Controllability of non-scalar parabolic systems: Some recent results and phenomena
Abstract
In this course we will deal with non-scalar systems which in fact are coupled parabolic scalar equations. We do not present results relating to the controllability problems of systems which come from fluid mechanics as Stokes, Navier-Stokes, ...
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Controllability of non-scalar parabolic systems: Some recent results and phenomena Manuel González-Burgos, UNIVERSIDAD DE SEVILLA Marrakesh Workshop on Control, Inverse Problems and Stabilization of Infinite Dimensional Systems Marrakesh, December 2016 M. González-Burgos Controllability of non-scalar parabolic systems
General objective: Study some null controllability problems for non-scalar parabolic systems. Non-scalar parabolic systems: arise in chemical reactions, when we model problems from the Biology and in a wide variety of physical situations. In this course we will deal with non-scalar systems which in fact are coupled parabolic scalar equations. We do not present results relating to the controllability problems of systems which come from fluid mechanics as Stokes, Navier-Stokes, ... M. González-Burgos Controllability of non-scalar parabolic systems
GOAL: 1Show the important differences between scalar and non-scalar problems. 2Give necessary and sufficient conditions (Kalman conditions) which characterize the controllability properties of these systems. 3Show some hyperbolic phenomena related to the controllability properties of these systems. We will only deal with 1Linear systems 2In general, “simple” Parabolic Systems. M. González-Burgos Controllability of non-scalar parabolic systems
Contents 1Introduction 2The parabolic scalar case The one-dimensional case: The moment method General case: Carleman Inequalities Final comments in the scalar case 3Finite-dimensional systems 4Distributed controllability of 2 ×2 linear systems 5Boundary controllability of a 2 ×2 linear system 6A generalization: Cascade systems 7The Kalman condition for a class of parabolic systems. Distributed controls 8The Kalman condition for a class of parabolic systems. Boundary controls 9New phenomena: Minimal time of controllability 10 New phenomena: Dependence on the position of the control set 11 Further results 12 Comments and open problems M. González-Burgos Controllability of non-scalar parabolic systems
1. Introduction M. González-Burgos Controllability of non-scalar parabolic systems
1. Introduction Let us fix T>0 and let Hand Ube two separable Hilbert spaces. Let us consider the autonomous system: (1) (y0=Ay+Buon (0,T), y(0) = y0∈H. Aand Bare “appropriate” operators, y0∈His the initial datum at t=0 and u∈L2(0,T;U)is the control (exerted by means of the operator B). Assume the problem is well-posed: ∀(y0,u)there exists a unique weak solution y∈C0([0,T]; H)to (1) which depends continuously on the data. Let us denote by y(t;y0,u)∈Hthe solution to the system at time t∈[0,T]. Example H=Rn(n≥1), U=Rm(m≥1), A∈ L(Rn)and B∈ L(Rm;Rn): ordinary differential system with nvariables and mcontrols. M. González-Burgos Controllability of non-scalar parabolic systems
1. Introduction Exact Controllability: System (1) is exactly controllable at time Tif ∀(y0,y1)∈H×H,there exists u∈L2(0,T;U)s.t. the solution yof (1) satisfies y(T;y0,u) = y1. Controllability to trajectories: System (1) is controllable to trajectories at time Tif ∀(y0,by0)∈H×Hand bu∈L2(0,T;U),there exists u∈L2(0,T;U)s.t. the corresponding weak solution to (1) satisfies y(T;y0,u) = y(T;by0,bu). Null Controllability: System (1) is null controllable at time Tif ∀y0∈Hthere exists u∈L2(0,T;U)s.t. y(T;y0,u) = 0. Linear case: Controllability to trajectories and null controllability are equivalent. Approximate Controllability: System (1) is approximately controllable at time Tif ∀(y0,y1)∈H×H, and every ε>0, there exists u∈L2(0,T;U)s.t. ky(T;y0,u)−y1kH≤ε. M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case Remark Problem (1) is linear. Then, System (1) is null controllable at time Tif and only if the system is exactly controllable to the trajectories at time T. Remark We will deal with parabolic problems. So, due to the regularizing effect of these problems, it is well-known that the exact controllability result fails. Therefore, in this course we will study null or approximate controllability results for the system under consideration. M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case Theorem Let us fix T >0. The following conditions are equivalent 1For any Ω⊂RN, bounded open set with Ωhaving a C2boundary, any ω⊂Ω, nonempty open subset, and any coefficients αij (1≤i,j≤N), D and c, satisfying (3) and (4), System (5) is null controllable in L2(Ω) at time T >0with distributed controls v∈L2(QT). 2For any Ω⊂RN, bounded open set with Ωhaving a C2boundary, any Γ0⊂∂Ω, nonempty relative open subset, and any coefficients αij (1≤i,j≤N), Dand c, satisfying (3) and (4), System (6) is null controllable in L2(Ω) at time T >0with boundary controls h∈L2(0,T;H1/2(∂Ω)). Proof: We will use in a fundamental way that the problem under consideration is scalar (in fact, same number of equations and controls). We follow some ideas from [BODART,G.-B.,PÉREZ-GARCÍA] Comm. PDE (2004) and [G.-B.,PÉREZ-GARCÍA] Asymp. Anal. (2006). ··· M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case Theorem Let us fix T >0. The following conditions are equivalent 1For any Ω⊂RN, bounded open set with Ωhaving a C2boundary, any ω⊂Ω, nonempty open subset, and any coefficients αij (1≤i,j≤N), D and c, satisfying (3) and (4), System (5) is null controllable in L2(Ω) at time T >0with distributed controls v∈L2(QT). 2For any Ω⊂RN, bounded open set with Ωhaving a C2boundary, any Γ0⊂∂Ω, nonempty relative open subset, and any coefficients αij (1≤i,j≤N), Dand c, satisfying (3) and (4), System (6) is null controllable in L2(Ω) at time T >0with boundary controls h∈L2(0,T;H1/2(∂Ω)). Proof: We will use in a fundamental way that the problem under consideration is scalar (in fact, same number of equations and controls). We follow some ideas from [BODART,G.-B.,PÉREZ-GARCÍA] Comm. PDE (2004) and [G.