On the boundary control of a parabolic system coupling KS-KdV and Heat equations

Transcription

1 SCIENTIA Series A: Mathematical Sciences, Vol , Universidad Técnica Federico Santa María Valparaíso, Chile ISSN c Universidad Técnica Federico Santa María 212 On the boundary control of a parabolic system coupling KS-KdV and Heat equations Eduardo Cerpa a, Alberto Mercado a and Ademir F. Pazoto b Abstract. This paper presents a control problem for a one-dimensional nonlinear parabolic system. The system consists of a Kuramoto-Sivashinsky-Korteweg de Vries equation coupled to a heat equation. The problem of boundary controllability is discussed. The local null-controllability of the system is proven. The proof is based on a Carleman estimates approach to deal with the linearized system around the origin. A local inversion theorem is applied to get the result for the nonlinear system. 1. Introduction A one-dimensional model for turbulence and wave propagation in reaction-diffusion systems is given by the Kuramoto-Sivashinsky KS equation, which reads as 1.1 u t + γu xxxx + au xx + uu x =. This equation, where γ and a are coefficients accounting for the long-wave instability and the short-wave dissipation respectively, was derived in various physical contexts [14, 15, 21]. By adding a linear third-order term, Benney in [5] takes into account dissipative effects in the Kuramoto-Sivashinsky-Korteweg-de Vries KS-KdV equation 1.2 u t + γu xxxx + u xxx + au xx + uu x =. Equation 1.2 allows to study various nonlinear dissipative waves. When looking for solitary-pulse solutions of 1.2 on the whole line, one finds out that they are unstable. In order to combine dissipative and dispersive features, and to simultaneously support stable solitary-pulse, a one-dimensional model consisting of a KS-KdV equation, linearly coupled to an extra dissipative equation, was proposed in [17]. The model has the form { ut + γu 1.3 xxxx + u xxx + au xx + uu x = v x, v t Γv xx + cv x = u x, 2 Mathematics Subject Classification. Primary 93B5 Secondary 35K41, 93C2. Key words and phrases. Parabolic system, boundary control, null-controllability. 55

2 56 E. CERPA, A. MERCADO AND A.F. PAZOTO where the dissipative parameter effective diffusion coefficient Γ > accounts for stabilization and c is the group-velocity mismatch between the wave modes. Generalizations of this system to the two-dimensional case has been considered. In [18], the authors use Zakharov-Kuznetsov equations and in [19] Kadomtsev-Petviashvili equations. In this work we are interested in the study of controllability properties of system 1.3 posed on a bounded interval with boundary control inputs h 1, h 2, h 3, and initial data u, v. Thus, the system considered is written as u t + γu xxxx + u xxx + au xx + uu x = v x, x, t, 1, T, v t Γv xx + cv x + v 2 = u x, x, t, 1, T, u, t = h 1 t, u1, t =, t, T, 1.4 u x, t = h 2 t, u x 1, t =, t, T, v, t = h 3 t, v1, t =, t, T, ux, = u x, vx, = v x, x, 1, where we have added a quadratic term into the heat equation. Here, we assume that 1.5 a, γ, and Γ are positive constants while c may have any sign. We address the problem of steering the solutions of system 1.4 to the rest. More precisely, given T > and an appropriate space X, we say that system 1.4 is null controllable if for any initial condition u, v X, there exist boundary controls h 1, h 2, h 3 such that the solution of 1.4 satisfies ut, = vt, =. We say that the local null controllability holds if we can find controls as above whenever u, v X is small enough. In this paper we will prove this last property for system 1.4, which couples a heat equation and a KS-like equation. On one hand, the null controllability for the heat equation is well known from [16] by Lebeau and Robbiano. The same result can be proven by using a global Carleman estimate as in [1] by Fursikov and Imanuvilov. On the other hand, recently the null controllability of the Kuramoto-Sivashinsky has been proven in [7] see also [6] about the linearized KS equation. The approach used is the same as that by Fursikov and Imanuvilov. In this paper we combine the results for these two equations in order to get the following theorem. Theorem 1.1. Let T >. H 2, 1 H 1, 1 with There exists δ > such that for any u, v u H 2,1 + v H 1,1 < δ, there exist h 1, h 2, h 3 L 2, T such that the corresponding solution u, v C[, T ], H 2, 1 H 1, 1 L 2, T ; L 2, 1 L 2, 1 of 1.4 satisfies ut, = vt, =. The paper is organized as follows. In Section 2, the well-posedness framework is stated for system 1.4. The proof of Theorem 1.1 is given in Section 3. The linearized system is studied in Section 3.1 by using a Carleman estimates approach. The final

