.4) with one control force, we need to show the following Carleman estimate. Theorem.7. Let T >. Moreover, assume that b 1 µ > on [, T ] ω 1, (.) for some ω 1 ω. Then there exist two positive constants C and s such that, for all s s, the solution (U, V ) W of (.1) satisfies T (sθ Ux s θ (x x 1) U ) e sϕ1 dx dt (sθa Vx s θ (x x ) V ) e sϕ dx dt ω U dx dt. a (.1) Theorem.7 is a consequence of Theorem.5 applied to ω 1 and of the following Lemma. Lemma.8. For each ε > there is C ε > such that T T s θ V e sφ(t,x) dx dt εj(v) C ε ω 1 where ε > is small enough, s is large enough and J(V ) = ω U dx dt, (sθa Vx s θ (x x ) V )e sϕ dx dt. Proof. The choice of the weight functions given in Lemma. will play a crucial role. We will adapt the technique used in []. Let χ C (, 1), such that supp(χ) ω and χ 1 on ω 1. Multiplying the first equation of system (.1) by s θ χe sφ(x) V and integrating over, we obtain s θ b 1 χe sφ(x) V dx dt = s θ χe sφ(x) U t V dx dt s θ χe sφ(x) ( U x ) x V dx dt (.) s θ b 11 χe sφ(x) UV dx dt. Integrating by parts and using the second equation in (.1), we obtain s θ χe sφ(x) V U t dx dt = s θ a χe sφ(x) U x V x dx dt s θ a (χe sφ(x) ) x UV x dx dt [ s θ b s 4 θ θψ (x) s θ θ] χe sφ (x) V U dx dt, a (.)

16 16 I. BOUTAAYAMOU, J. SALHI EJDE-16/5 and s θ χe sφ(x) ( U x ) x V dx dt = s θ χe sφ(x) V x U x dx dt s θ (χe sφ(x) ) x V x U dx dt s θ ( (χe sφ(x) ) x ) x V U dx dt. So, combining (.)-(.4), we obtain s θ b 1 χe sφ(x) V dx dt = I 1 I I, where I 1 = s θ ( a )χe sφ(x) U x V x dx dt, I = s θ ( a )(χe sφ(x) ) x UV x dx dt ( = s θ χ s 4 θ 4 Ψ,x (x)χ ) ( a )e sφ(x) UV x dx dt, [ I = s θ θ s θ (b 11 b ) s 4 θ θψ (x) ] χe sφ(x) UV dx dt s θ ( (χe sφ(x) ) x ) x UV dx dt. For ε >, we have I 1 = ( sθa e sϕ V x )((sθ) 5/ (a ) 1 (a1 a )χe s(φ(x)ϕ) U x ) dx dt (.4) ε sθa e sϕ Vx dx dt 1 s 5 θ 5 ( a ) χ e s(φ(x)ϕ) Ux dx dt. ε a }{{} L The integral L should be estimated by an integral in U. For this, we multiply the first equation in (.1) by s 5 θ 5 (a 1 a ) a χ e s(φ(x)ϕ) U and we integrate by parts, obtaining L = L 1 L L L 4, where L 1 = 1 s 5 (5θ 4 sθ 5 (Ψ (x) ψ )) θ ( a ) χ a e s(φ(x)ϕ) U dx dt, L = 1 s 5 θ 5 ( ( ( a ) χ e s(φ(x)ϕ) ) x ) x U dx dt, a L = s 5 θ 5 ( a ) χ b 11 e s(φ(x)ϕ) U dx dt, a L 4 = s 5 θ 5 ( a ) χ b 1 e s(φ(x)ϕ) UV dx dt. a