-B.,PÉREZ-GARCÍA] Asymp. Anal. (2006). ··· M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case Remark (Regularizing effect) The previous proof shows that if the distributed and boundary null controllability results for Systems (5) and (6) are valid with controls in L2(QT)and L2(0,T;H1/2(∂Ω)), then the previous systems are null controllable with controls in L∞(QT)and L∞(ΣT)(and even better for regular coefficients). Remark In the proof of Theorem 1 we have strongly used that the operator ∂t+L(t)is scalar. We will see that the previous equivalence is not valid for non-scalar parabolic operators. M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case From now on, we will concentrate on the distributed control problem (5). Let us introduce the adjoint problem (7) (−∂tϕ+L∗(t)ϕ=0 in QT, ϕ=0 on ΣT, ϕ(·,T) = ϕTin Ω, where ϕT∈L2(Ω) is given and L∗(t)is the operator given by L∗(t)ϕ=− N X i,j=1 ∂ ∂xiαij(x,t)∂ϕ ∂xj−∇·(Dϕ) + c(x,t)ϕa.e. in QT. This problem is also well-posed and the solution depends continuously on ϕT: there exists a constant e C>0 such that ∀ϕT∈L2(Ω) System (7) has only one solution ϕ∈L2(0,T;H1 0(Ω)) ∩C0([0,T]; L2(Ω)) and it satisfies kϕkL2(0,T;H1 0(Ω)) +kϕkC0([0,T];L2(Ω)) ≤e CkϕTkL2(Ω). M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case Theorem (Observability Inequality) Under the previous assumptions, System (5) is null controllable at time T >0 if and only if there exists a constant CT>0s.t. (8) kϕ(·,0)k2 L2(Ω) ≤CTZZω×(0,T)|ϕ|2dxdt,∀ϕT∈L2(Ω), where ϕis the solution of (7) associated to ϕT. Remark The Observability Inequality (8) in particular implies a better result: If (8) holds then, ∀y0∈L2(Ω) there is a distributed control v∈L2(QT)s.t. kvk2 L2(QT)≤CTky0k2 L2(Ω) and y(·,T) = 0, being ythe solution to (5) corresponding to y0and CT>0 the constant in (8). M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case Theorem (Observability Inequality) Under the previous assumptions, System (5) is null controllable at time T >0 if and only if there exists a constant CT>0s.t. (8) kϕ(·,0)k2 L2(Ω) ≤CTZZω×(0,T)|ϕ|2dxdt,∀ϕT∈L2(Ω), where ϕis the solution of (7) associated to ϕT. Remark The Observability Inequality (8) in particular implies a better result: If (8) holds then, ∀y0∈L2(Ω) there is a distributed control v∈L2(QT)s.t. kvk2 L2(QT)≤CTky0k2 L2(Ω) and y(·,T) = 0, being ythe solution to (5) corresponding to y0and CT>0 the constant in (8). M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case Remark (Control cost) The previous remark and inequality (8) provide an estimate of the cost of the control for system (5): If (8) holds at time T>0, then ZT(y0) := {v∈L2(QT) : y(T;y0,v) = 0} 6=∅,∀y0∈L2(Ω). We can then define the control cost for system (5) at time Tas K(T) = sup ky0kL2(Ω)=1inf v∈ZT(y0)kvkL2(QT),∀T>0. Thus, K(T)≤√CT. On the other hand, if ZT(y0)6=∅, for any y0∈L2(Ω), then, the observability inequality (8) for the adjoint system (7) holds with CT=K(T)2. It is then clear that K(T) = inf npCT:CT>0 is such that (8) holdso. M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 1. The one-dimensional case: The moment method We follow [FATTORINI,RUSSELL] Arch. Rat. Mech. Anal. (1971). M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 1. The one-dimensional case: The moment method Consider the boundary null controllability problem for the classical one-dimensional heat equation in (0, π)(for simplicity): (9) yt−yxx =0 in QT= (0, π)×(0,T), y(0,·) = v,y(π, ·) = 0 on (0,T), y(·,0) = y0in (0, π), with y0∈H−1(0, π)and v∈L2(0,T). The problem is well-posed and the solution (defined by transposition) depends continuously on the data y0and v. The operator −∂xx on (0, π)with homogenous Dirichlet boundary conditions admits a sequence of eigenvalues and normalized eigenfunctions given by λk=k2,φk(x) = r2 πsin kx,k≥1,x∈(0, π) which is a Hilbert basis of L2(0, π). In the sequel, we will use the notation yk= (y,φk)L2(0,π),∀y∈L2(0, π). M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 1. The one-dimensional case: The moment method The idea of the moment method is simple: Given y0∈H−1(0, π), ϕT∈H1 0(0, π)and v∈L2(0,T), then hy(·,T), ϕTi−hy0, ϕ(·,0)i=ZT 0 v(t)ϕx(0,t)dt. where yis the solution to (9) and ϕis the solution to the adjoint problem (−ϕt−ϕxx =0 in QT, ϕ=0 on {0,1}×(0,T), ϕ(·,T) = ϕTin (0, π). Property v∈L2(0, π)is a null control for system (9) (i.e., v∈L2(0,T)is a control s.t. the solution yto (9) satisfies y(·,T) = 0 in (0, π)) if and only if −hy0, ϕ(·,0)i=ZT 0 v(t)ϕx(0,t)dt,∀ϕT∈H1 0(0, π). M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 1. The one-dimensional case: The moment method Consequence: The previous result is valid for any nonempty bounded interval (a,b)and for any second order operator self-adjoint elliptic operator Ly=−(α(x)yx)x+c(x)y, with α∈C1([a,b]) and α>0 in (a,b), and c∈C0([a,b]).Then, if we apply Theorem 1, we also get a distributed controllability result for the problem yt+Ly=v1ωin QT= (a,b)×(0,T), y(a,·) = 0,y(b,·) = 0 on (0,T), y(·,0) = y0in (a,b), with y0∈L2(0, π)and ω⊆(a,b), a nonempty open subset. M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 2. General case: Carleman Inequalities We follow [FURSIKOV,IMANUVILOV] 1996 and [IMANUVILOV,YAMAMOTO] 2003. M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 2. General case: Carleman Inequalities We will consider the following parabolic equation: (10) −∂tz+L0(t)z=F0+ N X i=1 ∂Fi ∂xi in QT, z=0 on ΣT,z(·,T) = zTin Ω, with zT∈L2(Ω),Fi∈L2(QT),i=0,1,...,N, and L0(t)the self-adjoint parabolic operator given by L0(t)y=− N X i,j=1 ∂ ∂xiαij(x,t)∂y ∂xj with coefficients αij satisfying (3) (regularity) and (4) (uniform elliptic condition). M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 2. General case: Carleman Inequalities Lemma Let B⊂Ωbe a nonempty open subset and d ∈R. Then, ∃β0∈C2(Ω) (positive and only depending on Ωand B) and e C0,eσ0>0(only depending on Ω,Band d) s.t. for every zT∈L2(Ω), the solution z to (10)satisfies (11) I(d,z)≤e C0 sdZZB×(0,T) e−2sβγ(t)d|z|2 +sd−3ZZQT e−2sβγ(t)d−3|F0|2+sd−1 N X i=1ZZQT e−2sβγ(t)d−1|Fi|2!, ∀s≥es0=eσ0(T+T2);γ(t) = t−1(T−t)−1,β(x,t) = β0(x)/t(T−t) and I(d,z)≡sd−2ZZQT e−2sβγ(t)d−2|∇z|2+sdZZQT e−2sβγ(t)d|z|2. M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 2. General case: Carleman Inequalities Lemma When Fi≡0for 1≤i≤N, ∃e C1and eσ1(which only depend on Ω,Band d) s.t., ∀zT∈L2(Ω), the solution z to (10)satisfies (12) I1(d,z)≤e C1 sdZZB×(0,T) e−2sβγ(t)d|z|2+sd−3ZZQT e−2sβγ(t)d−3|F0|2!, for all s ≥es1=eσ1(T+T2)where I1(d,z)≡sd−4ZZQT e−2sβγ(t)d−4 |∂tz|2+ N X i,j=1 ∂2z ∂xi∂xj 2 +I(d,z). Proof: See [FURSIKOV,IMANUVILOV] 1996; [IMANUVILOV,YAMAMOTO] (2003) and [FERNÁNDEZ-CARA,GUERRERO] SICON (2006). M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 2. General case: Carleman Inequalities Recall that our objective is to prove a null controllability result at time Tfor (5) (∂ty+L(t)y=v1ωin QT, y=0 on ΣT,y(·,0) = y0in Ω, with L(t)given by: L(t)y=− N X i,j=1 ∂ ∂xiαij(x,t)∂y ∂xj+D(x,t)·∇y+c(x,t)y =L0(t)y+D(x,t)·∇y+c(x,t)y, with coefficients αij satisfying (3) and (4). We also know that this is equivalent to the observability inequality (8) kϕ(·,0)k2 L2(Ω) ≤CTZZω×(0,T)|ϕ|2dxdt,∀ϕT∈L2(Ω), for the solutions to the adjoint problem (7). M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 2. General case: Carleman Inequalities Corollary There exists a constant C0=C0(Ω,ω)>0such that ∀ϕT∈L2(Ω) and ϕthe corresponding solution to (7), the observability inequality (8) holds with CT=exp C01+1 T+kck2/3 ∞+Tkck∞+ (1+T)kDk2 ∞. Proof: We follow [FERNÁNDEZ-CARA,ZUAZUA] Ann. IHP (2000) and [DOUBOVA,FERNÁNDEZ-CARA,MG-B,ZUAZUA] SICON (2002). The Carleman inequality (11) applied to problem (7) implies (B≡ω,d=3 and −∂tϕ+L0(t)ϕ=∇·(Dϕ)−c(x,t)ϕ) that ∀s≥es0=eσ0(T+T2): sZZQT e−2sβγ(t)|∇ϕ|2+s3ZZQT e−2sβγ(t)3|ϕ|2 ≤e C0 s3ZZω×(0,T) e−2sβγ(t)3|ϕ|2 +kck2 ∞ZZQT e−2sβ|ϕ|2+s2kDk2 ∞ZZQT e−2sβγ(t)2|ϕ|2. M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 2. General case: Carleman Inequalities Corollary There exists a constant C0=C0(Ω,ω)>0such that ∀ϕT∈L2(Ω) and ϕthe corresponding solution to (7), the observability inequality (8) holds with CT=exp C01+1 T+kck2/3 ∞+Tkck∞+ (1+T)kDk2 ∞. Proof: We follow [FERNÁNDEZ-CARA,ZUAZUA] Ann. IHP (2000) and [DOUBOVA,FERNÁNDEZ-CARA,MG-B,ZUAZUA] SICON (2002). The Carleman inequality (11) applied to problem (7) implies (B≡ω,d=3 and −∂tϕ+L0(t)ϕ=∇·(Dϕ)−c(x,t)ϕ) that ∀s≥es0=eσ0(T+T2): sZZQT e−2sβγ(t)|∇ϕ|2+s3ZZQT e−2sβγ(t)3|ϕ|2 ≤e C0 s3ZZω×(0,T) e−2sβγ(t)3|ϕ|2 +kck2 ∞ZZQT e−2sβ|ϕ|2+s2kDk2 ∞ZZQT e−2sβγ(t)2|ϕ|2. M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 2. General case: Carleman Inequalities As a consequence we can prove that for s≥C1(T+T2+T2(kck2/3 ∞+kDk2 ∞)) (C1=C1(Ω,ω)) one has [sγ(t)]3−e C0kck2 ∞−e C0[sγ(t)]2kDk2 ∞≥1 2[sγ(t)]3. Consequently, for s=C1(T+T2+T2(kck2/3 ∞+kDk2 ∞)) that ZZQT e−2sβt−3(T−t)−3|ϕ|2≤e C1ZZω×(0,T) e−2sβt−3(T−t)−3|ϕ|2 and therefore ZZΩ×(T/4,3T/4)|ϕ|2≤eC(1+1/T+kck2/3 ∞+kDk2 ∞)ZZω×(0,T)|ϕ|2. This last inequality combined with energy estimates (C=C(a0)>0) d dt eC(kck∞+kDk2 ∞)tZΩ|ϕ|2(·,t)≥0∀t∈[0,T] implies (8) and the proof is complete. M. González-Burgos Controllability of non-scalar parabolic systems
2. The parabolic scalar case 2. General case: Carleman Inequalities Corollary Let us fix T >0,Ω⊂RN,ω⊆Ωand Γ0⊆∂Ω(arbitrary) as before. Then, there exist positive constants C0=C0(Ω,ω)and b C0=b C0(Ω,Γ0)s.t. 1∀y0∈L2(Ω) there is a control v∈L2(Ω) which satisfies kvk2 L2(QT)≤eC01+1/T+kck2/3 ∞+Tkck∞+(1+T)kDk2 ∞ky0k2 L2(Ω), and y(·,T) = 0in Ω, (y is the solution to (5) associated to y0and v). 2∀y0∈L2(Ω) there is a control h∈L2(0,T;H1/2(Ω)) which satisfies khk2 L2(0,T;H1/2(Ω)) ≤eb C01+1/T+kck2/3 ∞+Tkck∞+(1+T)kDk2 ∞ky0k2 L2(Ω), and y(·,T) = 0in Ω, (y is the solution to (6) associated to y0and vand, in fact, y ∈L2(0,T;H1(Ω)) ∩C0([0,T]; L2(Ω))). M. González-Burgos Controllability of non-scalar parabolic systems
3. Finite-dimensional systems M. González-Burgos Controllability of non-scalar parabolic systems
3. Finite-dimensional systems Let us consider the autonomous linear system (13) y0=Ay+Buon [0,T],y(0) = y0, where A∈ L(Cn)and B∈ L(Cm,Cn)are constant matrices, y0∈Cnand u∈L2(0,T;Cm)is the control. Problem: Given y0,yd∈Cn, is there a control u∈L2(0,T;Cm)such that the solution y to the problem satisfies y(T) = yd???? Let us define (controllability matrix) [A|B]=(B,AB ,A2B,··· ,An−1B)∈ L(Cnm;Cn). On the other hand, let {θl}1≤l≤ˆ p⊂Cbe the set of distinct eigenvalues of A∗. For l:1≤l≤ˆ p, we denote by mlthe geometric multiplicity of θl. The sequence {wl,j}1≤j≤mlwill denote a basis of the eigenspace associated to θl. M. González-Burgos Controllability of non-scalar parabolic systems
3. Finite-dimensional systems Let us consider the autonomous linear system (13) y0=Ay+Buon [0,T],y(0) = y0, where A∈ L(Cn)and B∈ L(Cm,Cn)are constant matrices, y0∈Cnand u∈L2(0,T;Cm)is the control. Problem: Given y0,yd∈Cn, is there a control u∈L2(0,T;Cm)such that the solution y to the problem satisfies y(T) = yd???? Let us define (controllability matrix) [A|B]=(B,AB ,A2B,··· ,An−1B)∈ L(Cnm;Cn). On the other hand, let {θl}1≤l≤ˆ p⊂Cbe the set of distinct eigenvalues of A∗. For l:1≤l≤ˆ p, we denote by mlthe geometric multiplicity of θl. The sequence {wl,j}1≤j≤mlwill denote a basis of the eigenspace associated to θl. M. González-Burgos Controllability of non-scalar parabolic systems
3. Finite-dimensional systems The following classical result can be found in R. KALMAN, Y.-CH. HO, K. NARENDRA,Controllability of linear dynamical systems,1963. and gives a complete answer to the problem of controllability of finite dimensional autonomous linear systems: Theorem Under the previous assumptions, the following conditions are equivalent 1System (13) is exactly controllable at time T, for every T >0. 2There exists T >0such that system (13) is exactly controllable at time T. 3rank [A|B] = n or ker[A|B]∗={0}(Kalman rank condition). 4Hautus test: rank A∗−θlIn B∗=n,∀l:1≤l≤ˆ p. 5rank [B∗wl,1,B∗wl,2,··· ,B∗wl,ml] = ml, for every l :1≤l≤ˆ p. M. González-Burgos Controllability of non-scalar parabolic systems
3. Finite-dimensional systems Remark 1The four controllability concepts (exact,exact to trajectories,null and approximate controllability) for System (13) are equivalent (finite-dimensional space). 2Observe that {B∗wl,1,B∗wl,2,...,B∗wl,ml} ⊂ Cm. Condition 5 in Theorem 4 says this set is linearly independent for any l:1≤l≤ˆ p. In particular, ml≤m∀l:1≤l≤ˆ p. 3Given the o.d.s. (adjoint problem) −ϕ0=A∗ϕin [0,T], ϕ(T) = ϕT∈Cn, it is not difficult to prove the following result: “System (13) is exactly controllable at time T if and only if the following property for the adjoint problem holds (unique continuation property) If B∗ϕ(·) = 0 on [0,T], then ϕT≡0." M. González-Burgos Controllability of non-scalar parabolic systems
3. Finite-dimensional systems Goal We have a complete characterization of the controllability results for finite-dimensional linear ordinary differential systems (a Kalman condition). Is it possible to obtain similar results for Partial Differentials Systems? We will focus on coupled linear parabolic systems. What are the possible generalizations to Systems of Parabolic Equations? M. González-Burgos Controllability of non-scalar parabolic systems