3 CONTROL OF A SYSTEM COUPLING KS-KDV AND HEAT EQUATIONS 57 result for the nonlinear system is obtained by means of a local inversion theorem in Section 3.2. Remark 1.2. As stated in the survey [4], the study of the controllability for systems of parabolic equations is rather recent. Let us mention, in the case of internal control of coupled reaction-diffusion equations, the articles [2, 3, 12, 13] and the references therein. In those papers, the main tool is a Carleman estimate. In the one-dimensional case, the boundary control of two coupled heat equations has been considered in [9]. The authors use the moment method and prove the existence of an appropriate biorthogonal family of L 2 -functions. 2. Well-posedness This section is devoted to the proof of well-posedness results for the equations we are concerned here. We state results for both the linear and nonlinear systems Linear homogeneous system. Next theorem states the existence and uniqueness of solutions for the linear system with homogeneous boundary data. Theorem 2.1. Let a, γ, Γ and c as in 1.5, u, v H 2, 1 H 1, 1 and f 1, f 2 L 2, T ; L 2, 1. Then, for any T > the linear system u t + γu xxxx + u xxx + au xx = v x + f 1, x, t, 1, T, v t Γv xx + cv x = u x + f 2, x, t, 1, T, u, t =, u1, t =, t, T, 2.1 u x, t =, u x 1, t =, t, T, v, t =, v1, t =, t, T, ux, = u x, vx, = v x, x, 1, has a unique solution u, v such that 2.2 u, v L 2, T ; H 4, 1 H 2, 1 C[, T ]; H 2, 1 H 1, 1. Moreover, there exists C > such that 2.3 u, v CH 2 H 1 L2 H 4 H 2 C { } f 1, f 2 L 2 L u, v H 2 H 1 for all u, v H 2, 1 H 1, 1 and f 1, f 2 L 2, T ; L 2, 1. Remark 2.2. We have introduced for m, s Z, the notation H m H s := H m, T ; H s, 1, CH s := C[, T ], H s, 1. The proof of Theorem 2.1 will be done by applying the Faedo-Galerkin method with a special basis formed by the solutions of the spectral problem associated with the operator 2. Therefore, the following result will be needed. Lemma 2.3. There exists a set of positive real numbers µ j j N such that the corresponding solutions w j j N of the problem { 2 w j x = µ j w j x, x, 1, 2.4 w j = w j = w j1 = w j 1 =,

4 58 E. CERPA, A. MERCADO AND A.F. PAZOTO form a basis in H 4, 1 H 2, 1, which is orthonormal in L 2, 1. We note that, since the operator 2 is simultaneously positive and self-adjoint, the assertions of Lemma 2.3 follow from classical results see, for instance, [8]. Proof of Theorem 2.1. We split this proof into five steps. Step 1: Approximate solution. Let w j j N be the basis of H 4, 1 H 2, 1 given by Lemma 2.3 and V N = w 1, w 2,..., w N the subspace spanned by the N first eigenfunctions w j. We formulate the approximate problem as follows. Find u N, v N C 1, T, V N, i.e., N N u N t = gi N tw i, and v N t = h N i tw i, i=1 where gi Nt and hn i t are solutions of the system of ordinary differential equations given by 2.5 u N t t, w j + γu N xxxxt, w j + u N xxxt, w j + au N xxt, w j = vx N t, w j + f 1, w j vt N t, w j Γvxxt, N w j + cvx N t, w j = u N x t, w j + f 2, w j u N = u N, v N = v N, for j = 1,..., N, where, denotes the inner product in L 2, 1 and u N N N, v N N N are sequences such that u N u and v N v as N, strongly in H 2, 1 and H 1, 1 respectively. According to Caratheodory s theorem, system 2.5 has a local solution on [, t N and its extension to [, T ] is a consequence of the estimates given below. Step 2: Energy estimates. We first replace w j by u N in the first equation of 2.5 and w j by v N in the second one. In order to make the reading easier we omit the indices and simply denote u N by u and v N by v. After integration by parts we can add both equations to obtain d 1 dt 2 u 2 + v 2 dx + γ u xx 2 dx + Γ = a i=1 v x 2 dx u xx udx + f 1 udx + f 2 vdx. Applying Hölder s inequality at the right hand side of the identity above it follows that d 1 dt 2 u 2 + v 2 dx + γ 2 u xx 2 dx + Γ v x 2 dx a 2 2γ + 1 u 2 + v 2 dx f f 2 2 dx.

5 CONTROL OF A SYSTEM COUPLING KS-KDV AND HEAT EQUATIONS 59 Then, from Gronwall s inequality and the convergences of u N and v N, we get 2.6 u 2 + v 2 dx + u xx 2 dxdt + v x 2 dxdt C 1 u, v 2 L 2 L 2 + f 1, f 2 2 L 2 L 2 2, where C 1 > does not depend on N and t [, T ]. Using w j = xµ 4 1 j w j in the first equation of 2.5, multiplying by gj N t and summing from j = 1 to N, we obtain the identity 2.7 u t + u xxx + au xx u xxxx dx + γ u xxxx 2 dx v x u xxxx dx = f 1 u xxxx dx. The next steps are devoted to estimate the terms appearing in the left hand side of the identity above. It will be done in several steps. i Performing integration by parts, we have u t u xxxx dx = d 1 dt 2 ii For any ε >, there exists cε > satisfying Then, for any δ >. u xx 2 dx. u xxx 2 ε u xxxx 2 + cε u 2. γ u xxx u xxxx dx 2δ + δε u xxxx 2 dx δcε 2γ 2γ iii The remaining terms can be estimated as follows au xx u xxxx dx γ 2δ v x u xxxx dx γ 2δ u xxxx 2 dx δa2 2γ u xxxx 2 dx δ 2γ u xx 2 dx v x 2 dx, u 2 dx, for any δ >. f 1 u xxxx dx γ 2δ u xxxx 2 dx δ 2γ f 2 1 dx,