17 EJDE-16/5 NULL CONTROLLABILITY OF DEGENERATE CASCADE SYSTEMS 17 Since θ θ and supp(χ) ω 1, the functions a i, a i, χ, ψ i, Ψ i and their derivatives are bounded on ω and also b 11, b 1 and b. We deduce that, for i {1,, } T L i s 7 θ 7 e s(φ(x)ϕ) U dx dt. ω For i = 4 we have [ (x x ) L 4 = (sθ) e sϕ V ][ (sθ) 7 b1 χ (a 1 a ) a a (x x ) e s(4φϕ) U ] dx dt ε s θ (x x ) e sϕ V dx dt Hence, a 1 4ε s 7 θ 7 b 1χ 4 (a 1 a ) a 1 a (x x ) e s(4φϕ) U dx dt ε s θ (x x ) T e sϕ V dx dt C ε s 7 θ 7 e s(4φϕ) U dx dt. a ω Furthermore T L ε ε T I 1 ε ω s 7 θ 7 e s(4φϕ) U dx dt s θ (x x ) e sϕ V dx dt. a ω s 7 θ 7 e s(4φϕ) U dx dt εj(v ). Using the fact that χ and χ are supported in ω and x ω, proceeding as before, one has I s 4 θ 4 (χ χ)e sφ UV x dx dt ( sθa e sϕ V x )((sθ) 7/ (a ) 1 (χ χ)e s(φϕ) U) dx dt ε I ε T sθa Vx e sϕ dx dt C ε s 5 θ 5 (χ χ χ)e sφ UV dx dt (s θ x x a e sϕ V )((sθ) 7 ω s 7 θ 7 e s(φϕ) U dx dt. a s θ (x x ) V e sϕ dx dt C ε a So, thanks to Lemma., we have (χ χ χ)e s(φϕ) U) dx dt x x T s 7 θ 7 e s(φϕ) U dx dt. ω e s(φϕ) e s(4φϕ) 1, sup s r θ r (t)e s(4φϕ) <, r R. (t,x)

18 18 I. BOUTAAYAMOU, J. SALHI EJDE-16/5 Then, for ε small enough and s large enough, we have T s θ b 1 χe sφ V dx dt ε U dx dt εj(v ). ω Finally, by the definition of χ and the previous inequality, it follows that T T µ s θ e sφ V dx dt s θ b 1 e sφ V dx dt ω 1 This completes the proof. ω 1 T ε s θ b 1 χe sφ V dx dt ω U dx dt εj(v ). 4. Observability and null controllability In this section we prove, as a consequence of the Carleman estimates established in the above section, observability inequalities for the adjoint problem (.1)-(.4). Theorem 4.1. Let T > be given. Then there exists a positive constant C T such that every (U, V ) solution of (.1)-(.4) satisfies 1 T [U (T, x) V (T, x)]dx T U (t, x) dx dt. ω To prove the above theorem we need the following result. Lemma 4.. Let T > be given. Then there exists a positive constant C T such that every (U, V ) W solution of (.1) satisfies 1 T [U (T, x) V (T, x)]dx T U (t, x) dx dt. ω Proof. Multiplying the first and the second equations in the system (.1) respectively by U t and V t. Integrating over (, 1) the sum of the new equations, we obtain = 1 [Ut Vt ] dx [ (x)u x U t ] 1 [a (x)v x V t ] 1 1 b V V t dx 1 Using the Young s inequality we obtain d dt 1 [ U x a V x ] dx b 1 V U t dx 1 d dt 1 1 is non-increasing on [, x i ) and non- By [, Lemma.1], the map x (x xi) a i(x) decreasing on (x i, 1], then ((x x i ) a i (x) b 11U dx b 11 UU t dx [ U x a V x ] dx. (b b 1)V dx, [U (t, x) V (t, x)] dx. ) 1/ {( x ) max i 1/, ((1 x i ) ) 1/ }. a i () a i (1)

19 EJDE-16/5 NULL CONTROLLABILITY OF DEGENERATE CASCADE SYSTEMS 19 Hence d dt 1 where [ U x a V x ] dx C 1 [ / 1 (x) (x x 1 ) U (t, x) a1/ (x) / (x x ) V (t, x) ] dx, / C := max {( x ) 1 1/, ((1 x 1 ) ) 1/, ( x ) 1/, ((1 x ) ) 1/ }. () (1) a () a (1) Moreover, by the Hardy-Poincaré inequality, and proceeding as in (.5), one has Hence d dt [U x a V x ]dx 1 1 [ U x a V x ] dx. d { 1 e C 1t [ Ux a Vx ] dx }. dt Consequently, the function t e C1t 1 [U x a V x ] dx is not increasing. Thus, 1 1 [ (x)ux(t, x) a (x)vx (T, x)] dx e C1T [ (x)ux(t, x) a (x)vx (t, x)] dx. Moreover, by the fact that inf ( T 4, T 4 ) (,1) sθe sϕi >, integrating over [T/4, T/4], and using the Carleman estimate (.1), we obtain 1 [ (x)u x(t, x) a (x)v x (T, x)] dx ec1t T T T/4 T/4 T T T/4 1 T/4 1 [ (x)u x(t, x) a (x)v x (t, x)] dx dt sθ[ (x)u xe sϕ1 a (x)v x e sϕ ] dx dt ω U (t, x) dx dt. On the other hand, applying the Hardy-Poincaré inequality one gets 1 [U (T, x) V (T, x)] dx 1 1 [ / 1 (x) (x x 1 ) U (T, x) a1/ (x) / (x x ) V (T, x) ] dx / [ (x)u x(t, x) a (x)v x (T, x)] dx, for a positive constant C. Combining (4.1) and (4.) the conclusion follows. (4.1) (4.) The proof of Theorem 4.1 is now standard using Lemma 4. and proceeding as in [, Proposition 4.1], but we give it for the reader s convenience.