4. Distributed controllability of 2×2 linear systems M. González-Burgos Controllability of non-scalar parabolic systems
4. Distributed controllability of 2 ×2 linear systems Let us consider the 2 ×2 linear reaction-diffusion system (QT= Ω ×(0,T)) (14) ∂ty1+L1 0(t)y1+a11y1+a12y2=v1ωin QT, ∂ty2+L2 0(t)y2+a21y1+a22y2=0 in QT, yi=0 on ΣT=∂Ω×(0,T),yi(·,0) = yi 0in Ω,1≤i≤2, where Ω,ωand Tare as before, aij =aij(x,t)∈L∞(QT)(1 ≤i,j≤2), yi 0∈L2(Ω) (1 ≤i≤2) and Lk 0(t)is, for every 1 ≤k≤2, the second order operator Lk 0(t)y=− N X i,j=1 ∂ ∂xiαk ij(x,t)∂y ∂xjwhere αk ij satisfy (3) and (4). Remark System (14) is controlled by means of a scalar distributed control exerted on the right-hand side of the first equation. The second equation is indirectly controlled by the coupling term a21y1.Necessary condition a21 6≡ 0 (a21 ∈L∞(QT)). M. González-Burgos Controllability of non-scalar parabolic systems
4. Distributed controllability of 2 ×2 linear systems Equivalently, the previous system can be written as (15) (∂ty+b L(t)y+Ay=Bv1ωin QT, y=0 on ΣT,y(·,0) = y0in Ω, where b L(t)is the matrix operator given by b L(t) = diag (L1 0(t),L2 0(t)), y= (yi)1≤i≤2is the state and where (y0= (yi 0)1≤i≤2∈L2(Ω; Rn),A(·,·)=(aij(·,·))1≤i,j≤2∈L∞(QT;L(Rn)), and B≡e1= (1,0)∗∈R2 are given. Let us observe that, for each y0∈L2(Ω; R2)and v∈L2(QT), System (15) admits a unique weak solution y∈L2(0,T;H1 0(Ω; R2)) ∩C0([0,T]; L2(Ω; R2)). M. González-Burgos Controllability of non-scalar parabolic systems
4. Distributed controllability of 2 ×2 linear systems Assumption We assume that the coupling coefficient a21 ∈L∞(QT)satisfies (16) a21 ≥c0>0 or −a21 ≥c0>0 in ω0×(0,T), with ω0⊆ωa new open subset. As in the scalar case, the controllability result for system (15) is equivalent to the observability inequality:∃CT>0 such that kϕ1(·,0)k2 L2+kϕ2(·,0)k2 L2≤CTZZω×(0,T)|ϕ1(x,t)|2dx dt, where ϕis the solution associated to ϕ0∈L2(Ω; R2)of the adjoint problem: (17) −ϕt+b L(t)ϕ+A∗ϕ=0 in QT, ϕ=0 on ΣT, ϕ(·,T) = ϕ0in Ω. M. González-Burgos Controllability of non-scalar parabolic systems
4. Distributed controllability of 2 ×2 linear systems Summarizing We have proved that the solutions to the adjoint system (17) −ϕt+b L(t)ϕ+A∗ϕ=0 in QT, ϕ=0 on ΣT, ϕ(·,T) = ϕ0in Ω. satisfy the Carleman inequality C0=C0(Ω,ω0,c0,ka21k∞,d) I1(d+3, ϕ1) + I1(d, ϕ2)≤C0sd+4ZZω×(0,T) e−2sαγ(t)d+4|ϕ1|2, ∀s≥s0=σ0hT+T2+T2ka11k2/3 ∞+ka12k1/3 ∞+ka22k2/3 ∞i. (C0=C0(Ω,ω0,c0,ka21k∞,d)and σ0=σ0(Ω,ω0,c0,ka21k∞,d)are positive constants). M. González-Burgos Controllability of non-scalar parabolic systems
4. Distributed controllability of 2 ×2 linear systems As in the scalar case, combining the previous result and energy inequalities satisfied by the solutions of the adjoint system it is possible to prove an observability inequality for the adjoint system and deduce: Corollary Let us assume (16). Then, there exists a positive constant C(only depending on Ω,ω,c0and ka21k∞) such that for every y0∈L2(Ω; R2)there is a control v∈L2(Ω) which satisfies kv|k2 L2(QT)≤eCHky0k2 L2(Ω;R2), and y(·,T) = 0in Ω, with y the solution to (15) associated to y0and v. In the previous inequality, His given by H ≡ 1+T+1 T+ka11k2/3 ∞+ka12k1/3 ∞+ka22k2/3 ∞+Tmax 1≤i,j≤2kaijk∞. M. González-Burgos Controllability of non-scalar parabolic systems
4. Distributed controllability of 2 ×2 linear systems Remark System (14) is always controllable if we exert a control in each equation (two controls). The controllability result for system (14) is independent of the operators L1 0(t)and L2 0(t). We will see that the situation is more intricate if in the system a general control vector B∈R2is considered. The same result can be obtained for the distributed approximate controllability at time T. Therefore, approximate and null controllability are equivalent concepts (distributed case). Using a different technique (fictitious controls), it is possible to prove a null controllability result as in the previous corollary when the coupling matrix A∈L∞(QT;L(R2)) satisfies: There exist an open subset ω0⊂⊂ ωand a positive constant a0s.t. |a21(x,t)| ≥ a0>0 in ω0×(0,T). M. González-Burgos Controllability of non-scalar parabolic systems
4. Distributed controllability of 2 ×2 linear systems References 1L. DE TERESA,Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations 25 (2000), no. 1–2, 39–72. 2F. AMMAR KHODJA, A. BENABDALLAH, C. DUPAIX ET I. KOSTIN, Controllability to the trajectories of phase-field models by one control force, SIAM J. Control Optim. 42 (2003), no. 5, 1661-1689. 3M. G.-B., R. PÉREZ-GARCÍA,Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot. Anal. 46 (2006), no. 2, 123–162. 4M. G.-B., L. DE TERESA,Controllability results for cascade systems of m coupled parabolic PDEs by one control force, Port. Math. 67 (2010), no. 1, 91–113. M. González-Burgos Controllability of non-scalar parabolic systems
5. Boundary controllability of a 2×2 linear system M. González-Burgos Controllability of non-scalar parabolic systems
5. Boundary controllability of a 2 ×2 linear system Let us now consider the boundary controllability problem for the one-dimensional linear reaction-diffusion system: (18) yt−Dyxx =Ayin QT= (0, π)×(0,T), y|x=0=1 0v,y|x=π=0 on (0,T), y(·,0) = y0in (0, π), with y0∈H−1(0, π;R2),v∈L2(0,T)is the control and D=d10 0d2,d1,d2>0,(d16=d2),and A=0 0 1 0 . Existence and uniqueness For any y0∈H−1(0, π;R2)and v∈L2(0,T), system (18) has a unique solution y∈L2(QT)∩C0([0,T]; H−1(0, π;R2)) defined by transposition. M. González-Burgos Controllability of non-scalar parabolic systems
5. Boundary controllability of a 2 ×2 linear system Let us now consider the boundary controllability problem for the one-dimensional linear reaction-diffusion system: (18) yt−Dyxx =Ayin QT= (0, π)×(0,T), y|x=0=1 0v,y|x=π=0 on (0,T), y(·,0) = y0in (0, π), with y0∈H−1(0, π;R2),v∈L2(0,T)is the control and D=d10 0d2,d1,d2>0,(d16=d2),and A=0 0 1 0 . Question Are the controllability properties of system (18) independent of d1and d2??? NO. M. González-Burgos Controllability of non-scalar parabolic systems