6 6 E. CERPA, A. MERCADO AND A.F. PAZOTO Then, using previous computations from i up to iii in 2.7 we deduce that d 1 1 u xx 2 dx + γ 1 2 dt 2 δ εδ 2γ 2 u xxxx 2 dx δ 2γ v x 2 dx + δcε 2γ u 2 dx + a2 δ 2γ u xx 2 dx + δ 2γ We choose ε = 4γ2 δ, with δ sufficiently large. We can combine 2.6 and Gronwall s 2 inequality to obtain f 2 1 dx. 2.8 u xx 2 dx + u xxxx 2 dxdt C 2 u, v 2 H 2 L2 + f 1, f 2 2 L 2 L 2 2, where C 2 is a positive constant that does not depend on N. Now, we obtain a uniform bound for the term v N, which is simply denoted by v. In order to do that, we replace w j by v xx in the second equation of 2.5 and perform integration by parts to obtain d 1 dt 2 v x 2 dx + Γ Thus, for any δ >, we get d 1 1 v x 2 dx + Γ 3δ dt 2 2 v xx 2 dx = c v xx 2 dx c2 2δ v x v xx dx v x 2 dx + 1 2δ u x v xx dx f 2 v xx dx. u x 2 dx + 1 2δ Consequently, choosing δ > sufficiently small and using Gronwall s inequality the previous estimates allow us to conclude that f 2 2 dx. 2.9 v x 2 dx + C u, v 2 L 2 H 1 v xx 2 dxdt + f 1, f 2 2 L 2 L 2 2, where C is a positive constant independent on N. Step 3: Convergence of the approximate solutions. According to 2.6, 2.8 and 2.9, the sequences u N N 1 and v N N 1 satisfy u N N 1 is bounded in L, T ; H 2, 1 L 2, T ; H 4, 1, u N t N 1 is bounded in L 2, T ; L 2, 1, 2.1 v N N 1 is bounded in L, T ; H 1, 1 L 2, T ; H 2, 1, vt N N 1 is bounded in L 2, T ; L 2, 1,

7 CONTROL OF A SYSTEM COUPLING KS-KDV AND HEAT EQUATIONS 61 for all T >. The boundeness given by 2.1 allows to extract subsequences of u N N 1 and v N N 1, still denoted by the the same index N, such that u N u weakly in L, T ; H 2, 1 L 2, T ; H 4, 1, u N t u t weakly in L 2, T ; L 2, 1, 2.11 v N v weakly in L, T ; H 1, 1 L 2, T ; H 2, 1, vt N v t weakly in L 2, T ; L 2, 1, as N. Observe that 2.11 allows to pass to the limit in the linear terms of the approximate problem 2.5 to deduce that u, v satisfies 2.2 and u t t + u xxx t + au xx t + γu xxxx t v x t f 1, wt L2,1dt = v t t + cv x t Γv xx t v x t f 2, wt L 2,1dt =, for all w L 2, T ; L 2, 1. Thus, the above identities hold for w D, T, 1 and since u t + u xxx + au xx + γu xxxx v x f 1, and v t + cv x Γv xx v x f 2, belong to L 2, T, 1 the above identities hold a. e. in, 1, T, for any T >. Step 4: Verifications of the initial data. From the previous steps, we obtain some subsequences of u N N 1 and v N N 1, still denoted by the the same index N, such that 2.12 and 2.13 u N t, wt dt ut, wt L 2,1 L 2,1 dt u N t t, wt dt u L 2,1 t t, wt L2,1 dt, for any w L 2, T ; L 2, 1, as N. Let θ C 1 [, T ], such that θ = 1 and θt =. For any z L 2, 1, we take wt = θ tz in 2.12 and wt = θtz in Thus, we obtain d dt [un t, z L 2,1θt] dt d dt [ut, z L 2,1θt] dt, as N. Consequently, for any z L 2, 1, 2.14 u N, z u, z L 2,1 L 2,1, as N. From 2.5, we know that u N = u N and that u N u as N. By using 2.14 we conclude that u = u. Analougously, we obtain v = v. Step 5: Uniqueness. Let u 1, v 1 and u 2, v 2 be two solutions of the problem, corresponding to the same initial data u, v and sources terms f 1, f 2. Then,

