Some remarks on the exact controllability to trajectories for the nonlinear heat equation
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Some remarks on the exact controllability to trajectories for the nonlinear heat equation M. González-Burgos Workshop on Control and Inverse Problems Besançon, June 2010 M. González-Burgos Remarks on the controllability for the nonlinear heat equation
Contents 1Introduction. Statement of the problem 2Null Controllability of the linear problem with regular controls First approach Second approach Third approach 3The “best” null control M. González-Burgos Remarks on the controllability for the nonlinear heat equation
1. Introduction. Statement of the problem Let Ω⊂RNbe a bounded domain, N≥1, with boundary ∂Ωof class C2. Let ω⊆Ωbe an open subset anf let us fix T>0. We consider the linear and nonlinear problems for the heat equation: (1) ∂ty−∆y+ay=v1ωin Q= Ω ×(0,T), y=0 on Σ = ∂Ω×(0,T), y(·,0) = y0in Ω, (2) (∂ty−∆y+F(y) = v1ωin Q, y=0 on Σ,y(·,0) = y0in Ω. In (1) and (2), 1ωrepresents the characteristic function of the set ω, y(x,t)is the state, y0is the initial datum and is given in an appropriate space, and vis the control function (which is localized in ω -distributed control-). In (1), a∈L∞(Q)is given. We will assume that F:R→Ris a given function. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
1. Introduction. Statement of the problem Let Ω⊂RNbe a bounded domain, N≥1, with boundary ∂Ωof class C2. Let ω⊆Ωbe an open subset anf let us fix T>0. We consider the linear and nonlinear problems for the heat equation: (1) ∂ty−∆y+ay=v1ωin Q= Ω ×(0,T), y=0 on Σ = ∂Ω×(0,T), y(·,0) = y0in Ω, (2) (∂ty−∆y+F(y) = v1ωin Q, y=0 on Σ,y(·,0) = y0in Ω. In (1) and (2), 1ωrepresents the characteristic function of the set ω, y(x,t)is the state, y0is the initial datum and is given in an appropriate space, and vis the control function (which is localized in ω -distributed control-). In (1), a∈L∞(Q)is given. We will assume that F:R→Ris a given function. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
1. Introduction. Statement of the problem Let Ω⊂RNbe a bounded domain, N≥1, with boundary ∂Ωof class C2. Let ω⊆Ωbe an open subset anf let us fix T>0. We consider the linear and nonlinear problems for the heat equation: (1) ∂ty−∆y+ay=v1ωin Q= Ω ×(0,T), y=0 on Σ = ∂Ω×(0,T), y(·,0) = y0in Ω, (2) (∂ty−∆y+F(y) = v1ωin Q, y=0 on Σ,y(·,0) = y0in Ω. In (1) and (2), 1ωrepresents the characteristic function of the set ω, y(x,t)is the state, y0is the initial datum and is given in an appropriate space, and vis the control function (which is localized in ω -distributed control-). In (1), a∈L∞(Q)is given. We will assume that F:R→Ris a given function. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