5. Boundary controllability of a 2 ×2 linear system As before, system (18) is null controllable at time Tif and only if the observability inequality kϕ1(·,0)k2 H1 0(0,π)+kϕ2(·,0)k2 H1 0(0,π)≤CTZT 0|ϕ1,x(0,t)|2dt, holds. Again ϕis the solution associated to ϕ0∈H1 0(0, π;R2)of the adjoint problem: (19) −ϕt−Dϕxx =A∗ϕin QT, ϕ|x=0=ϕ|x=π=0 on (0,T), ϕ(·,T) = ϕ0in (0, π). Let us see that, in general, this inequality fails (even if a21 =16=0!!!!!). M. González-Burgos Controllability of non-scalar parabolic systems
5. Boundary controllability of a 2 ×2 linear system A necessary condition: Proposition Assume that system (18) is null controllable at time T (d16=d2). Then (λk=k2), d1λk6=d2λj,∀k,j≥1(⇐⇒ pd1/d26∈ Q). Proof: By contradiction, assume that d1λk=d2λjfor some k,jand take K=max{k,j}. The idea is transforming system (19) into an o.d.s. Recall that λkand φkare the eigenvalues and normalized eigenfunctions of −∂xx on (0, π)with homogenous Dirichlet boundary conditions: λk=k2, φk(x) = r2 πsin kx,k≥1,x∈(0, π). Idea: Take ϕ0∈XK={ϕ0=PK `=1a`φ`:a`∈R2} ⊂ H1 0(0, π;R2). M. González-Burgos Controllability of non-scalar parabolic systems
5. Boundary controllability of a 2 ×2 linear system Consider also BK= B . . . B ∈R2K,(B=1 0)and L∗ K=diag (−λ1D+A∗,−λ2D+A∗,··· ,−λKD+A∗)∈ L(R2K). Taking in (19) arbitrary initial data ϕ0,K=PK `=1a`φ`∈H1 0(0, π;R2)where a`∈R2,it is not difficult to see that system (19) is equivalent to the o.d. system (20) −Z0=L∗ KZon [0,T],Z(0) = Z0∈R2K. From the observability inequality for system (19) we deduce the unique continuation property for the solutions to (20): B∗ KZ(·) = 0 in (0,T)=⇒Z≡0. M. González-Burgos Controllability of non-scalar parabolic systems
6. A generalization: Cascade systems We consider the linear parabolic system ∂ty1+L1 0(t)y1+ n X j=1 C1j·∇yj+ n X j=1 a1jyj=v1ωin QT= Ω ×(0,T), ∂ty2+L2 0(t)y2+ n X j=1 C2j·∇yj+ n X j=1 a2jyj=0 in QT, ··· ∂tyn+Ln 0(t)yn+ n X j=1 Cnj ·∇yj+ n X j=1 anjyj=0 in QT, yi=0 on ΣT=∂Ω×(0,T),yi(·,0) = yi 0in Ω,1≤i≤n, where aij =aij(x,t)∈L∞(QT),Cij =Cij(x,t)∈L∞(QT;RN)(1 ≤i,j≤n), yi 0∈L2(Ω) (1 ≤i≤n) and Lk 0(t)is, for every 1 ≤k≤n, the second order operator Lk 0(t)y=− N X i,j=1 ∂ ∂xiαk ij(x,t)∂y ∂xjwhere αk ij satisfy (3) and (4) for every k. M. González-Burgos Controllability of non-scalar parabolic systems
6. A generalization: Cascade systems Objective Controllability properties of the system: nequations controlled with a unique distributed control. Equivalently, the previous system can be written as (21) (∂ty+b L(t)y+C·∇y+Ay=Bv1ωin QT, y=0 on ΣT,y(·,0) = y0in Ω, where b L(t)is the matrix operator given by b L(t) = diag (L1 0(t),··· ,Ln 0(t)), y= (yi)1≤i≤nis the state and ∇y= (∇yi)1≤i≤n, and where (y0= (yi 0)1≤i≤n∈L2(Ω; Rn),A(·,·)=(aij(·,·))1≤i,j≤n∈L∞(QT;L(Rn)), C(·,·)=(Cij(·,·))1≤i,j≤n∈L∞(QT;L(Rn;RNn)) and B≡e1= (1,0, ..., 0)∗ are given. Let us observe that, for each y0∈L2(Ω; Rn)and v∈L2(QT), System (21) admits a unique weak solution y∈L2(0,T;H1 0(Ω; Rn)) ∩C0([0,T]; L2(Ω; Rn)). M. González-Burgos Controllability of non-scalar parabolic systems
6. A generalization: Cascade systems By cascade system we mean that matrices Aand Chave the following structure: A= a11 a12 a13 ... a1n a21 a22 a23 ... a2n 0a32 a33 ... a3n . . .. . ........ . . 0 0 ... an,n−1ann ,C= C11 C12 ... C1n 0C22 ... C2n . . .. . ..... . . 0 0 ... Cnn with aij ∈L∞(QT)and Cij ∈L∞(QT;RN)and the coefficients ai,i−1satisfy ai,i−1≥c0>0 or −ai,i−1≥c0>0 in ω0×(0,T),∀i:2≤i≤n, with ω0⊆ωa new open subset. Remark It is natural to assume that ai,i−16≡ 0 for any i:2≤i≤n. The previous assumption is stronger but will provide the controllability result. M. González-Burgos Controllability of non-scalar parabolic systems
6. A generalization: Cascade systems In this case, the corresponding adjoint problem has the form −∂tϕi+Li 0(t)ϕi− i X j=1 [∇·(Cjiϕj)−ajiϕj] = −ai+1,iϕi+1in QT, ··· (1≤i≤n−1), −∂tϕn+Ln 0(t)ϕn− n X j=1 [∇·(Cjnϕj)−ajnϕj] = 0 in QT, ϕi=0 on ΣT, ϕi(·,T) = ϕi,Tin Ω,1≤i≤n, where ϕi,T∈L2(Ω) (1 ≤i≤n). Again, the null controllability of System (21) (with L2-controls) at time Tis equivalent to the existence of a constant CT>0 such that the so-called observability inequality kϕ(·,0)k2 L2(Ω;Rn)≤CTZZω×(0,T)|ϕ1(x,t)|2 holds for every solution ϕ= (ϕ1, . . . , ϕn)∗to the adjoint problem. M. González-Burgos Controllability of non-scalar parabolic systems
6. A generalization: Cascade systems Theorem Under the previous assumptions, let M0=max2≤i≤nkai,i−1k∞. Then, there exist a positive function α0∈C2(Ω) (only depending on Ωand ω0), two positive constants C0and σ0(only depending on Ω,ω0,c0, M0and d) and l≥0(only depending on n) such that, for every ϕT∈L2(QT;Rn), the solution ϕto the adjoint problem satisfies n X i=1I(d+3(n−i), ϕi)≤C0sd+lZZω0×(0,T) e−2sαγ(t)d+l|ϕ1|2, ∀s≥s0=σ0T+T2+T2max i≤jkaijk 2 3(j−i)+3 ∞+kCijk 2 3(j−i)+1 ∞. In the previous inequality, γ(t) = t−1(T−t)−1,α(x,t) = α0(x)/t(T−t)and I(d,z)is given in Lemma 2.3 (with αinstead of β). M. González-Burgos Controllability of non-scalar parabolic systems
6. A generalization: Cascade systems Combining the previous result and energy inequalities satisfied by the solutions of the adjoint system it is possible to prove an observability inequality for the adjoint system (as in the scalar case). Summarizing, we get Corollary Under assumptions of the previous result, there exists a positive constant C (only depending on Ω,ω,n,c0and M0) such that for every y0∈L2(Ω; Rn) there is a control v∈L2(Ω) which satisfies kv|k2 L2(QT)≤eCHky0k2 L2(Ω;Rn), and y(·,T) = 0in Ω, with y the solution to (21) associated to y0and v. In the previous inequality, His given by H ≡ 1+T+1 T+max i≤jkaijk 2 3(j−i)+3 ∞+kCijk 2 3(j−i)+1 ∞+Tkaijk∞+kCijk2 ∞. M. González-Burgos Controllability of non-scalar parabolic systems
6. A generalization: Cascade systems Sketch of the proof of Theorem 6.1: Given ω0⊂ω, we choose ω1⊂⊂ ω0. Let α0∈C2(Ω) be the function provided by Lemma 2.3 and associated to Ω and B≡ω1. We will do the proof in two steps: Step 1. Let ϕbe the solution to adjoint system associated to ϕT. Each component satisfies −∂tϕi+Li 0(t)ϕi= i X j=1 [∇·(Cjiϕj)−ajiϕj]−ai+1,iϕi+1. We begin applying inequality (11) with B=ω1to each function ϕiwith L0≡Li 0,d=d+3(n−i)and the corresponding right-hand side. Now if we take s≥s0=σ0T+T2+T2max i≤jkaijk 2 3(j−i)+3 ∞+kCijk 2 3(j−i)+1 ∞, with σ0=σ0(Ω,ω0,c0,M0)>0, we obtain the existence of a positive constants C1=C1(Ω,ω0,c0,M0)such that if s≥s0, then M. González-Burgos Controllability of non-scalar parabolic systems