8 62 E. CERPA, A. MERCADO AND A.F. PAZOTO u = u 1 u 2 and v = v 1 v 2 satisfy u t + u xxx + au xx + γu xxxx = v x, x, t, 1, T, v t + cv x Γv xx = u x, x, t, 1, T, 2.15 u, t = u x, t = u1, t = u x 1, t =, t, T, v, t = v1, t =, t, T, ux, =, vx, =, x, 1. Then, combining Gronwall s inequality we can proceed as in the previous estimates to deduce that u = v =, i. e., u 1 = u 2 and v 1 = v 2. Finally, inequality 2.3 follows directly from estimates 2.8 and 2.9. This completes the proof of Theorem Adjoint system. A time-backward equation will appear in the next section, when applying the transposition method. Moreover, this equation will be used when studying control properties of system 1.4. It is called adjoint equation and reads as 2.16 ϕ t + γϕ xxxx + aϕ xx ϕ xxx = ψ x + g 1, ψ t Γψ xx cψ x = ϕ x + g 2, ϕt, =, ϕt, 1 =, ϕ x t, =, ϕ x t, 1 =, ψt, =, ψt, 1 =, ϕt, x = ϕ T, ψt, x = ψ T. By performing the change T t by t, we obtain system 2.1 and we can apply Theorem 2.1. Thus, we get the well posedness of the adjoint system. Proposition 2.4. Let ϕ T, ψ T H 2, 1 H 1, 1 and g 1, g 2 L 2, T ; L 2, 1 2. System 2.16 has a unique solution ϕ, ψ CH 2 H 1 L 2 H 4 H 2. Moreover, there exists C > such that 2.17 ϕ, ψ CH 2 H 1 L2 H 4 H 2 C { g 1, g 2 L2 L ϕ T, ψ T H 2 H 1 }. In order to deal with L 2 -regular boundary data, we present the next regularity result. Taking into account the continuous embeddings H 2, 1 C 1 [, 1] and H 4, 1 C 3 [, 1], from Proposition 2.4, we directly obtain: Proposition 2.5. There exists C > such that any ϕ, ψ solution of 2.16 satisfies 2.18 ψ x, L 2,T + ϕ xxx, L 2,T + ϕ xx, L 2,T } C { g 1, g 2 L 2 L ϕ T, ψ T H 2,1 H 1, Linear non-homogeneous system. We have to deal with boundary data in the space L 2, T. Let us define what we mean by a solution of the linear control

9 CONTROL OF A SYSTEM COUPLING KS-KDV AND HEAT EQUATIONS 63 system 2.19 u t + γu xxxx + u xxx + au xx = v x + f 1, x, t, 1, T, v t Γv xx + cv x = u x + f 2, x, t, 1, T, u, t = h 1 t, u1, t =, t, T, u x, t = h 2 t, u x 1, t =, t, T, v, t = h 3 t, v1, t =, t, T, ux, = u x, vx, = v x, x, 1. Definition 2.6. Let u H 2, 1, v H 1, 1, h 1, h 2, h 3 L 2, T, f 1 L 1 W 1,1, f 2 L 1 L 1. A solution of the system 2.19 is a couple u, v L 2, T ; L 2, 1 2 such that for any g 1, g 2 L 2, T ; L 2, 1, 2.2 γ ut, xg 1 t, x+ vt, xg 2 t, x = u, ϕ, x H 2,H 2 + v, ψ, x H 1,H 1 h 1 tϕ xxx t, dt + γ h 2 tϕ xx t, dt + Γ h 3 tψ x t, dt + f 1 t, x, ϕt, x L1 W 1,1,L W 1, + f 2 t, x, ψt, x L1 L 1,L L, where ϕ, ψ is the solution of ϕ t + γϕ xxxx + aϕ xx ϕ xxx = ψ x + g 1, ψ t Γψ xx cψ x = ϕ x + g 2, 2.21 ϕt, =, ϕt, 1 =, ψt, =, ψt, 1 =, ϕt, x =, ψt, x =. Remark 2.7. As usual,, X,Y stands for the duality product between two spaces X and Y. Next theorem establishes existence and uniqueness of solutions of system Theorem 2.8. Let u H 2, 1, v H 1, 1, h 1, h 2, h 3 L 2, T, and f 1 L 1 W 1,1, f 2 L 1 L 1. Then, there exists a unique solution u, v L 2, T ; L 2, 1 2 of system Moreover u, v C[, T ], H 2, 1 H 1, 1. Proof. The right-hand side of 2.2 defines a linear bounded functional L h : g 1, g 2 L 2, T ; L 2, 1 2 L hg 1, g 2 R, and therefore, from the Riesz representation theorem, we obtain the existence and uniqueness of a solution u, v L 2, T ; L 2, 1 2. We can prove that Proposition 2.4 and all the above results on system 2.16 still hold if g 1, g 2 L 1, T ; H 2, 1 H 1, 1. Thus, the right hand side of 2.2 also defines a linear bounded functional L h : g 1, g 2 L 1, T ; H 2, 1 H 1, 1 L hg 1, g 2 R, which gives the extra regularity stated in the theorem.

10 64 E. CERPA, A. MERCADO AND A.F. PAZOTO 2.4. Nonlinear system. Theorem 2.9. There exists a positive real number r such that for any u H 2, 1, v H 1, 1, h 1, h 2 and h 3 L 2, T satisfying 2.22 u, v H 2 H 1,1 + h 1, h 2, h 3 L2,T 3 r, the nonlinear equation 1.4 has a unique solution u, v L 2, T ; L 2, 1 2. Moreover u, v C[, T ], H 2, 1 H 1, 1. Proof. Let us consider u, v, h 1, h 2 and h 3 satisfying 2.22 for r > to be chosen later. We define the following map 2.23 Π : l, m L 2, T ; L 2, 1 2 u, v L 2, T ; L 2, 1 2 where u, v is the solution of u t + γu xxxx + u xxx + au xx = v x ll x, x, t, 1, T, v t Γv xx + cv x = u x m 2, x, t, 1, T, u, t = h 1 t, u1, t =, t, T, 2.24 u x, t = h 2 t, u x 1, t =, t, T, v, t = h 3 t, v1, t =, t, T, ux, = u x, vx, = v x, x, 1. Let us notice that ũ, ṽ is a fixed point of this map Π if and only if ũ, ṽ is a solution of our nonlinear control system 1.4. From Theorem 2.8 and by using and ll x L1 W 1,1 1 2 l 2 L 2 L 2 we get m 2 L1 L 1 = m 2 L 2 L 2 Πl, m L2 L 2 2 C u H 2,1 + v H 1,1 + h 1, h 2, h 3 L 2,T 3 + l 2 L 2 L 2 + m 2 L 2 L 2 For each R >, let us denote the ball of radius R and centered at the origin by B, R := {l, m L 2, T ; L 2, 1 2 ; l, m L 2 L 2 2 R}. We see that if r > and R > are chosen such that Cr+2R 2 R, we obtain that Π B,R B, R. Let us verify that we can choose R such that Π is a contraction. Let l, m and l, m L 2, T ; L 2, 1 2. The couple û, ˆv = Π l, m Πl, m is the solution of.