1. Introduction. Statement of the problem Remark In this talk we are interested in studying the controllability properties of systems (1) and (2) (controllability to trajectories). Linear Problem: For every ωand Tsystem (1) is null controllable (equivalently exactly controllable to trajectories): For every y0∈L2(Ω) there is v∈L2(Q)s.t. the solution yto (1) satisfies y(T)≡0 in Ω. 1H.O. FATTORINI, D.L. RUSSELL,Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal. 43 (1971), 272–292. 2G. LEBEAU, L. ROBBIANO,Contrôle exact de l’équation de la chaleur, Comm. P.D.E. 20 (1995), no. 1-2, 335–356. a≡0: v∈C∞ 0(ω×(0,T)). 3O. YU. IMANUVILOV,Controllability of parabolic equations, (Russian) Mat. Sb. 186 (1995), no. 6, 109–132; translation in Sb. Math. 186 (1995), no. 6, 879–900. a∈L∞(Q):v∈L2(Q). M. González-Burgos Remarks on the controllability for the nonlinear heat equation
1. Introduction. Statement of the problem Remark In this talk we are interested in studying the controllability properties of systems (1) and (2) (controllability to trajectories). Linear Problem: For every ωand Tsystem (1) is null controllable (equivalently exactly controllable to trajectories): For every y0∈L2(Ω) there is v∈L2(Q)s.t. the solution yto (1) satisfies y(T)≡0 in Ω. 1H.O. FATTORINI, D.L. RUSSELL,Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal. 43 (1971), 272–292. 2G. LEBEAU, L. ROBBIANO,Contrôle exact de l’équation de la chaleur, Comm. P.D.E. 20 (1995), no. 1-2, 335–356. a≡0: v∈C∞ 0(ω×(0,T)). 3O. YU. IMANUVILOV,Controllability of parabolic equations, (Russian) Mat. Sb. 186 (1995), no. 6, 109–132; translation in Sb. Math. 186 (1995), no. 6, 879–900. a∈L∞(Q):v∈L2(Q). M. González-Burgos Remarks on the controllability for the nonlinear heat equation
1. Introduction. Statement of the problem Remark In this talk we are interested in studying the controllability properties of systems (1) and (2) (controllability to trajectories). Linear Problem: For every ωand Tsystem (1) is null controllable (equivalently exactly controllable to trajectories): For every y0∈L2(Ω) there is v∈L2(Q)s.t. the solution yto (1) satisfies y(T)≡0 in Ω. 1H.O. FATTORINI, D.L. RUSSELL,Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal. 43 (1971), 272–292. 2G. LEBEAU, L. ROBBIANO,Contrôle exact de l’équation de la chaleur, Comm. P.D.E. 20 (1995), no. 1-2, 335–356. a≡0: v∈C∞ 0(ω×(0,T)). 3O. YU. IMANUVILOV,Controllability of parabolic equations, (Russian) Mat. Sb. 186 (1995), no. 6, 109–132; translation in Sb. Math. 186 (1995), no. 6, 879–900. a∈L∞(Q):v∈L2(Q). M. González-Burgos Remarks on the controllability for the nonlinear heat equation
1. Introduction. Statement of the problem Nonlinear Problem: Under appropriate assumptions on the function F(which has a superlinear growth at infinity) system (2) is exactly controllable to trajectories at time T: 1E. FERNÁNDEZ-CARA,Null controllability of the semilinear heat equation, ESAIM Control Optim. Calc. Var. 2 (1997), 87–103. F(s)∼ |s|log(1+|s|). 2E. FERNÁNDEZ-CARA, E. ZUAZUA,Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 5, 583–616. F(s)∼ |s|logp(1+|s|),p∈[0,3/2). 3V. BARBU,Exact controllability of the superlinear heat equation, Appl. Math. Optim. 