6. A generalization: Cascade systems n X i=1I(d+3(n−i), ϕi)≤C1 n X i=1 ss+3(n−i)ZZω1×(0,T) e−2sαγ(t)s+3(n−i)|ϕi|2. Step 2. Thanks to the assumption ai,i−1≥c0>0 or −ai,i−1≥c0>0 in ω0×(0,T),∀i:2≤i≤n, with ω0⊆ωan open subset, and the cascade structure ai,i−1ϕi=∂tϕi−1−Li−1 0(t)ϕi−1+ i−1 X j=1 [∇·(Cj,i−1ϕj)−aj,i−1ϕi−1]in QT, can eliminate the local terms for 2 ≤i≤n. In order to carry this process out, we will need the following result: M. González-Burgos Controllability of non-scalar parabolic systems
6. A generalization: Cascade systems Lemma Under assumptions of Theorem 6.1 and given l ∈N,ε > 0, k ∈ {2, ..., n}and two open sets O0and O1such that ω1⊂O1⊂⊂ O0⊂ω0, there exist a constant Ck(only depending on Ω,O0,O1,c0and M0) and lkj ∈N, 1≤j≤k−1(only depending on l, n, k and j), such that, if s ≥s0, one has slZZO1×(0,T) e−2sαγ(t)l|ϕk|2≤ε[I(d+3(n−k), ϕk) + I(d+3(n−k−1), ϕk+1)] +Ck1+1 εk−1 X j=1 slkj ZZO0×(0,T) e−2sαγ(t)lkj |ϕj|2. (In this inequality we have taken ϕk+1≡0when k =n). The proof of Theorem 6.1 is a consequence of this Lemma 6.3. For the details, see [DE TERESA], Comm. PDE (2000), [G.-B., PÉREZ-GARCÍA], Asymp. Anal. (2006) and [G.-B., DE TERESA], Port. Math. (2010). M. González-Burgos Controllability of non-scalar parabolic systems
6. A generalization: Cascade systems Remark 1Cascade systems appear in the context of existence of insensitizing controls for a scalar parabolic equation: Equivalent to a null controllability result for a 2 ×2 parabolic system (n=2) with one equation forward in time and the other one backward. The coupling coefficient a21 is 1Owith O⊆Ωan open set and O∩ω6=∅. 2The previous proof uses the assumption ai,i−1≥c0>0 or −ai,i−1≥c0>0 in ω0×(0,T),∀i:2≤i≤n, in a crucial way. When ai,i−1are constant, this assumption is necessary. Is this condition necessary in the general case??? No. 3Is it possible to provide a necessary and sufficient (Kalman condition) condition for the null controllability of non-scalar systems? YES in some constant coefficient systems. M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Let us consider {λk}k≥1the sequence of eigenvalues for L0with homogeneous Dirichlet boundary conditions and {φk}k≥0the corresponding normalized eigenfunctions. Theorem (A Necessary Condition) If system (22) is null controllable at time T then (24) rank [−λkD+A|B] = n,∀k≥1. where [−λkD+A|B]=[B,(−λkD+A)B,(−λkD+A)2B,··· ,(−λkD+A)n−1B]. Proof: Reasoning by contradiction: ∃k≥1 such that rank [−λkD+A|B]<n. Then the o.d.s. −Z0= (−λkD+A∗)Zin (0,T),is not B∗-observable at time T. M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems There exists Z0∈Rn,Z06=0, such that the solution Zto the previous system satisfies B∗Z(·) = 0 on (0,T). But ϕ(x,t) = Z(t)φk(x)is the solution to adjoint problem −∂tϕ+DL0ϕ=A∗ϕin QT, ϕ=0 on ΣT, ϕ(·,T) = ϕ0in Ω, associated to ϕ0(x) = Z0φk6≡ 0 and B∗ϕ(·,·)≡0 in QT. Then, the observability inequality kϕ(·,0)k2 L2(Ω) ≤CTZZω×(0,T)|B∗ϕ(x,t)|2, fails and the system is not null controllable at time T. Remark If condition (24) is not satisfied, then system (22) is neither approximately controllable nor null controllable at time T(for any T>0) even if ω≡Ω. M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Question: Is condition (24) rank [−λkD+A|B] = n,∀k≥1, a sufficient condition for the null controllability of system (22)??? Let us now introduce the unbounded matrix operator K= [DL0+A|B] = [B,(−DL0+A)B,··· ,(−DL0+A)n−1B], (K:D(K)⊂L2(Ω; Rnm)→L2(Ω; Rn),with D(K) := {y∈L2(Ω; Rnm) : Ky∈L2(Ω; Rn)}. Then, Proposition ker K∗={0}if and only if condition (24),rank [−λkD+A|B] = n, ∀k≥1, holds. M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Question: Is condition (24) rank [−λkD+A|B] = n,∀k≥1, a sufficient condition for the null controllability of system (22)??? Let us now introduce the unbounded matrix operator K= [DL0+A|B] = [B,(−DL0+A)B,··· ,(−DL0+A)n−1B], (K:D(K)⊂L2(Ω; Rnm)→L2(Ω; Rn),with D(K) := {y∈L2(Ω; Rnm) : Ky∈L2(Ω; Rn)}. Then, Proposition ker K∗={0}if and only if condition (24),rank [−λkD+A|B] = n, ∀k≥1, holds. M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems (22) (∂ty+DL0y=Ay+Bv1ωin QT, y=0 on ΣT,y(·,0) = y0(·)in Ω, Theorem (Kalman condition) System (22) is exactly controllable to trajectories at time T if and only if System (22) is approximately controllable at time T if and only if ker K∗={0}(⇐⇒ rank [−λkD+A|B] = n, ∀k≥1). Remark One can prove, either there exists k0≥1 such that rank [−λkD+A|B] = n,∀k≥k0 or rank [−λkD+A|B]<n,∀k≥1. M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Controllability (outside a finite dimensional space) if and only if the algebraic Kalman condition rank [−λkD+A|B] = nis satisfied for one frequency k≥1. Remark System (22) can be exactly controlled to the trajectories with one control force (m=1 and B∈Rn) even if A≡0 . Indeed, let us assume that B= (bi)1≤i≤n∈Rn. Then, [(−λkD+A)|B] = b1(−λkd1)b1··· (−λkd1)n−1b1 b2(−λkd2)b2··· (−λkd2)n−1b2 . . .. . ..... . . bn(−λkdn)bn··· (−λkdn)n−1bn ∈ L(Rn), and (24) holds if and only if bi6=0 for every iand diare distinct. M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Idea of the proof: We have proved the necessary condition. Therefore, let us prove that rank [−λkD+A|B] = n, for any k, is a sufficient condition for the null controllability at time Tof the system. Then, the objective is to prove the observability inequality: kϕ(·,0)k2 L2(Ω) ≤CZZω×(0,T)|B∗ϕ(x,t)|2, for the solutions to the adjoint problem. To this end we use two arguments: Prove a global Carleman estimate for a scalar parabolic equation of order nin time. Prove a coercivity property for the Kalman operator K. M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Let us fix ϕ0∈D(Li 0),∀i≥0 and consider ϕthe corresponding solution to the adjoint system (23) −∂tϕ+DL0ϕ=A∗ϕin QT, ϕ=0 on ΣT, ϕ(·,T) = ϕ0in Ω. Let us take Φ = n X i=1 aiϕi,with ai∈R(1 ≤i≤n). Then, Φis a regular solution (Li 0∂j tΦ∈L2(QT),∀i,j) to the linear parabolic scalar equation of order nin time det (Id∂t−DL0+A∗) Φ = 0 in QT, Li 0Φ = 0 on ΣT,∀i≥0. The key point is to prove a Carleman inequality for the solutions to the previous problem. Fix ω0⊂⊂ ωa nonempty open subset. Recall Lemmas 2.3 and 2.4: M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Lemma There exist a α0∈C2(Ω) (positive), and two constants C0,σ0>0(only depending on Ω,ω0and d) s.t. I1(d, φ)≡ZZQT e−2sα[sγ(t)]d−4|φt|2+|L0φ|2 +ZZQT e−2sα[sγ(t)]d−2|∇φ|2+ZZQT e−2sα[sγ(t)]d|φ|2 ≤C0 ZZω0×(0,T) e−2sα[sγ(t)]d|φ|2+ZZQT e−2sα[sγ(t)]d−3|φt±L0φ|2!, ∀s≥s0=σ0(Ω,ω)(T+T2),∀φ∈L2(0,T;H1 