11 CONTROL OF A SYSTEM COUPLING KS-KDV AND HEAT EQUATIONS 65 û t + γû xxxx + û xxx + aû xx = ˆv x + ll x l l x, x, t, 1, T, ˆv t Γˆv xx + cˆv x = û x + m 2 m 2, x, t, 1, T, û, t =, û1, t =, t, T, û x, t =, û x 1, t =, t, T, ˆv, t =, ˆv1, t =, t, T, ûx, =, ˆvx, =, x, 1. From Theorem 2.8, we get Π l, m Πl, m L2 L 2 C By using that { } l l x ll x L1 W 1,1 + m 2 m 2 L1 L 1. l l x ll x L 1 W 1,1 = 1 2 l 2 l 2 L 1 L l + l L 2 L 2 l l L 2 L 2 and m 2 m 2 L 1 L 1 m + m L 2 L 2 m m L 2 L 2 we obtain { } Π l, m Πl, m L 2 L 2 2CR l l L 2 L 2 + m m L 2 L 2 and therefore the map Π is a contraction if 2CR < 1. By applying the Banach fixed point theorem, we can conclude that Π has a unique fixed point which is the solution of equation 1.4 in L 2, T ; L 2, 1 2. Now, once we have u, v the solution of equation 1.4, we see that u, v satisfies equation 2.19 with source terms f 1 = uu x, and f 2 = v 2, which belong to L 1, T ; W 1,1, 1 and L 1, T ; L 1, 1 respectively. We apply Thereom 2.8 and we get the extra regularity u, v C[, T ], H 2, 1 H 1, Null controllability 3.1. Linear control system. In this section, we study the boundary control of the linear system Let us take a well-posedness framework U 1, U 2, U 3, X 1, X 2, Y, Z for this system. By this we mean that given h j U j for j = 1, 2, 3, f 1, f 2 Y = Y 1 Y 2, u X 1 and v X 2 then there exists a unique u, v Z = Z 1 Z 2 solution of equation This system is said to be null controllable if for any state u X 1, v X 2 and for any f 1, f 2 Y 1 Y 2, one can find controls h k U k, k = 1, 2, 3, such that the solution u, v of 2.19 satisfies ut = vt =. It is a well-known fact that by duality, this null-controllability property is equivalent to the existence of a constant C > such that 3.1 ϕ, ψ Y + ϕ, x X 1 + ψ, x X 2 C g 1, g 2 Z + ϕ xxx t, U 1 + ϕ xx t, U 2 + ψ x t, U 3

12 66 E. CERPA, A. MERCADO AND A.F. PAZOTO for every ϕ T X1, ψ T X2 and g Z, where stands for dual space and ϕ, ψ is the solution of the adjoint linear system Inequality 3.1 is called an observability inequality for system In this section we prove a Carleman estimate for system Then, we use it in order to prove the observability inequality 3.1 within an appropriate well-posedness framework. Thus, we get the null-controllability of system We shall use an abbreviated notation for the derivatives and integrals. We write, and instead of for k integer, w kx instead of k w x k in the last case. We take a function β C 2 [, 1] satisfying, avoiding the symbols dxdt 3.2 < δ dk β x, dxk x [, 1], for k =, 1 and 3.3 d 2 β x δ <, dx2 x [, 1], for some positive constant δ. Carleman estimates for heat equation are well known see [1] for example. We recall a one-parameter estimate, which holds because we are in the one-dimensional case, in the following theorem. Theorem 3.1. See [1]. Let µt, x = βx tt t with β C2 [, 1] satisfying 3.2 and 3.3. There exist C > and s > such that 3.4 s 3 ψ 2 e 2sµ µ 3 + s ψ x 2 e 2sµ µ C Lψ 2 e 2sµ + s ψ x, t 2 e 2sµ,t dt for every s s and ψ C [, T ] [, 1], where Lψ = ψ t Γψ xx cψ x. On the other hand, a Carleman estimate for the KS equation has been studied in [7]. Theorem 3.2. See [7]. Let ηt, x = βx tt t with β C4 [, 1] satisfying 3.2 and 3.3. There exist C > and λ > such that 3.5 λ 7 ϕ 2 e 2λη η 7 +λ 5 ϕ x 2 e 2λη η 5 +λ 3 ϕ 2x 2 e 2λη η 3 +λ ϕ 3x 2 e 2λη η C P ϕ 2 e 2λη + λ 3 ϕ 2x, t 2 e 2λη,t ηx, 3 tdt + λ ϕ 3x, t 2 e 2λη,t η x, tdt for every λ λ and ϕ C [, T ] [, 1], where P ϕ = ϕ t + γϕ xxxx + aϕ xx ϕ xxx ϕ x.