42 (2000), no. 1, 73–89. F(s)∼ |s|logp(1+|s|)(p∈[0,3/2)), 1 ≤N<6 and a dissipativity condition on the the nonlinearity: sF(s)≥ −µo|s|2(µ0≥0). M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2. Linear null controllability result with regular controls We consider the distributed controllability problem for the linear system: (1) (∂ty−∆y+ay=v1ωin Q, y=0 on Σ,y(·,0) = y0in Ω, where ω⊂Ωis an open subset, v∈L2(Q)is the control and y0is given in L2(Ω). Let us fix ϕ0∈L2(Ω) and consider the adjoint problem (3) (−∂tϕ−∆ϕ+aϕ=0 in Q, ϕ=0 on Σ, ϕ(T) = ϕ0in Ω. It is well known: M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2. Linear null controllability result with regular controls Theorem The following conditions are equivalent: 1There exists Cs.t. ∀y0∈L2(Ω), there is v∈L2(Q), with kvk2 L2(Q)≤Cky0k2 L2(Ω), s.t. the solution yvto (1) associated to y0and vsatisfies yv(T) = 0in L2(Ω). 2There exists C>0s.t. (observability inequality) kϕ(0)k2 L2(Ω) ≤CZZω×(0,T) |ϕ(x,t)|2dx dt, holds for every solution ϕto the adjoint problem (3) associated to ϕ0∈L2(Ω). M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2. Linear null controllability result with regular controls The observability inequality for the adjoint problem with an explicit expression of Cwith respect to the data can be obtained from a global Carleman inequalities for the linear parabolic problem: (4) (−∂tϕ−∆ϕ=F0in Q, ϕ=0 on Σ, ϕ(·,T) = ϕ0in Ω, with F0∈L2(Q)and ϕ0∈L2(Ω) are given. In V. A. FURSIKOV, O. YU. IMANUVILOV,Controllability of Parabolic Equations, Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Seoul, 1996, E. FERNÁNDEZ-CARA, E. ZUAZUA,The cost of approximate controllability for heat equations: the linear case. Adv. Differential Equations 5 (2000), no. 4-6, 465–514, it is proved: M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2. Linear null controllability result with regular controls The observability inequality for the adjoint problem with an explicit expression of Cwith respect to the data can be obtained from a global Carleman inequalities for the linear parabolic problem: (4) (−∂tϕ−∆ϕ=F0in Q, ϕ=0 on Σ, ϕ(·,T) = ϕ0in Ω, with F0∈L2(Q)and ϕ0∈L2(Ω) are given. In V. A. FURSIKOV, O. YU. IMANUVILOV,Controllability of Parabolic Equations, Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Seoul, 1996, E. FERNÁNDEZ-CARA, E. ZUAZUA,The cost of approximate controllability for heat equations: the linear case. Adv. Differential Equations 5 (2000), no. 4-6, 465–514, it is proved: M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2. Linear null controllability result with regular controls Lemma There exist a regular and strictly positive function, α0, and two constants C0yσ0(only depending on Ωand ω) s.t. I(ϕ)≡s−1ZZQ e−2sαt(T−t)|∂tϕ|2+|∆ϕ|2 +sZZQ e−2sαt−1(T−t)−1|∇ϕ|2+s3ZZQ e−2sαt−3(T−t)−3|ϕ|2 ≤C0 s3ZZω×(0,T) e−2sαt−3(T−t)−3|ϕ|2+ZZQ e−2sα|F0|2!, ∀s≥s0=σ0(Ω,ω)(T+T2), (ϕis the solution to (4)associated to ϕ0∈L2(Ω)). The function α=α(x,t)is given by α(x,t) = α0(x)/t(T−t). M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2. Linear null controllability result with regular controls Coming back to the adjoint problem (3) −∂tϕ−∆ϕ+aϕ=0 in Q, ϕ=0 on Σ, ϕ(·,T) = ϕ0in Ω. Lemma There exist C1>0and σ1>0(only depending on Ωand ω) s.t. (5) I(ϕ) = s−1ZZQ e−2sαt(T−t)|∂tϕ|2+|∆ϕ|2 +sZZQ e−2sαt−1(T−t)−1|∇ϕ|2+s3ZZQ e−2sαt−3(T−t)−3|ϕ|2 ≤C1s3ZZω×(0,T) e−2sαt−3(T−t)−3|ϕ|2, ∀s≥s1=σ1(Ω,ω)T+T2+T2kak2/3 ∞. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2. Linear null controllability result with regular controls Coming back to the adjoint problem (3) −∂tϕ−∆ϕ+aϕ=0 in Q, ϕ=0 on Σ, ϕ(·,T) = ϕ0in Ω. Lemma There exist C1>0and σ1>0(only depending on Ωand ω) s.t. (5) I(ϕ) = s−1ZZQ e−2sαt(T−t)|∂tϕ|2+|∆ϕ|2 +sZZQ e−2sαt−1(T−t)−1|∇ϕ|2+s3ZZQ e−2sαt−3(T−t)−3|ϕ|2 ≤C1s3ZZω×(0,T) e−2sαt−3(T−t)−3|ϕ|2, ∀s≥s1=σ1(Ω,ω)T+T2+T2kak2/3 ∞. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.1. First approach We follow: E. FERNÁNDEZ-CARA, E. ZUAZUA,Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 17, No. 5, (2000), 583–616. From the previous global Carleman inequality one has: Theorem For every a∈L∞(Q)and ϕ0∈L2(Ω) one has (observability inequality) kϕ(0)k2 L2(Ω) ≤exp [CM(T,kak∞)] ZZω×(0,T) |ϕ|2, (ϕsolution to (3)) with C=C(Ω,ω)>0and M given by: M(T,kak∞) = 1+1 T+Tkak∞+kak2/3 ∞. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.1. First approach We follow: E. FERNÁNDEZ-CARA, E. ZUAZUA,Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 17, No. 5, (2000), 583–616. From the previous global Carleman inequality one has: Theorem For every a∈L∞(Q)and ϕ0∈L2(Ω) one has (observability inequality) kϕ(0)k2 L2(Ω) ≤exp [CM(T,kak∞)] ZZω×(0,T) |ϕ|2, (ϕsolution to (3)) with C=C(Ω,ω)>0and M given by: M(T,kak∞) = 1+1 T+Tkak∞+kak2/3 ∞. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.1. First approach Remark This inequality shows the null controllability result for the linear system (1) with a control vin L2(Q)(in fact, Supp v⊂ω×(0,T)) and provides the following estimate for kvkL2(Q): kvk2 L2(Q)≤exp [CM(T,kak∞)]ky0k2, with M given as before. Is it possible to solve this problem with a control v∈L∞(Q)?YES. The key point is a better observability inequality with a weaker norm on the right hand-side: M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.1. First approach Coupled parabolic systems: Let us consider a “simple” coupled parabolic system (∂ty−∆y=Ay+Bv1ωin Q, y=0 on Σ,y(0) = y0in Ω,(∂tϕ+ ∆ϕ=−A∗ϕin Q, ϕ=0 on Σ, ϕ(T) = ϕ0in Ω, with A=0 0 1 0 ,B=1 0(one control force) and y0∈L2(Ω)2. The particular structure of Aand B(cascade system) gives: kϕ(0)k2 L2(Ω) ≤CZZω0×(T/4,3T/4) |ϕ1|2, for a constant C>0. Then, there is v∈L2(Q)s.t. yv(T) = 0 in Ωand kvk2 L2(Ω) ≤Cky0k2 L2(Ω)2.Control in L∞(Q)?? M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.1. First approach Coupled parabolic systems: Let us consider a “simple” coupled parabolic system (∂ty−∆y=Ay+Bv1ωin Q, y=0 on Σ,y(0) = y0in Ω,(∂tϕ+ ∆ϕ=−A∗ϕin Q, ϕ=0 on Σ, ϕ(T) = ϕ0in Ω, with A=0 0 1 0 ,B=1 0(one control force) and y0∈L2(Ω)2. The particular structure of Aand B(cascade system) gives: kϕ(0)k2 L2(Ω) ≤CZZω0×(T/4,3T/4) |ϕ1|2, for a constant C>0. Then, there is v∈L2(Q)s.t. yv(T) = 0 in Ωand kvk2 L2(Ω) ≤Cky0k2 L2(Ω)2.Control in L∞(Q)?? M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.1. First approach Following this technique, does the following inequality ZZω0×(T/4,3T/4) |ϕ1|2≤C ZZω×(0,T) |ϕ1|!2 hold?? NO. Remark This first approach cannot be applied to the previous coupled system since the local regularizing effect of the linear adjoint problem involves the functions ϕ1and ϕ2while the corresponding “refined” observability inequality should only involve ϕ1(recall that the control v only appears in first equation of the direct problem). M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.2. Second approach We follow V. BARBU,Exact controllability of the superlinear heat equation, Appl. Math. Optim. 