20 I. BOUTAAYAMOU, J. SALHI EJDE-16/5 Proof of Proposition 4.1. Let (U, V ) L (, 1) L (, 1) and let (U, V ) be the solution of (.1) associated to (U, V ). Since D(A i ) is densely defined in L (, 1), there exists a sequence (U n, V n ) n D(A 1) D(A ) which converge to (U, V ) in L (, 1) L (, 1). Now, consider the solution (U n, V n ) associated to (U n, V n ). Since the semigroup generated by A is analytic, hence A is closed (e.g., see [19, Theorem I.1.4]), thus, by [19, Theorem II.6.7], we obtain that (U n, V n ) n converges to a certain (U, V ) in C([, T ]; H), so that lim 1 n 1 lim n T lim n U n(t, x) dx = V n (T, x) dx = 1 1 T U (T, x) dx V (T, x) dx Un(t, x) dx dt = U (t, x) dx dt. ω ω But by Lemma 4. we know that 1 T [Un(T, x) Vn (T, x)]dx T Un(t, x) dx dt. ω Thus Theorem 4.1 is now proved. By Theorem 4.1, using a standard technique (e.g., see [4, Section 7.4]), one can deduce the following controllability result. Theorem 4.. If the assumption. is satisfied, then the cascade degenerate parabolic system (1.1) (1.4) with one control force is null controllable. 5. Appendix As in [1, ], we give the proof of the Caccioppoli s inequality for linear cascade systems with two interior degeneracies. Lemma 5.1 (Caccioppoli s inequality). Let ω and ω two open subintervals of (, 1) such that ω ω ω (, 1) and x i ω. Then, there exist two positive constants C and s such that every solution (U, V ) W of the adjoint problem (.1) satisfies T [Ux(t, x) Vx (t, x)]e sφ dx dt ω T (5.1) s θ [U (t, x) V (t, x)]e sφ dx dt, ω for all s s. Proof. Define a smooth cut-off function ξ C (, 1) such that supp(ξ) ω and ξ 1 on ω. Since (U, V ) solves (.1), we have = T = d dt [ 1 ξ e sφ (U V ) dx]dt s Φ ξ e sφ (U V ) dx dt (ξ e sφ ) x (x)uu x dx dt ξ e sφ (x)u x dx dt ξ e sφ b (x)v dx dt

21 EJDE-16/5 NULL CONTROLLABILITY OF DEGENERATE CASCADE SYSTEMS 1 ξ e sφ b 11 U dx dt ξ e sφ a (x)v x dx dt Then, integration by parts yields = = ξ e sφ[ Ux a Vx ] dxdt 1 s Φ ξ e sφ (U V ) dx dt ξ e sφ (b 11 U b V ) dx dt s Φ ξ e sφ (U V ) dx dt 1 ) ((ξ e sφ ) x a V dx dt ξ e sφ b 1 UV dx dt. x ξ e sφ b 1 UV dx dt (ξ e sφ ) x a (x)v V x dx dt. (ξ e sφ ) x ( UU x a V V x )dx dt ξ e sφ b 1 UV dx dt ((ξ e sφ ) x ) x U dx dt ξ e sφ (b 11 U b V ) dx dt Since x i ω, supp(ξ) ω, ξ 1 on ω and θ cθ then, using the Young inequality, we obtain T min x ω {(x), a (x)} e sφ [Ux Vx ] dx dt ω T T and the proof is complete. ξ e sφ [ U x a V x ] dx dt (1 s θ s θ )[U V ]e sφ dx dt ω s θ [U V ]e sφ dx dt, ω References [1] E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj, L. Maniar; Carleman estimates and null controllability of coupled degenerate systems, Evol. Equ. Control Theory, (1), [] E. M. Ait Benhassi, F. Ammar Khodja, A. Hajjaj, L. Maniar; Null controllability of degenerate parabolic cascade systems, Portugal. Math., 68 (11), [] F. Alabau-Boussouira, P. Cannarsa, G. Fragnelli; Carleman estimates for degenerate parabolic operators with applications to null controllability, J. evol.equ. 6 (6) [4] K. Beauchard, P. Cannarsa, R. Guglielmi; Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS), 16 (14), no. 1, [5] K. Beauchard, E. Zuazua; Some controllability results for the D Kolmogorov equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (9), [6] I. Boutaayamou, G. Fragnelli, L. Maniar; Lipschitz stability for linear parabolic systems with interior degeneracy, Electron. J. Differential Equations, 14 (14), no. 167, 1-6. [7] I. Boutaayamou, A. Hajjaj, L. Maniar; Lipschitz stability for degenerate parabolic systems, Electron. J. Differential Equations, 149, (14) 1-15.