0(Ω)) s.t. φt±L0φ∈L2(QT). γ(t) = t−1(T−t)−1,α(x,t) = α0(x)/t(T−t). M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Theorem Let n,k1,k2∈Nand d ∈R. There exist two constants Cand σ(only depending on Ω,ω,n,D,A, k1, k2and d), and r0=r0(n)∈Nsuch that k1 X i=0 k2 X j=0J(d−4(i+j),Li 0∂j tΦ) ≤CZZω×(0,T) e−2sα[sγ(t)]3+r0|Φ|2, , ∀s≥s=σ(Ω,ω)(T+T2),Φsolution to the previous problem and J(τ, z) := I1(τ+3(n−1),z) + n X i=1I1(τ+3(n−2),Piz) + n−1 X p=2X 1≤i1<···<ip≤nI1(τ+3(n−p−1),Pip···Pi1z). (Pi≡∂t−diL0) M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Conclusion If ϕis a regular solution to the adjoint problem −∂tϕ+DL0ϕ=A∗ϕin QT, ϕ=0 on ΣT, ϕ(·,T) = ϕ0in Ω, then, any linear combination Φ = Pn i=1aiϕisatisfies Theorem 10. In particular any component of B∗ϕ. Recall K= [DL0+A|B]=[B,(−DL0+A)B,··· ,(−DL0+A)n−1B],then K∗ϕ(·,t) = [B∗ϕ , B∗(−DL0+A∗)ϕ , ··· ,B∗(−DL0+A∗)n−1ϕ]tr(·,t) = [B∗ϕ , −∂t(B∗ϕ),··· ,(−1)n−1∂n−1 t(B∗ϕ)]tr(·,t)∈Rnm. We apply Theorem 10 with k1=n−1 and k2=k≥0. Then, after some computations, we deduce (d=3) M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Conclusion If ϕis a regular solution to the adjoint problem −∂tϕ+DL0ϕ=A∗ϕin QT, ϕ=0 on ΣT, ϕ(·,T) = ϕ0in Ω, then, any linear combination Φ = Pn i=1aiϕisatisfies Theorem 10. In particular any component of B∗ϕ. Recall K= [DL0+A|B]=[B,(−DL0+A)B,··· ,(−DL0+A)n−1B],then K∗ϕ(·,t) = [B∗ϕ , B∗(−DL0+A∗)ϕ , ··· ,B∗(−DL0+A∗)n−1ϕ]tr(·,t) = [B∗ϕ , −∂t(B∗ϕ),··· ,(−1)n−1∂n−1 t(B∗ϕ)]tr(·,t)∈Rnm. We apply Theorem 10 with k1=n−1 and k2=k≥0. Then, after some computations, we deduce (d=3) M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Then, after some computations, we deduce (d=3) ZT 0 e −2sM0 t(T−t)[sγ(t)]3kLk 0K∗ϕk2 L2(Ω)nm ≤CZZω×(0,T) e−2sα[sγ(t)]3+r0|B∗ϕ|2 for every s≥σT+T2. In this inequality, M0=maxΩα0and r0≥0 is an integer only depending on n. Remark The previous inequality is a partial observability estimate. It is valid even if the Kalman condition does not hold, i.e., even if kerK∗6={0}. M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems The coercivity property of K∗: Theorem Assume that ker K∗={0}and consider k = (n−1)(2n−1). Then there exists C>0such that if z ∈L2(Ω)nsatisfies K∗z∈D(Lk 0)nm, one has kzk2 L2(Ω)n≤CkLk 0K∗zk2 L2(Ω)nm . So, from the previous inequality we get ZT 0 e −2sM0 t(T−t)[sγ(t)]3kϕk2 L2(Ω)nm ≤CZZω×(0,T) e−2sα[sγ(t)]3+r0|B∗ϕ|2 and the observability inequality: kϕ(·,0)k2 L2(Ω) ≤CZZω×(0,T)|B∗ϕ(x,t)|2. M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Summarizing 1We have established a Kalman condition ker K∗={0} which characterizes the controllability properties of system (22). 2The Kalman condition for system (22) kerK∗={0}generalizes the algebraic Kalman condition ker[A|B]∗={0}for o.d.s. 3This Kalman condition is also equivalent to the approximate controllability of system (22) at time T. Again, approximate and null controllability are equivalent concepts for system (22). M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems References 1F. AMMAR-KHODJA, A. BENABDALLAH, C. DUPAIX, M. G.-B.,A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differ. Equ. Appl. 1(2009), no. 3, 139–151. D=Id,A=A(t)and B=B(t). 2F. 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. Ddiagonal matrix,Aand Bconstant matrices. 3E. FERNÁNDEZ-CARA, M. G.-B, L. DE TERESA,Controllability of linear and semilinear non-diagonalizable parabolic systems, ESAIM Control Optim. Calc. Var. 21 (2015), no. 4, 1178–1204. Dnon-diagonalizable matrix with Jordan blocks of dimension ≤4, Aand Bconstant matrices. M. González-Burgos Controllability of non-scalar parabolic systems
7. The Kalman condition for a class of parabolic systems Open problems Null controllability properties of (22) (∂ty+DL0y=A(t)y+B(t)v1ωin QT, y=0 on ΣT,y(·,0) = y0(·)in Ω, when A(t)and B(t)depend on t(for instance, A∈C∞([0,T]; L(Rn)) and B∈C∞([0,T]; L(Rm,Rn))) and D=diag (d1,d2,··· ,dn)∈ L(Rn) with di>0. Null controllability properties of (22) (∂ty+DL0y=Ay+Bv1ωin QT, y=0 on ΣT,y(·,0) = y0(·)in Ω, when Aand Bare constant matrices and Dis a general non-diagonalizable matrix (definite positive). M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls [AMMAR-KHODJA,BENABDALLAH,G.-B.,DE TERESA], J. Math. Pures Appl. (2011). M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls Let us consider the boundary controllability problem: (25) yt=yxx +Ayin QT= (0, π)×(0,T), y(0,·) = Bv,y(π, ·) = 0 on (0,T), y(·,0) = y0in (0, π), where A∈ L(Cn)and B∈ L(Cm;Cn)are two given matrices and y0∈H−1(0, π;Cn)is the initial datum. In system (25), v∈L2(0,T;Cm)is the control function (to be determined). Simpler problem: One-dimensional case and D=Id. This problem has been studied in the case n=2: E. FERNÁNDEZ-CARA, M. G.-B., L. DE TERESA,Boundary controllability of parabolic coupled equations, J. Funct. Anal. 259 (2010), no. 7, 1720–1758. M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls We consider again {λk}k≥1the sequence of eigenvalues for −∂xx in (0, π) with homogenuous Dirichlet boundary conditions and {φk}k≥0the corresponding normalized eigenfunctions: λk=k2, φk(x) = r2 πsin kx,k≥1,x∈(0, π). Theorem (n=2, m=1) Let A∈ L(C2)and B∈C2be given and let us denote by µ1and µ2the eigenvalues of A∗. Then (25) is exactly controllable to the trajectories at any time T >0if and only if rank [A|B] = 2and λk−λj6=µ1−µ2∀k,j∈Nwith k 6=j. M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls Remark (One control, m=1) When m=1, the Kalman condition (27) is equivalent to rank [A|B] = n and λk−λl6=µi−µjfor any k,l∈Nand 1 ≤i,j≤pwith (k,i)6= (l,j), where {µi}1≤i≤p⊂Cis the set of distinct eigenvalues of A∗. We generalize the results of [FERNÁNDEZ-CARA,G.-B.,DE TERESA], J. Funct. Anal. (2010). One control, m=1 We have imposed two conditions: 1rank [A|B] = n: System (25) is not decoupled. 