42 (2000), no. 1, 73–89. We recall the global Carleman inequality (∂Ω∈C2): I(ϕ) = s−1ZZQ e−2sαt(T−t)|∂tϕ|2+|∆ϕ|2 +sZZQ e−2sαt−1(T−t)−1|∇ϕ|2+s3ZZQ e−2sαt−3(T−t)−3|ϕ|2 ≤C1s3ZZω×(0,T) e−2sαt−3(T−t)−3|ϕ|2, ∀s≥s1=σ1(Ω,ω)T+T2+T2kak2/3 ∞, where C1=C1(Ω,ω)>0 and ϕthe solution to (3) −∂tϕ−∆ϕ+aϕ=0 in Q, ϕ=0 on Σ, ϕ(·,T) = ϕ0(·)in Ω. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.2. Second approach We follow V. BARBU,Exact controllability of the superlinear heat equation, Appl. Math. Optim. 42 (2000), no. 1, 73–89. We recall the global Carleman inequality (∂Ω∈C2): I(ϕ) = s−1ZZQ e−2sαt(T−t)|∂tϕ|2+|∆ϕ|2 +sZZQ e−2sαt−1(T−t)−1|∇ϕ|2+s3ZZQ e−2sαt−3(T−t)−3|ϕ|2 ≤C1s3ZZω×(0,T) e−2sαt−3(T−t)−3|ϕ|2, ∀s≥s1=σ1(Ω,ω)T+T2+T2kak2/3 ∞, where C1=C1(Ω,ω)>0 and ϕthe solution to (3) −∂tϕ−∆ϕ+aϕ=0 in Q, ϕ=0 on Σ, ϕ(·,T) = ϕ0(·)in Ω. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.2. Second approach In this work, a control in Lp(Q), with p=p(N), is obtained from the previous global Carleman inequality (we fix s=s1=σ1(Ω,ω)T+T2+T2kak2/3 ∞). First Step: Lemma For every a∈L∞(Q)and ϕ0∈L2(Ω) one has (observability inequality) kϕ(0)k2 L2(Ω) ≤exp [CM(T,kak∞)] ZZω×(0,T) e−2s1αt−3(T−t)−3|ϕ|2, (ϕsolution to (3)) with C=C(Ω,ω)>0and M given by: M(T,kak∞) = 1+1 T+Tkak∞+kak2/3 ∞. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.2. Second approach In this work, a control in Lp(Q), with p=p(N), is obtained from the previous global Carleman inequality (we fix s=s1=σ1(Ω,ω)T+T2+T2kak2/3 ∞). First Step: Lemma For every a∈L∞(Q)and ϕ0∈L2(Ω) one has (observability inequality) kϕ(0)k2 L2(Ω) ≤exp [CM(T,kak∞)] ZZω×(0,T) e−2s1αt−3(T−t)−3|ϕ|2, (ϕsolution to (3)) with C=C(Ω,ω)>0and M given by: M(T,kak∞) = 1+1 T+Tkak∞+kak2/3 ∞. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.2. Second approach Second Step: From this observability inequality we deduce Proposition ∀y0∈L2(Ω), there is v∈Lp(N)(Q), with p(N)<∞if N =2and p(N) = 2(N+2) N−2if N ≥3, and kvk2 Lp(N)(Q)≤e[CM(T,kak∞)]ky0k2 L2(Ω), s.t. the solution yvto (1) associated to y0and vsatisfies yv(T) = 0in L2(Ω). Sketch of the proof:1.- We consider the optimal control problem min v∈L2(Q)1 2ZZQ e2s1αt3(T−t)3|v(x,t)|2dx dt +1 2εkyv(T)k2 L2(Ω), (yv∈L2(Q)2is the solution of (1) associated to y0and v). M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.2. Second approach Second Step: From this observability inequality we deduce Proposition ∀y0∈L2(Ω), there is v∈Lp(N)(Q), with p(N)<∞if N =2and p(N) = 2(N+2) N−2if N ≥3, and kvk2 Lp(N)(Q)≤e[CM(T,kak∞)]ky0k2 L2(Ω), s.t. the solution yvto (1) associated to y0and vsatisfies yv(T) = 0in L2(Ω). Sketch of the proof:1.- We consider the optimal control problem min v∈L2(Q)1 2ZZQ e2s1αt3(T−t)3|v(x,t)|2dx dt +1 2εkyv(T)k2 L2(Ω), (yv∈L2(Q)2is the solution of (1) associated to y0and v). M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.2. Second approach This problem has a unique solution vε∈L2(Q)and, using the optimality system, it is characterized: vε=e−2s1αt−3(T−t)−3ϕε1ω and (∂tyε−∆yε+ayε=vε1ωin Q, yε=0 sobre Σ,yε(·,0) = y0in Ω, −∂tϕε−∆ϕε+aϕε=0 in Q, ϕε=0 on Σ, ϕε(·,T) = −1 εyε(·,T)in Ω. The previous observability inequality (Lemma 7) gives: ZZω×(0,T) e−2s1αt−3(T−t)−3|ϕε|2+1 εkyε(T)k2 L2(Ω) ≤e[CM(T,kak∞)]ky0k2 L2(Ω). M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.3. Third approach We follow O. BODART, M. G.