13 CONTROL OF A SYSTEM COUPLING KS-KDV AND HEAT EQUATIONS 67 In order to get an estimate for system 2.16 let us recall that, if ϕ, ψ is a solution, then P ϕ = ψ x + g 1 and Lψ = ϕ x + g 2. We apply the Carleman estimates 3.4 and 3.5 with the same weight function and s = λ to the corresponding equation in the system, and we add both estimates. Taking the parameter λ large enough, the integrals involving the coupling terms ψ x and ϕ x are absorbed by the left hand side of the inequality. Hence we obtain the following result. Theorem 3.3. Let β and η be as in Theorem 3.2. There exist C > and λ > such that 3.6 λ 7 ϕ 2 e 2λη η 7 +λ 5 ϕ x 2 e 2λη η 5 + λ 3 λ 3 ψ 2 e 2λη η 3 +λ ϕ 2x 2 e 2λη η 3 +λ ψ 2 e 2λη η C g g 2 2 e 2λη ϕ 3x 2 e 2λη η +λ 3 ϕ 2x, t 2 e 2λη,t ηx, 3 tdt+λ ϕ 3x, t 2 e 2λη,t η x, tdt +λ ψ x, t 2 e 2sη,t dt for every λ λ and g j L 2, T ; L 2, 1, j = 1, 2, where ϕ, ψ is the solution of equation We prove the following energy estimate for equation Lemma 3.4. If g 1, g 2 L 2, T ; L 2, 1 and ϕ, ψ are the solutions of system 2.16 then 3.7 d dt for every t [, T ]. ϕx, t 2 + ψx, t 2 dx C g g 2 2dx ϕx, t 2 + ψx, t 2 dx+ Proof. Multiplying the first and the second equations of system 2.16 by ϕ and ψ respectively and integrating in, 1 we obtain d 2 dt and d 2 dt ϕx, t 2 dx + γ ψx, t 2 dx + Γ ϕ xx x, t 2 dx+a = ψ x x, t 2 dx = ψ x x, tϕx, tdx + ϕ xx x, tϕx, tdx ϕ x x, tψx, tdx ϕx, tg 1 x, tdx ψx, tg 2 x, tdx

14 68 E. CERPA, A. MERCADO AND A.F. PAZOTO for each t [, T ]. Adding 3.8 and 3.9 and using that a ϕ xx ϕ γ ϕ xx 2 + a2 ϕ 2, γ we get 3.7. Lemma 3.5. If g 1, g 2 L 2, T ; L 2, 1 and ϕ, ψ are the solutions of system 2.16, we define Φt = 3.1 ϕt 2 + ψt 2 dx. Then, 3T Φ L, T 2 C Φ 2 L 2 T 2, 3T g g 2 2dxdt Proof. Let z C, T be such that zt = 1 for all t [, T/2] and zt = for all t [3T/4, T ]. Multiplying 3.7 by z we obtain 3.11 d dt zφ CzΦ z tφ + z g g 2 2dx. For each t [, T ] we apply Gronwall inequality in [t, T ]. Since zt = we obtain 3.12 ztφt C where ht = zt g g 2 2dx z t t From 3.12 we deduce 3.1. t hsds g g 2 2dx. We introduce a weight function ηx, t = βxφt where β C 4 [, 1] satisfies hypothesis and φ is defined by 4 φt = T 2 if t < T/2, 1 if T/2 t T. tt t Proposition 3.6. There exist λ, C > such that for every λ λ the solution ϕ, ψ of 2.16 satisfies ψ 2 e 2λ η η 3 + ψx, 2 dx + ϕx, 2 dx C g g 2 2 e 2λ η + ϕ 2 e 2λ η η 7 + ϕ 3x, t 2 e 2λ η,t η x, tdt + for every g 1, g 2 such that e 2λ η g1 2 + g2 2 <. ϕ 2x, t 2 e 2λ η,t η x, t 3 dt. ψ x, t 2 e 2s η,t dt

15 CONTROL OF A SYSTEM COUPLING KS-KDV AND HEAT EQUATIONS 69 Proof. From Lemma 3.5 and the fact that e 2λ η δ > for t [, 3T/4] we obtain /2 ϕx, 2 + ψx, 2 dx + /2 η 7 e 2λ η ϕx, t 2 dxdt η 3 e 2λ η ψx, t 2 dxdt ϕ, ψ L,T/2;L 2,1 2 C ϕ, ψ L T/2,3T/4;L 2,1 2 + C g 1, g 2 L,3T/4;L 2,1 2 C + C 3T/4 T/2 3T 4 1 ϕx, t 2 + ψx, t 2 dxdt e 2λ η g g 2 2dxdt. On the other hand, we have ηx, t = ηx, t if t [T/2, T ] and ηx, t ηx, t if t [, T/2]. Using that η, k e 2λ η, = for k = 3, 7 and Carleman estimate 3.6 we deduce that T/2 ψ 2 e 2λ η η 3 + T/2 ϕ 2x, t 2 e 2λ η,t η x, t 3 dt + ϕ 2 e 2λ η η 7 C g g 2 2 e 2λ η Inequality 3.13 is obtained from 3.14 and ϕ 3x, t 2 e 2λ η,t η x, tdt + ψ x, t 2 e 2s η,t dt. We will assume an additional property of the weight function. We will ask the function βx to satisfy 3.16 max βx < 2 min βx, x [,1] x [,1] as well as, the stated conditions 3.2 and 3.3. Thus, we define 3.17 r 1 := min x [,1] βx, r 2 := max x [,1] βx and we take r R such that r 2 < r < 2r 1. Through the rest of the paper we will denote by ρ the function defined by 3.18 ρt = e r T t for t, T.

