DSpace at VNU: Construction of a Control for the Cubic Semilinear Heat Equation

Vietnam J Math DOI 10.1007/s10013-015-0171-x Construction of a Control for the Cubic Semilinear Heat Equation Thi Minh Nhat Vo1,2,3 Received: 28 June 2014 / Accepted: 18 June 2015 © Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015 Abstract In this article, we consider the null controllability problem for the cubic semilinear heat equation in bounded domains of Rn , n ≥ with Dirichlet boundary conditions for small initial data A constructive way to compute a control function acting on any nonempty open subset ω of is given such that the corresponding solution of the cubic semilinear heat equation can be driven to zero at a given final time T Furthermore, we provide a quantitative estimate for the smallness of the size of the initial data with respect to T that ensures the null controllability property Keywords Null controllability · Cubic semilinear heat equation · Linear heat equation Mathematics Subject Classification (2010) Primary 35K58 · Secondary 93B05 Introduction and Main Result Many systems in physics, mechanics, or more recently in biology or medical sciences are described by partial differential equations (PDEs) It is necessary to control the character- This work was written while the author was visiting the University of Orleans (France) She thanks the MAPMO department of mathematics of the University of Orleans The author also wishes to acknowledge Region Centre for its financial support Thi Minh Nhat Vo vtmnhat@hcmpreu.edu.vn Universit´e Paris 13, Sorbonne Paris Cit´e, LAGA, CNRS UMR 7539, Institut Galil´ee, 99, Avenue J.-B Cl´ement 93430 Villetaneuse Cedex, France Universit´e d’Orl´eans, Laboratoire MAPMO, CNRS UMR 7349, F´ed´eration Denis Poisson, FR CNRS 2964, Bˆatiment de Math´ematiques, B.P 6759, 45067 Orl´eans Cedex 2, France Ho Chi Minh City University of Natural Science, Ho Chi Minh City, Vietnam T.M.N Vo istic variables, such as the speed of a fluid or the temperature of a device, etc to guarantee that a bridge will not collapse or the temperature is at the desired level for example In the specific words, given a time interval (0, T ), an initial state and a final one, we have to find a suitable control such that the solution matches both the initial state at time t = and the final one at time t = T Let be a bounded connected open set in Rn (n ≥ 3) with a boundary ∂ of class C ; ω be a nonempty open subset in Consider the cubic semilinear heat equation complemented with initial and Dirichlet boundary conditions, which has the following form: ⎧ × (0, T ), ⎨ ∂t y − y + γ y = 1|ω u in y=0 on ∂ × (0, T ), (1) ⎩ in , y(·, 0) = y where γ ∈ {1, −1} Well-posedness property and blow-up phenomena for the cubic semilinear heat equation are now well-known results (see, e.g., [2, 4]) It will be said that (1) is null controllable at time T if there exists a control function u such that the corresponding initial boundary problem possesses a solution y which is null at final time T The basic discussion of this article is how to construct a control function that leads to the null controllability property of system (1) Our main result is the following: Theorem There exists a constant G > such that for any T > 0, any y ∈ H01 ( ) satisfying y 2H ( ) ≤ max √ G, [0;T ] G(1 + t)2 te t there exists a control function u ∈ L2 (ω × (0, T )) such that the solution of (1) satisfies y(·, T ) = Furthermore, the control can be computed explicitly and the construction of the control is given below Remark 1 Theorem ensures the local null controllability of (1) for any control set ω, any small enough initial data y ∈ H01 ( ), at any time T It is well-known that the system (1) without control function blows up in finite time for the case γ = −1 But thanks to an appropriate control function, Theorem affirms that the blow-up phenomena can be prevented for very specific initial data This issue (i.e., the null controllability for semilinear heat equations) has been extensively studied (see, e.g., [1, 5–7] and the references therein) Obviously, the result is not new from the point of view of null controllability, but the method completely differs from others An important achievement of our result is that we can construct the control function An outline of the construction is described as follows: firstly, we remind the construction of the control for the linear heat equation with an estimate of the cost (see, e.g, [13] or [5]); secondly, from the previous result, we similarly when adding an outside force using the method of Liu et al in [9] The solution will be forced to be null at time T by adding an exponential weight function; lastly, thanks to an appropriate iterative fixed point process and linearization by replacing the outside force by cubic function, the desired control is constructed, but the result is only local, i.e., the initial condition must be small enough The precise construction of the control function is found in the proof of this Theorem Another main achievement of our result is to give a quantitative estimate for the smallness of the size of the initial condition with respect to the control time T The upper Construction of a Control for the Cubic Semilinear Heat Equation bound of initial data is a function with respect to the final control time T , which obviously increases to a certain value and then keeps to be a constant until T tends to ∞ Background We now review the achievements of controllability for the heat equations which has been intensively studied in the past Consider the heat equation in the following form: ⎧ × (0, T ), ⎨ ∂t y − y + c(t, x)y + f (t, x, y) = 1|ω u + g in y=0 on ∂ × (0, T ), (2) ⎩ in y(·, 0) = y For linear case with f ≡ 0, (2) is null controllable with no restriction on y , T , and ω, which means the global null controllability holds There are at least two ways to approach such result The first one is due to Lebeau and Robbiano [8], who connect null controllability to an interpolation estimate for elliptic system The second one is due to Fursikov and Imanuvilov [7], and is based on a global Carleman inequality which is an estimate with an exponential weight function and on a minimization technique to construct the control function For nonlinear case, Fursikov and Imanuvilov [7] also give us the proof of global null controllability when f (t, x, s) satisfies the global Lipschitz condition in s variable with f (t, x, 0) ≡ by means of Schauder’s fixed point theorem, and assert the local null controllability when f (t, x, s) satisfies the superlinear growth