-B., R. PÉREZ-GARCÍA,Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Comm. P.D.E 29 (2004), no. 7-8, 1017–1050. ASSUMPTION Given y0∈L2(Ω), there is e v∈L2(Q), with Supp e v⊂ω0and ω0⊂⊂ ω, such that the solution to (1) e ysatisfies e y(·,T)≡0 in Ω. One has e y∈W(0,T) = {y∈L2(0,T;H1 0(Ω)) : ∂ty∈L2(0,T;H−1(Ω))}and a explicit estimate ke ykW(0,T)≤exp (C(1+T)kak∞)ky0k2+ke vk2. The function e yis regular except near t=0 and near ω0. The idea is eliminate these irregular parts of e y. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.3. Third approach We follow O. BODART, M. G.-B., R. PÉREZ-GARCÍA,Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Comm. P.D.E 29 (2004), no. 7-8, 1017–1050. ASSUMPTION Given y0∈L2(Ω), there is e v∈L2(Q), with Supp e v⊂ω0and ω0⊂⊂ ω, such that the solution to (1) e ysatisfies e y(·,T)≡0 in Ω. One has e y∈W(0,T) = {y∈L2(0,T;H1 0(Ω)) : ∂ty∈L2(0,T;H−1(Ω))}and a explicit estimate ke ykW(0,T)≤exp (C(1+T)kak∞)ky0k2+ke vk2. The function e yis regular except near t=0 and near ω0. The idea is eliminate these irregular parts of e y. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.3. Third approach We follow O. BODART, M. G.-B., R. PÉREZ-GARCÍA,Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Comm. P.D.E 29 (2004), no. 7-8, 1017–1050. ASSUMPTION Given y0∈L2(Ω), there is e v∈L2(Q), with Supp e v⊂ω0and ω0⊂⊂ ω, such that the solution to (1) e ysatisfies e y(·,T)≡0 in Ω. One has e y∈W(0,T) = {y∈L2(0,T;H1 0(Ω)) : ∂ty∈L2(0,T;H−1(Ω))}and a explicit estimate ke ykW(0,T)≤exp (C(1+T)kak∞)ky0k2+ke vk2. The function e yis regular except near t=0 and near ω0. The idea is eliminate these irregular parts of e y. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.3. Third approach Let us now introduce two cut-off functions η∈C∞([0,T]) and θ∈C∞(Ω) such that (η≡1 in [0,T 4],η≡0 in [3T 4,T],0≤η≤1 in [0,T],|η0(t)| ≤ C/T,∀t; θ≡1 in ω0,0≤θ≤1 in Ωand Supp θ⊂ω. Let Ybe the solution to system (1) corresponding to v≡0: (∂tY−∆Y+aY=0 in Q, Y=0 on Σ,Y(·,0) = y0(·)in Ω, We now take (y= (1−θ)e y+ηθYin Q, v= (∂t−∆ + a)y. It is clear that Supp v(·,t)⊆Supp θ⊂ω,yis the solution to (1) corresponding to the control vand, taking into account that e y(T)≡0 in Ω, we get y(·,T)≡0 in Ω. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.3. Third approach Let us now introduce two cut-off functions η∈C∞([0,T]) and θ∈C∞(Ω) such that (η≡1 in [0,T 4],η≡0 in [3T 4,T],0≤η≤1 in [0,T],|η0(t)| ≤ C/T,∀t; θ≡1 in ω0,0≤θ≤1 in Ωand Supp θ⊂ω. Let Ybe the solution to system (1) corresponding to v≡0: (∂tY−∆Y+aY=0 in Q, Y=0 on Σ,Y(·,0) = y0(·)in Ω, We now take (y= (1−θ)e y+ηθYin Q, v= (∂t−∆ + a)y. It is clear that Supp v(·,t)⊆Supp θ⊂ω,yis the solution to (1) corresponding to the control vand, taking into account that e y(T)≡0 in Ω, we get y(·,T)≡0 in Ω. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.3. Third approach In fact vis a regular control and its regularity properties are independent of y0and e v. Indeed, we can express yand vas y≡(1−θ)q+η(t)Y,v≡θη0Y+2∇θ· ∇q+ (∆θ)q, where qis given by q=e y−ηYand, therefore, satisfies (∂tq−∆q+aq=e v1ω−η0Yin Q, q=0 on Σ,q(·,0) = 0 in Ω. Let us fix δ∈(0,T/4),p∈[2,∞)and O0,O1⊂⊂ Ωsuch that O1⊂⊂ Ω\ω0(and, in particular, O1∩Supp e v=∅). If we denote by (Xp 0={y∈Lp(δ,T;W2,p(O0)) : ∂ty∈Lp(O0×(δ,T))}, Xp 1={y∈Lp(0,T;W2,p(O1)) : ∂ty∈Lp(Oi×(0,T))} then, Y∈Xp 0,q∈Xp 1and v∈Lp(0,T;W1,p 0(Ω)). M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.3. Third approach In fact vis a regular control and its regularity properties are independent of y0and e v. Indeed, we can express yand vas y≡(1−θ)q+η(t)Y,v≡θη0Y+2∇θ· ∇q+ (∆θ)q, where qis given by q=e y−ηYand, therefore, satisfies (∂tq−∆q+aq=e v1ω−η0Yin Q, q=0 on Σ,q(·,0) = 0 in Ω. Let us fix δ∈(0,T/4),p∈[2,∞)and O0,O1⊂⊂ Ωsuch that O1⊂⊂ Ω\ω0(and, in particular, O1∩Supp e v=∅). If we denote by (Xp 0={y∈Lp(δ,T;W2,p(O0)) : ∂ty∈Lp(O0×(δ,T))}, Xp 1={y∈Lp(0,T;W2,p(O1)) : ∂ty∈Lp(Oi×(0,T))} then, Y∈Xp 0,q∈Xp 1and v∈Lp(0,T;W1,p 0(Ω)). M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.3. Third approach In fact, we can obtain something better: if p>N+2, one has Xp 0,→C1+α,(1+α)/2(O0×[δ,T]) and Xp 1,→C1+α,(1+α)/2(O1×[0,T]) with α=1−(N+2)/p. Thus, v∈C0 0(Q)and kvkC0≤eC(1+T+Tkak∞)ke ykW(0,T) with C=C(Ω,T)>0. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.2. Third approach. Remarks I 1The previous regularity result for vis independent of the initial datum y0, the control e vand the regularity of the boundary ∂Ω. We have only used the local regularity properties of the operator L≡∂t−∆ + a. In the case in which a≡0, we obtain v∈C∞(Q) (as in the paper of Lebeau-Robbiano). 2In fact we have proved: “Let us fix y0∈L2(Ω) and assume that there exists e v∈L2(Q)such that the solution e y to the linear problem (1) satisfies e y(T)≡0in Ω. Then, there exists e v∈C0 0(Q) s.t. the solution yvof (1) also satisfies yv(T)≡0in Ω”. 3This technique can be applied if we consider a linear parabolic problem with a first order term B· ∇yobtaining the same regularity result. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
2.2. Third approach. Remarks II 4When Ωand ωare unbounded open sets we can obtain the same result: L. DE TERESA, M. G.-B.,Some results on controllability for linear and nonlinear heat equations in unbounded domains, Adv. Diff. Eq. 12 (2007), no. 11, 1201–1240. 5This approach also works in the case of systems of two coupled parabolic equations. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
3. The “best” null control We take ψ=s−5/2e−sα∗(t)t5/2(T−t)5/2ϕ=ρ0(t)ϕ. Then, (∂tψ+ ∆ψ=aρ0(t)ϕ+∂tρ0(t)ϕin Q, ψ=0 on Σ, ψ(·,T) = 0 in Ω. If s≥s1=σ1T+T2+T2kak2/3 ∞, we have ∂tρ0(t)ϕ∈H2,1(Q)and k∂tρ0(t)ϕk2 H2,1≤Cs−1ZZQ e−2sαt(T−t)|∂tϕ|2+|∆ϕ|2. But, H2,1(Q),→Lp(N)(Q)with p(N) = 2(N+2) N−2. Thus, k∂tρ0(t)ϕkLp(N)(Q)≤Ck∂tρ0(t)ϕkH2,1 We can also prove that aρ0(t)ϕ∈Lp(N)(Q)and kaρ0(t)ϕk2 Lp(N)(Q)≤Cs−1ZZQ e−2sαt(T−t)|∂tϕ|2+|∆ϕ|2. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