16 7 E. CERPA, A. MERCADO AND A.F. PAZOTO Proposition 3.7. There exists C > such that the solutions ϕ, ψ of 2.16, satisfy 3.19 ρϕ, ρψ L H 2 L H 1 C g g 2 2 e r 1 T t + ψ x t, 2 e 2r 1 T t dt + ϕ 2x t, 2 e 2r 1 T t T t 3 dt + ϕ 3x t, 2 e 2r 1 T t T t 1 dt for every g such that g 2 e 2r T t <. Proof. Define ϕ = ρϕ, ψ = ρψ. Notice that ϕ, ψ satisfy system 2.16 with ϕ T = ψ T = and with the functions ρg 1 ρ t ϕ and ρg 2 ρ t ψ at the right-hand side instead of g 1 and g 2, respectively. Thanks to Proposition 2.4 we have ρϕ, ρψ L H 2 L H 1 C ρg 1, ρg 2 L2 L 2 + ρ t ϕ, ρ t ψ L2 L 2 We can easily check the existence of some positive constants C 1 and C 2 such that ρg j 2 C g j 2 e 2r T t, ρ t ϕ 2 C ϕ 2 e 2r T t T t 4 C ϕ 2 e 2r 2 T t. and in the same manner ρ t ψ 2 C ψ 2 e 2r 2 T t. Therefore, by using 3.13, we get Inequality 3.19 directly implies an observability inequality like 3.1 in some weighted spaces. In order to precise that, we introduce the following notations. Definition 3.8. Given T > and a function η :, T R +, we denote { } L 2 t η := f ; ft 2 ηt dt < and { L 2 txη := f ; endowed with their natural norms. } fx, t 2 ηt dxdt < Taking into account the continuous embeddings H 1, 1 L, 1 and H 2, 1 W 1,, 1, inequality 3.19 gives us the observability 3.1 in the spaces U 1 = L 2 t e 2r 1 T t, U 2 = L 2 t e 2r 1 T t T t, U 3 = L 2 t e 2r 1 T t T t 3, X 1 = H 2, 1, X 2 = H 1, 1, Z = L 2 txe 2r 1 T t 2,

17 CONTROL OF A SYSTEM COUPLING KS-KDV AND HEAT EQUATIONS Y 1 = {y; ρ 1 y L 1, T ; W 1,1, 1} and Y 2 = {y; ρ 1 y L 1, T ; L 1, 1}. Thus, by duality we obtain the null controllability result in this functional framework. Next proposition gives extra information on the decay of the controlled trajectories, which will be useful later in dealing with the nonlinear equation. Proposition 3.9. For each f 1 Y 1, f 2 Y 2, u H 2, 1, v H 1, 1 there exist controls h j U j for j = 1, 2, 3, such that the solution u, v of system 2.19 belongs to L 2 txe 2r 1 T t 2 and satisfies In particular, ut = vt =. T t 2 e r 1 T t u, v C[, T ]; H 2, 1 H 1, 1. Proof. The existence of the solution u, v L 2 txe 2r 1 T t 2 is a direct consequence of the inequality 3.19 and the controllability-observability duality. On the other hand, let us define ũ = T t 2 e r 1 T t u, ṽ = T t 2 e r 1 T t v, fj = T t 2 e r 1 T t fj for j = 1, 2 and h j = T t 2 e r 1 T t hj for j = 1, 2, 3. Then we have that h j L 2, T and ũ, ṽ satisfy 3.21 ũ t + γũ xxxx + aũ xx + ũ xxx = ṽ x + f 1 2T te r 1 T t u + r1 e r 1 T t u, ṽ t Γṽ xx + cṽ x = ũ x + f 2 2T te r 1 T t v + r1 e r 1 T t v. By using that u, v L 2 txe 2r 1 T t 2 and that f j Y j for j = 1, 2, we obtain recall that r 1 < r 2 < r < 2r 1 f 1, f 2 L 1, T ; W 1,1, 1 L 1, T ; L 1, 1 and that 2T te r 1 T t u + r1 e r 1 T t u 2T te r 1 T t v + r1 e r 1 T t L 2, T ; L 2, 1. Hence, by applying Theorem 2.8 we obtain the desired result Nonlinear control system. We shall prove the null controllability of the nonlinear system. We will deduce that result from the null controllability of the linear equation by using a local inversion theorem. In order to obtain Theorem 1.1, we apply the following result. Theorem 3.1. see [1] Let E and G be two Banach spaces and let Λ : E G satisfy Λ C 1 E; G. Assume that ê E, Λê = ĝ, and Λ ê : E G is surjective. Then, there exists δ > such that, for every g G satisfying g ĝ G < δ, there exists some e E solution of the equation Λe = g.