22 I. BOUTAAYAMOU, J. SALHI EJDE-16/5 [8] H. Brezis; Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer ScienceBusiness Me-dia, LLC, 11. [9] J. M. Buchot, J. P. Raymond, linearized model for boundary layer equations, in Optimal control of complex structures (Oberwolfach, ), Internat. Ser. Numer. Math., 19 (), Birkhauser, Basel, 1-4. [1] P. Cannarsa, G. Fragnelli, D. Rocchetti, Controllability results for a class of one dimensional degenerate parabolic problems in nondivergence form, J. evol. equ., 8 (8), [11] P. Cannarsa, G. Fragnelli, D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, (7), no. 4, [1] P. Cannarsa, P. Martinez, J. Vancostenoble; The cost of controlling degenerate parabolic equations by boundary controls, arxiv: [1] P. Cannarsa, P. Martinez, J. Vancostenoble; Global Carleman estimates for degenerate parabolic operators with applications, Memoirs of the American Mathematical Society, to appear. [14] P. Cannarsa, P. Martinez, J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM, J. Control Optim., 47 (8), [15] P. Cannarsa, L. De Teresa; Controllability of 1-d coupled degenerate parabolic equations, Electron. J. Differential Equations, 7 (9), 1-1. [16] P. Cannarsa, J. Tort, M. Yamamoto; Unique continuation and approximate controllability for a degenerate parabolic equation, Appl. Anal., 91, No. 8, (1). [17] J.-M. Coron; Control and Nonlinearity, Mathematical Surveys and Monographs, Vol. 16, Providence, RI: Amer. Math. Soc., 7. [18] H. Emamirad, G. R. Goldstein, J. A. Goldstein; Chaotic solution for the Black-Scholes equation, Proc. Amer. Math. Soc., 14 (1), 4-5. [19] K. J. Engel, R. Nagel; One-Parameter Semigroups for Linear Evolution Equations, Springer- Verlag, New York, (). [] W. H. Fleming, M. Viot; Some measure-valued Markov processes in population genetics theory, Indiana Univ. Math. J., 8 (1979), [1] G. Fragnelli, D. Mugnai; Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Mem. Amer. Math. Soc., in press, 4 (16), arxiv: [] G. Fragnelli, D. Mugnai; Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Advances in Nonlinear Analysis, (1), [] M. Gueye; Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim, Vol 5 (14), No 4, p [4] J. Le Rousseau, G. Lebeau; On carleman estimates for elliptic and parabolic operators. applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (1), [5] J. L. Lions; Contrôlabilité exacte perturbations et stabilisation de systèmes distribués, (Tome 1, Contrôlabilité exacte; Tome, Perturbations), Recherches en Mathematiques Appliquées, Masson, [6] X. Liu, H. Gao, P. Lin; Null controllability of a cascade system of degenerate parabolic equations, Acta Math. Sci. Ser. A Chin. Ed., 8, (8), [7] P. Martinez, J. Vancostenoble; Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., 6 (6), no., 5-6. [8] N. Shimakura, Partial Differential Operators of elliptic type, Translations of Mathematical Monographs 99 (199), American Mathematical Society, Providence, RI. Idriss Boutaayamou Département de Mathématiques Informatiques et Gestion, Faculté Polydisciplinaires de Ouarzazate, Université Ibn Zohr, B.P. 68, Ouarzazate 45, Morocco address: dsboutaayamou@gmail.com Jawad Salhi Faculté des Sciences et Techniques, Université Hassan 1er, Laboratoire MISI, B.P. 577, Settat 6, Morocco address: sj.salhi@gmail.com