condition in s by means of the implicit function theorem In [7], Fursikov and Imanuvilov point out that null controllability works in case the initial data is small enough but without an explicit formula In addition, Anita and Tataru [1] improve the result of Fursikov and Imanuvilov by providing sharp estimates for the controllability time in terms of the size of the initial data A little bit different from this document, in [6], Fern´andez-Cara and Zuazua establish the first result in the literature on the null controllability of blowing-up semilinear heat equation In detail, they prove that the system is null-controllable at any time provided a globally defined and bounded trajectory exists and the nonlinear term f (x, t, s) is such that |f (s)| grows slower than |s| log (1 + |s|) as |s| → ∞ Furthermore, they observe that it is not possible to obtain a global controllability result for a cubic nonlinear term More recently, the controllability of a parabolic system with a cubic coupling term has been studied by Coron et al in [3] Another interesting problem is to study the case where the blow-up phenomena will not occur, for example when γ = Our method gives the following result: Corollary There exists a constant G > such that for any T > 0, any y ∈ L2 ( ) satisfying y0 L2 ( ) ≤ max [0;T ] T √ G, G(1 + t)2 te t there exists a control function u ∈ L2 (ω × (0, T )) such that the solution of (1) with γ = satisfies y(·, T ) = The article is organized as follows In Section 2, we deal with the linear heat equation The construction of the control for the linear heat equation with outside force is described there In Section 3, we apply this construction with a fixed point argument in order to prove the main results: Theorem 1; Corollary T.M.N Vo Linear Cases In this section, we survey the null controllability properties for the linear heat equation 2.1 Basic Linear Case Now we recall the results about the null controllability and observability for linear heat equation Theorem For any T > and any z0 ∈ L2 (ω × (0, T )) such that the solution z of ⎧ ⎨ ∂t z − z = 1|ω u z=0 ⎩ z(·, 0) = z0 satisfies z(·, T ) = in holds: L2 ( ), there exists a control function u ∈ in × (0, T ), on ∂ × (0, T ), in , Furthermore, u can be chosen such that the following estimate u C L2 (ω×(0,T )) ≤ Ce T z0 L2 ( ) for some positive constant C = C( , ω) The positive constant C is given in the following equivalent theorem (observability estimate for the heat equation) Theorem There exists a constant C > such that, for any T > 0, for each φT ∈ L2 ( ), the associated solution of the system ⎧ × (0, T ), ⎨ ∂t φ + φ = in φ=0 on ∂ × (0, T ), ⎩ φ(·, T ) = φT in satisfies φ(·, 0) C L2 ( ) ≤ Ce T φ L2 (ω×(0,T )) The two above results are quite an old subject which started at least from the works of [8] and [7] Many improvements are given in [5, 6, 10–13] We turn now to study the null controllability problem for linear case, but with an outside force 2.2 Linear Case with the Outside Force Consider the linear heat equation with the outside force, which has the following form: ⎧ × (0, T ), ⎨ ∂t y − y = f + 1|ω u in y=0 on ∂ × (0, T ), (3) ⎩ in y(·, 0) = y For the moment, we choose y ∈ L2 ( ) and f ∈ L2 ( × (0, T )) Construction of a Control for the Cubic Semilinear Heat Equation Let {Tk }k≥0 be the sequence of real positive numbers given by Tk = T − T , ak (4) where a > Put fk = 1|(Tk ,Tk+1 ) f We start to describe the algorithm to construct the control: we initiate with z0 = y and w−1 = Define the sequences {zk }k≥0 , {uk }k≥0 , {vk }k≥0 , {wk }k≥0 as follows Let vk be the solution of ⎧ in × (Tk , Tk+1 ), ⎨ ∂t vk − vk = fk vk = on ∂ × (Tk , Tk+1 ), (5) ⎩ vk (·, Tk ) = wk−1 (·, Tk ) in Introduce zk+1 = vk (·, Tk+1 ) (6) Let C uk = −Ce Tk+1 −Tk ϕk , where (7) ⎧ in × (Tk , Tk+1 ), ⎨ ∂t ϕk + ϕk = ϕk = on ∂ × (Tk , Tk+1 ), ⎩ T ϕk (·, Tk+1 ) = ϕk k+1 in T Here, ϕk k+1 is the unique minimizer (see the proof of Theorem 1.1, page 1399, [5]) of the following functional depending on εk > 0: Jεk : L2 ( ) → R given by C Jεk T φk k+1 Ce Tk+1 −Tk = Tk+1 Tk |φk |2 dxdt + ω εk T |φk k+1 |2 dx + φk (·, Tk )zk dx, where C is the constant in Theorem and ⎧ in × (Tk , Tk+1 ), ⎨ ∂t φk + φk = φk = on ∂ × (Tk , Tk+1 ), ⎩ T φk (·, Tk+1 ) = φk k+1 ∈ L2 ( ) Let wk be the solution of ⎧ × (Tk , Tk+1 ), ⎨ ∂t wk − wk = 1|ω uk in wk = on ∂ × (Tk , Tk+1 ), ⎩ wk (·, Tk ) = zk in (8) Therefore (see, e.g., [5]) T wk (·, Tk+1 ) = εk ϕk k+1 in (9) and Ce C Tk+1 −Tk Tk+1 Tk |uk |2 dxdt + ω εk |wk (·, Tk+1 )|2 dx ≤ zk L2 ( ) (10) T.M.N Vo Finally, put yk = vk + wk , then it solves ⎧ × (T , T1 ), ⎨ ∂t y0 − y0 = f0 + 1|ω u0 in on ∂ × (T , T1 ), y0 = ⎩ in y0 (·, 0) = y and ⎧ ⎨ ∂t yk+1 − yk+1 = fk+1 + 1|ω uk+1 in × (Tk+1 , Tk+2 ), on ∂ × (Tk+1 , Tk+2 ), yk+1 = ⎩ yk+1 (·, Tk+1 ) = wk (·, Tk+1 ) + zk+1 in Notice that yk (·, Tk+1 ) = yk+1 (·, Tk+1 ), therefore the functions y = k≥0 1|[Tk ,Tk+1 ] yk and u = k≥0 1|[Tk ,Tk+1 ] uk satisfy (3) Now we are able to state our result: (recall that a and εk are needed in (4) and (9) respectively) Theorem Let C be the constant in Theorem There are λ > 0, a > and a sequence {εk }k≥0 of real positive numbers such that for any y ∈ H01 ( ) and any f ∈ L2 ( function 3λC × (0, T )) such that f e T −t ∈ L2 ( u= × (0, T )), the above constructed control 1|[Tk ,Tk+1 ] uk k≥0 is in L2 (ω × (0, T )) and drives the solution of (3) to y(·, T ) = Furthermore, there exists a positive constant K such that the following estimate holds: √ λC 3λC 3λC ≤ K + T e T ∇y L2 ( ) +K(1+T ) f e T −t ∇ye T −t C([0,T ];L ( )) L ( ×(0,T )) Now, we come to the proof of Theorem 2.3 Proof of Theorem M Our strategy to prove Theorem is as follows: we want to get ye T −t C([0,T ];L2 ( )) < +∞ for some suitable constant M > in order to deduce that y(·, T ) = To so, since y = k≥0 1|[Tk ,Tk+1 ] yk and yk = vk + wk is given by (5)–(8), we start to estimate vk C([Tk ,Tk+1 ];L2 ( )) and wk C([Tk ,Tk+1 ];L2 ( )) In the same time, we also derive B an inequality for ue T −t L2 (ω×(0,T )) for some suitable constant B > in order to get D × (0, T )) Finally, we will focus on estimating ∇ye T −t C([0,T ];L2 ( )) for some u∈ suitable constant D > By the classical energy estimate for the heat equation with outside force, one has from (5)–(8) √ v0 C([T0 ,T1 ];L2 ( )) ≤ T f0 L2 ( ×(T0 ,T1 )) , √ vk+1 C([Tk+1 ,Tk+2 ];L2 ( )) ≤ T fk+1 L2 ( ×(Tk+1 ,Tk+2 )) + wk (·, Tk+1 ) L2 ( ) L2 (ω and wk C([Tk ,Tk+1 ];L2 ( )) ≤ √ T uk L2 (ω×(Tk ,Tk+1 )) + zk L2 ( ) Construction of a Control for the Cubic Semilinear Heat Equation By using the following estimates, which are implied by (10): uk ≤ L2 (ω×(Tk ,Tk+1 )) √ C Ce Tk+1 −Tk zk and wk (·, Tk+1 ) L2 ( ) L2 ( ) ≤ √ εk zk L2 ( ) , (11) we get vk+1 C([Tk+1 ,Tk+2 ];L2 ( )) ≤ √ T fk+1 L2 ( ×(Tk+1 ,Tk+2 )) √ + εk zk (12) L2 ( ) and wk C([Tk ,Tk+1 ];L2 ( )) ≤ √ C CT e Tk+1 −Tk zk L2 ( ) + zk (13) L2 ( ) Since by (6) vk+1 (·, Tk+2 ) = zk+2 , it implies using (12) that zk+2 ≤ L2 ( ) √ T fk+1 L2 ( ×(Tk+1 ,Tk+2 )) + √ εk zk L2 ( ) As a result, for any constant A > 0, we get A e T −Tk+1 zk L2 ( ) k≥0 A A = e T −T1 z0 L2 ( ) + e T −T2 z1 A A ≤ e T −T1 y L2 ( ) + e T −T2 A √ + T e T −Tk+3 fk+1 k≥0 A ≤ e T −T1 y L2 ( ) A ≤ e T −T1 y L2 ( ) + + √ √ √ L2 ( ) A + e T −Tk+3 zk+2 L2 ( ) k≥0 T f0 L2 ( ×(0,T1 )) L2 ( ×(Tk+1 ,Tk+2 )) A + e T −Tk+3 √ εk zk L2 ( ) k≥0 A e T −Tk+2 fk T L2 ( ×(Tk ,Tk+1 )) A + k≥0 e T −Tk+3 √ εk zk L2 ( ) k≥0 aA e T −Tk+1 fk T L2 ( ×(Tk ,Tk+1 )) k≥0 a2 A + e T −Tk+1 √ εk zk (14) L2 ( ) k≥0 Choose εk = a2 A in order that e T −Tk+1 k≥0 (15) A √ εk ≤ 12 e T −Tk+1 , then (14) becomes A e T −Tk+1 zk −1) − 2A(a e T −Tk+1 , A L2 ( ) ≤ 2e T −T1 y L2 ( ) √ +2 T aA e T −Tk+1 fk k≥0 L2 ( ×(Tk ,Tk+1 )) (16) T.M.N Vo On one hand, for any constant M > 0, we obtain by (12), (13), and (15) M e T −Tk+1 yk C([Tk ,Tk+1 ];L2 ( )) k≥0 M ≤ e T −Tk+1 vk k≥0 M ≤ e T −T1 √ T f0 C([Tk ,Tk+1 ];L2 ( )) M + e T −Tk+1 wk k≥0 L2 ( ×(T0 ,T1 )) M + e T −Tk+1 √ T fk C([Tk ,Tk+1 ];L2 ( )) L2 ( ×(Tk ,Tk+1 )) k≥1 M + e T −Tk+2 k≥0 εk zk √ T fk M ≤ √ e T −Tk+1 L2 ( ) M + 1+ e T −Tk+1 C √ CT e Tk+1 −Tk zk L2 ( ) k≥0 L2 ( ×(Tk ,Tk+1 )) k≥0 + e aM−A(a −1) T −Tk+1 zk L2 ( ) M + e T −Tk+1 zk k≥0 L2 ( ) k≥0 √ + CT e C M+ 2(a−1) T −Tk+1 zk L2 ( ) k≥0 M ≤ e T −Tk+1 √ T fk L2 ( ×(Tk ,Tk+1 )) + k≥0 where N = max{aM − A(a − 1), M, M + N ≤ A with (16) that M e T −Tk+1 yk C([Tk ,Tk+1 ];L2 ( )) ≤ 1+ k≥0 √ √ + CT C 2(a−1) }, N e T −Tk+1 zk L2 ( ) , k≥0 which implies under the condition A CT e T −T1 y √ √ +3 T + CT L2 ( ) aA e T −Tk+1 fk L2 ( ×(Tk ,Tk+1 )) k≥0 Therefore M ye T −t C([0,T ];L2 ( )) M ≤ e T −Tk+1 yk C([Tk ,Tk+1 ];L2 ( )) k≥0 ≤ 1+ √ CT A e T −T1 y √ √ +3 T + CT L2 ( ) a2 A f e T −t L2 ( ×(0,T )) On the other hand, by the first inequality in (11), one has for any constant B > B e T −Tk+1 uk L2 (ω×(Tk ,Tk+1 )) B ≤ k≥0 e T −Tk+1 √ C Ce Tk+1 −Tk zk L2 ( ) k≥0 ≤ √ C e k≥0 C B+ 2(a−1) T −Tk+1 zk L2 ( ) , (17) Construction of a Control for the Cubic Semilinear Heat Equation which implies under the condition B + ≤ A with (16), that √ A ≤ Ce T −T1 y B e T −Tk+1 uk C 2(a−1) L2 (ω×(Tk ,Tk+1 )) k≥0 √ +2 CT L2 ( ) aA e T −Tk+1 fk L2 ( ×(Tk ,Tk+1 )) k≥0 Therefore B ue T −t L2 (ω×(0,T )) B ≤ e T −Tk+1 uk √ A ≤ Ce T −T1 y By taking B = M = C 2(a−1) and A = C ye 2(a−1) T −t ≤c 1+ √ L2 (ω×(Tk ,Tk+1 )) k≥0 C a−1 , L2 ( (18) L2 (ω×(0,T )) √ √ )+c T 1+ T aC T e a−1 T y L2 ( ×(0,T )) + ue 2(a−1) T −t )) √ a2 A + CT f e T −t we conclude from (17) and (18) that C C([0,T ];L2 ( L2 ( ) a2 C f e a−1 T −t (19) L2 ( ×(0,T )) for some constant c We turn now to the case y ∈ H01 ( ) For any constant D > 0, put D p = p(t) = e T −t and g = py then g satisfies the following system ⎧ × (0, T ), ⎨ ∂t g − g = p y + p(1|ω u − f ) in g=0 on ∂ × (0, T ), D ⎩ g(·, 0) = e T y in Applying classical energy estimate, one has D ∇g ≤ e T ∇y C([0,T ];L2 ( )) + pu L2 ( ) + py L2 ( ×(0,T )) L2 ( ×(0,T )) + pf L2 ( ×(0,T )) , which implies, for any ρ ∈ (1, 3/2) the existence of Kρ > such that D ∇ye T −t ρD D C([0,T ];L2 ( )) ≤ e T ∇y + ue ρD T −t L2 ( ) + Kρ ye T −t L2 ( ×(0,T )) L2 ( ×(0,T )) + fe 3D T −t L2 ( ×(0,T )) (20) Take a= in order that a > 1, ρD = (20) that D ∇ye T −t C([0,T ];L2 ( )) 2ρ C 2(a−1) ≤K 1+ for some constant K With λ = proof of Theorem and C D= 2ρ and a2 C a−1 2ρ( 2ρ −1) −1 (21) , = 3D Then, it implies by combining (19) and √ 3D T e T ∇y 2ρ 3D L2 ( ) + K(1 + T ) f e T −t L2 ( ×(0,T )) (22) in order that D = λC, we have completed the T.M.N Vo Proof of Main Results This section focuses on the proof of the main results, Theorem and Corollary 1, which ensures that system (1) is null controllable with the different conditions of the initial data First, we start with the proof of Theorem 3.1 Proof of Theorem The idea of the proof of Theorem is as follows: first, by applying the result in Theorem 4, we construct a control sequence um ∈ L2 ( × (0, T )) such that the solution of ⎧ = 1|ω um in × (0, T ), ⎨ ∂t ym − ym + γ ym−1 ym = on ∂ × (0, T ), ⎩ in ym (·, 0) = y satisfies ym (·, T ) = in H01 ( ); secondly, by proving ym converges to y and um converges to u, we will get the desired result Now, we start the first step by checking that the function f = −γ ym−1 satisfies the condition of Theorem Denote D = λC First take y0 such 3D D × (0, T )), for example y0 = e− T −t e T y Now by that y0 (·, 0) = y and γ y03 e T −t ∈ L2 ( 3D T −t 3e induction, we will prove that γ ym ym−1 e 3D T −t ∈ × (0, T )) for any m ≥ Indeed, suppose × (0, T )), by Theorem 4, ym verifies L2 ( D ∇ym (·, t)e T −t ∈ L2 ( D L2 ( ) ≤ K(1 + √ T )e 3D T ∇y 3D L2 ( ) + K(1 + T ) ym−1 e T −t L2 ( ×(0,T )) Using Sobolev embedding, we obtain 3D T −t e ym L2 ( ×(0,T )) T ≤c D ∇ym (·, t)e T −t 1+ ≤ cKT √ L2 ( ) dt 3D K T e T ∇y L2 ( T −t ) + (1 + T ) ym−1 e L2 ( ×(0,T )) 1, then ym converges to y in C([0, T ], H01 ( )) and um converges to u in L2 (ω × (0, T )) This completes the proof 3.2 Proof of Corollary Now, we prove Corollary Consider the following system: ⎧ ⎨ ∂t y − y + y = in × (0, T /2), y=0 on ∂ × (0, T /2), ⎩ in y(·, 0) = y Recall that no blow-up phenomena occurs We can establish by classical energy estimate that y(·, T /2) ∈ H01 ( ) Furthermore, one has y(·, T /2) H01 ( ) ≤ y T L2 ( ) ≤ max [0;T ] √ G G(1 + t)2 te t Consequently, applying Theorem 1, we obtain the existence of u ∈ L2 ( × (T /2, T )) such that the solution of ⎧ ⎨ ∂t y − y + y = 1|ω u in × (T /2, T ), y=0 on ∂ × (T /2, T ), ⎩ y(·, T /2) = y(·, T /2) in , satisfies y(·, T ) = 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Handbook of Differential Equations: Evolutionary Equations, vol 3, pp 527–621 Elsevier/North-Holland, Amsterdam (2007) ... given a time interval (0, T ), an initial state and a final one, we have to find a suitable control such that the solution matches both the initial state at time t = and the final one at time... condition with respect to the control time T The upper Construction of a Control for the Cubic Semilinear Heat Equation bound of initial data is a function with respect to the final control time T ,... in case the initial data is small enough but without an explicit formula In addition, Anita and Tataru [1] improve the result of Fursikov and Imanuvilov by providing sharp estimates for the controllability