18 72 E. CERPA, A. MERCADO AND A.F. PAZOTO Let us define some appropriate spaces E, G and a map Λ whose surjectivity is equivalent to the null controllability for the KS equation. We denote L 1 u, v = u t + γu xxxx + u xxx + au xx v x, L 2 u, v = v t Γv xx + cv x u x. Keeping in mind 3.2 we define the spaces E := and { } u, v Z : L 1 u, v Y 1, L 2 u, v Y 2, and T t 2 e r 1 T t u, v CH 2 H 1. The map Λ is given by G := H 2, 1 Y 1 H 1, 1 Y 2. Λ : E G, u, v u,, L 1 u, v + uu x, v,, L 2 u, v + v 2. We now prove that Λ is well-defined. First, we have to verify that uu x Y 1 for each u, v E. We have uu x Y 1 e r T t uux L 1, T ; W 1,1, 1 e r T t u 2 L 1, T ; L 1, 1 r u 2 e T t dxdt <. Therefore, as r < 2r 1 see and u L 2 xte 2r 1 T t, we see that u, v E implies uu x Y 1. In the same way, let us see that v 2 Y 2 for each u, v E. In fact, v 2 e r T t dxdt v 2 e 2r 1 T t dxdt < and then v 2 e 2r 1 T t L 1, T ; L 1, 1 L 1, T ; W 1,1, 1, which means that v 2 Y 2. Notice that each one of the maps u 1, u u 1u 2 x Y 1, and v 1, v 2 v 1 v 2 Y 2 is a bilinear continuous map and consequently Λ is a C 1 map. As the functions u, v E satisfy ut = vt =, the local surjectivity of Λ around the origin is equivalent to the local null controllability of system 1.4. Thus, by Theorem 3.1, the proof of Theorem 1.1 will be ended if we prove that the map Λ is surjective. Proposition The map Λ : E G is surjective. Proof. It is easy to see that this map is given by Λ : E G, u, v u,, L 1 u, v, v,, L 2 u, v,

19 CONTROL OF A SYSTEM COUPLING KS-KDV AND HEAT EQUATIONS 73 and therefore its surjectivity is equivalent to the null controllability of the linearized equation with source terms lying in Y 1 Y 2. This control property was proved in Proposition 3.9. Acknowledgments. This work has been partially supported by Fondecyt E. Cerpa, Fondecyt A. Mercado, CNPq A. F. Pazoto, MathAmsud CIP-PDE and CMM-Basal grants. References [1] V. M. Alekseev, V. M. Tikhomirov, S. V. Fomin, Optimal Control, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1987, Translated from the Russian by V. M. Volosov. [2] F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parbolic systems, J. Evol. Equ. 9 29, no. 2, pp [3] F. Ammar-Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Null-controllability of some systems of parabolic type by one control force, ESAIM Control Optim. Calc. Var , no. 3, pp [4] F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: a survey, Mathematical Control and Related Fields 1 211, no. 3, pp [5] D. J. Benney, Long waves in liquid film, J. Math. and Phys , pp [6] E. Cerpa, Null controllability and stabilization of a linear Kuramoto-Sivashinsky equation, Commun. Pure Appl. Anal. 9 21, pp [7] E. Cerpa and A. Mercado, Local exact controllability to the trajectories of the 1-D Kuramoto- Sivashinsky equation, J. Differential Equations , pp [8] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Tata MacGraw-Hill, New Delhi, [9] E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations, J. Funct. Anal , no. 7, pp [1] A. Fursikov, O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes 34, Seoul National University, Korea, [11] O. Glass and S. Guerrero, On the controllability of the fifth-order Korteweg-de Vries equation, Ann. Inst. H. Poincar Anal. Non Linaire 26 29, no. 6, pp [12] M. González-Burgos and L. de Teresa, Controllability results for cascade systems of mm coupled parabolic PDEs by one control force, Port. Math , no. 1, pp [13] S. Guerrero, Null controllability of some systems of two parabolic equations with one control force, SIAM J. Control Optim , no. 2, pp [14] Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems, Theor. Phys , pp [15] Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys , pp [16] G. Lebeau and L. Robbiano, Contrôle exact de l équation de la chaleur, Comm. Partial Differential Equations , pp [17] B. A. Malomed, B.-F. Feng and T. Kawahara, Stabilized Kuramoto-Sivashinsky system, Phys. Rev. E 64 21, [18] B. A. Malomed, B.-F. Feng and T. Kawahara, Cylindrical solitary pulses in a twodimensional stabilized Kuramoto-Sivashinsky system, Phys. D , pp [19] B. A. Malomed, B.-F. Feng and T. Kawahara, Stable two-dimensional solitary pulses in linearly coupled dissipative Kadomtsev-Petviashvili equations, Phys. Rev. E 66 22,

20 74 E. CERPA, A. MERCADO AND A.F. PAZOTO [2] J. Simon, Compact sets in the L p, T ; B spaces, Ann. Mat. Pura Appl , pp [21] G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames I Derivation of basic equations, Acta Astronaut , pp a Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 168, Valparaíso, Chile. address: eduardo.cerpa@usm.cl Received revised a Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 168, Valparaíso, Chile. address: alberto.mercado@usm.cl b Instituto de Matemática, Universidade Federal do Rio de Janeiro, UFRJ, P.O. Box 6853, CEP , RJ, Brasil address: ademir@im.ufrj.br