3. The “best” null control We take ψ=s−5/2e−sα∗(t)t5/2(T−t)5/2ϕ=ρ0(t)ϕ. Then, (∂tψ+ ∆ψ=aρ0(t)ϕ+∂tρ0(t)ϕin Q, ψ=0 on Σ, ψ(·,T) = 0 in Ω. If s≥s1=σ1T+T2+T2kak2/3 ∞, we have ∂tρ0(t)ϕ∈H2,1(Q)and k∂tρ0(t)ϕk2 H2,1≤Cs−1ZZQ e−2sαt(T−t)|∂tϕ|2+|∆ϕ|2. But, H2,1(Q),→Lp(N)(Q)with p(N) = 2(N+2) N−2. Thus, k∂tρ0(t)ϕkLp(N)(Q)≤Ck∂tρ0(t)ϕkH2,1 We can also prove that aρ0(t)ϕ∈Lp(N)(Q)and kaρ0(t)ϕk2 Lp(N)(Q)≤Cs−1ZZQ e−2sαt(T−t)|∂tϕ|2+|∆ϕ|2. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
3. The “best” null control The maximal parabolic regularity for the heat equation (∂Ω∈C2) gives ψ=s−5/2e−sα∗(t)t5/2(T−t)5/2ϕ∈W2,1 p(N)(Q)and kψk2 W2,1 p(N)(Q)≤Cs−1ZZQ e−2sαt(T−t)|∂tϕ|2+|∆ϕ|2 ≤C2s3ZZω×(0,T) e−2sαt−3(T−t)−3|ϕ|2. Conclusion We have obtained a new Carleman inequality for the problem (3) ks−5/2e−sα∗(t)t5/2(T−t)5/2ϕk2 W2,1 p(N)(Q)+I(ϕ) ≤C2s3ZZω×(0,T) e−2sαt−3(T−t)−3|ϕ|2, ∀s≥s1=σ1T+T2+T2kak2/3 ∞. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
3. The “best” null control The maximal parabolic regularity for the heat equation (∂Ω∈C2) gives ψ=s−5/2e−sα∗(t)t5/2(T−t)5/2ϕ∈W2,1 p(N)(Q)and kψk2 W2,1 p(N)(Q)≤Cs−1ZZQ e−2sαt(T−t)|∂tϕ|2+|∆ϕ|2 ≤C2s3ZZω×(0,T) e−2sαt−3(T−t)−3|ϕ|2. Conclusion We have obtained a new Carleman inequality for the problem (3) ks−5/2e−sα∗(t)t5/2(T−t)5/2ϕk2 W2,1 p(N)(Q)+I(ϕ) ≤C2s3ZZω×(0,T) e−2sαt−3(T−t)−3|ϕ|2, ∀s≥s1=σ1T+T2+T2kak2/3 ∞. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
3. The “best” null control Corollary ∀y0∈L2(Ω), there is v∈W2,1 p(N)(Q), with p(N)<∞if N =2and p(N) = 2(N+2) N−2if N ≥3, and kvk2 W2,1 p(N) ≤e[CM(T,kak∞)]ky0k2 L2(Ω), s.t. the solution yvto (1) associated to y0and vsatisfies yv(T) = 0in L2(Ω). Remark We can apply a boot-strap argument and deduce that the previous result is valid for every p ∈[2,∞). In this case the constant Calso depends on p. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
3. The “best” null control Corollary ∀y0∈L2(Ω), there is v∈W2,1 p(N)(Q), with p(N)<∞if N =2and p(N) = 2(N+2) N−2if N ≥3, and kvk2 W2,1 p(N) ≤e[CM(T,kak∞)]ky0k2 L2(Ω), s.t. the solution yvto (1) associated to y0and vsatisfies yv(T) = 0in L2(Ω). Remark We can apply a boot-strap argument and deduce that the previous result is valid for every p ∈[2,∞). In this case the constant Calso depends on p. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
3. The “best” null control Reference V. BARBU,Controllability of parabolic and Navier-Stokes equations, Sci. Math. Jpn. 56 (2002), no. 1, 143–211. F. AMMAR-KHODJA, A. BENABDALLAH, C. DUPAIX, M. G.-B.,A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems, J. Evol. Equ. 9 (2009), no. 2, 267–291. M. G.-B., S. GUERRERO, J.-P. PUEL,Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation, Commun. Pure Appl. Anal. 8 (2009), no. 1, 311–333. M. González-Burgos Remarks on the controllability for the nonlinear heat equation
3. The “best” null control Reference V. BARBU,Controllability of parabolic and Navier-Stokes equations, Sci. Math. Jpn. 56 (2002), no. 1, 143–211. F. AMMAR-KHODJA, A. BENABDALLAH, C. DUPAIX, M. G.-B.,A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems, J. Evol. Equ. 9 (2009), no. 2, 267–291. M. G.-B., S. GUERRERO, J.-P. PUEL,Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation, Commun. Pure Appl. Anal. 8 (2009), no. 1, 311–333. M. González-Burgos Remarks on the controllability for the nonlinear heat equation