2λk−λl6=µi−µj: The adjoint system can be written (R0=Id∂xx +A∗) (26) −ϕt=R0ϕin QT, ϕ=0 on ΣT, ϕ(·,T) = ϕ0in (0, π), and the eigenvalues of R0are simple. M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls Before proving the result, let us analyze the Kalman condition (27) rank Kk=nk,∀k≥1: Proposition Let us denote by {µi}1≤i≤p⊂Cthe set of distinct eigenvalues of A∗. Then, 1There exists an integer k0=k0(A)∈N, only depending on A, such that, λk−λl6=µi−µj,∀k>k0,l≥1,k6=l,and 1≤i,j≤p. 2The following conditions are equivalent: (a) rankKk=nk for every k ≥1. (b) rankKk=nk for every k :1≤k≤k0. (c) rankKk0=nk0. M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls Necessary implication. We reason as before: if rank Kk<nk, for some k≥1, then the o.d.s. −Z0=L∗ kZon (0,T),Z(T) = Z0∈Cnk is not B∗ k-observable on (0,T), i.e., there exists Z06=0 s.t. B∗ kZ(t) = 0 for every t∈(0,T). From Z0it is possible to construct ϕ0∈H1 0(0, π;Cn)with ϕ06≡ 0 such that the corresponding solution to the adjoint problem (27) satisfies B∗ϕx(0,t) = 0∀t∈(0,T). As a consequence: The unique continuation property and the previous observability inequality for the adjoint problem fail: Neither approximate nor null controllability at any Tfor system (25). M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls Sufficient implication. For the proof we follow the ideas from H.O. FATTORINI, D.L. RUSSELL,Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal. 43 (1971), 272–292. Two “big” steps: (I) We reformulate the null controllability problem for system (25) as a vector moment problem. (II) Existence and bounds of a family biorthogonal to appropriate complex matrix exponentials. M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls (I) The vector moment problem: As in the scalar case, v∈L2(0,T;Cm)is a null control for system (25) yt=yxx +Ayin QT, y(0,·) = Bv,y(π, ·) = 0 on (0,T), y(·,0) = y0in (0, π), (i.e., the solution yto (25) satisfies y(·,T) = 0 in (0, π))⇐⇒ vsatisfies −hy0, ϕ(·,0)i=ZT 0 (v(t),B∗ϕx(0,t))Cmdt,∀ϕ0∈H1 0(0, π;Cn), where ϕis the solution to the adjoint problem (26) −ϕt=ϕxx +A∗ϕin QT, ϕ(0,·) = ϕ(π, ·) = 0 on (0,T), ϕ(·,T) = ϕ0in (0, π). M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls (I) The vector moment problem: Thus, the idea is to take firstly ϕ0∈Xk0, (Xk0={ϕ0:ϕ0=Pk0 i=1aiφiwith ai∈Cn}) and then ϕ0=aφk, with k>k0 and a∈Cn. Therefore, we want v∈L2(0,T;Cm)s.t. ZT 0 (v(T−t),B∗ k0eL∗ k0tΦ0)Cmdt =F(Y0,Φ0),∀Φ0∈Cnk0, ZT 0 (v(T−t),B∗e(−λkId+A∗)ta)Cmdt =fk(y0,a),∀a∈Cn,∀k>k0, In some sense, vhas to solve an infinite number of null controllability problems for appropriate o.d. systems: Y0=Lk0Y+Bk0von (0,T),Y(0) = Y0; Z0= (−λkId+A)Z+Bvon (0,T),Z(0) = y0k:= (y0, φk),∀k>k0. M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls (II) Biorthogonal families to appropriate complex matrix exponentials. From the previous step, we have obtained the complex matrix exponentials eL∗ k0tand {e(−λkId+A∗)t}k>k0. Let us denote {γ`}1≤`≤e p⊂Cthe set of distinct eigenvalues of L∗ k0and recall that {µi}1≤i≤p⊂Cis the set of distinct eigenvalues of A∗. Then, the set Λ={γ`}1≤`≤e p∪{−λk+µi}k>k0,1≤i≤pis the set of eigenvalues of the operator ∂xxId +A∗. Thus, our next purpose is: Objective As in the scalar case, construction of a biorthogonal family in L2(0,T;C)to ntjeγ`t,tje(−λk+µi)t:1≤`≤ep,1≤i≤p,0≤j≤η−1,k>k0o, which satisfies appropriate bounds (see (22)). In the previous expression, ηis the maximal dimension of the Jordan blocks associated to γ`and µi. M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls (II) Biorthogonal families to appropriate complex matrix exponentials. Let us fix η≥1, an integer, T∈(0,∞]and {Λk}k≥1⊂C+a sequence s.t. Λk6=Λj,∀k,j≥with k6=j. Let us recall that the family {qk,j}k≥1,0≤j≤η−1⊂L2(0,T;C)is biorthogonal to {tje−Λkt}k≥1,0≤j≤η−1if one has ZT 0 tje−Λktq∗ l,i(t)dt =δklδij,∀(k,j),(l,i) : k,l≥1,0≤i,j≤η−1. In addition, we want the family {qk,j}k≥1,0≤j≤η−1⊂L2(0,T;C)to satisfy the property: For any ε>0, there is C(ε,T)>0 s.t. kqk,jkL2(0,T;C)≤C(ε,T)eε<Λk, ∀k≥1 and 0 ≤j≤η−1. M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls (II) Biorthogonal families to appropriate complex matrix exponentials. Theorem Let us fix T ∈(0,∞]and assume that for two positive constants δand ρone has <Λk≥δ|Λk|,|Λk−Λl| ≥ ρ|k−l|,∀k,l≥1, X k≥1 1 |Λk|<∞. Then, ∃{qk,j}k≥1,0≤j≤η−1biorthogonal to tje−Λktk≥1,0≤j≤η−1such that, for every ε>0, there exists C(ε,T)>0satisfying kqk,jkL2(0,T;C)≤C(ε,T)eε<Λk,∀(k,j) : k≥1,0≤j≤η−1. M. González-Burgos Controllability of non-scalar parabolic systems
8. The Kalman condition for a class of parabolic systems. Boundary controls (II) Biorthogonal families to appropriate complex matrix exponentials. Proof: The proof of this result is very technical. It can be found in [AMMAR-KHODJA,BENABDALLAH,G.-B.,DE TERESA], The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl. (2011). M. González-Burgos Controllability of non-scalar parabolic systems
9. New phenomena: Minimal time of controllability We are going to revisited problem (18). With a slightly change of notations, this problem is: (18) yt−Dyxx +A0y=0 in QT= (0, π)×(0,T), y(0,·) = Bv,y(π, ·) = 0 on (0,T), y(·,0) = y0in (0, π), where D=diag (1,d),A0=0 1 0 0 ,B=0 1.When d=1 (i.e., D=Id), we saw Theorem (d=1) Let A0∈ L(C2)and B∈C2be given and let us denote by µ1and µ2the eigenvalues of A∗ 0. Then (18) is approximate and null controllable at any time T >0if and only if rank [A|B] = 2and (λk=k2) λk−λj6=µ1−µ2∀k,j∈Nwith k 6=j. M. González-Burgos Controllability of non-scalar parabolic systems
9. New phenomena: Minimal time of controllability (18) yt−Dyxx +A0y=0 in QT= (0, π)×(0,T), y(0,·) = Bv,y(π, ·) = 0 on (0,T), y(·,0) = y0in (0, π), where D=diag (1,d),A0=0 1 0 0 ,B=0 1. Theorem (d6=1) Under the previous assumptions, system (18) is approximate controllable at time T >0if and only if √d6∈ Q. Therefore: 1If d=1, (18) is approximate and null controllable at any T>0. 2If d6=1, we only know that system (18) is approximate controllable at time T>0if and only if √d6∈ Q. M. González-Burgos Controllability of non-scalar parabolic systems
9. New phenomena: Minimal time of controllability (18) yt−Dyxx +A0y=0 in QT, y(0,·) = Bv,y(π, ·) = 0 on (0,T), y(·,0) = y0in (0, π), where D=diag (1,d),A0=0 1 0 0 ,B=0 1 Assumption In the sequel, D=diag (1,d)with d6=1 and √d6∈ Q. Goal Analyze the null controllability properties at time T>0 of system (18). M. González-Burgos Controllability of non-scalar parabolic systems
9. New phenomena: Minimal time of controllability (18) yt−Dyxx +A0y=0 in QT, y(0,·) = Bv,y(π, ·) = 0 on (0,T), y(·,0) = y0in (0, π), Let ϕbe a solution of the adjoint problem: −ϕt−Dϕxx +A∗ 0ϕ=0 in QT, ϕ(0,·) = ϕ(π, ·) = 0 on (0,T), ϕ(·,T) = ϕ0∈H1 0(0, π)2in (0, π). If yis a solution of the direct problem, then hy(T), ϕ0i−hy0, ϕ(0)i=ZT 0 v(t)B∗Dϕx(0,t)dt Thus y(T) = 0⇐⇒ ∃v∈L2(0,T)such that ZT 0 v(t)B∗Dϕx(0,t)dt =−hy0, ϕ(0)i,∀ϕ0∈H1 0(0, π;R2) M. González-Burgos Controllability of non-scalar parabolic systems
9. New phenomena: Minimal time of controllability (18) yt−Dyxx +A0y=0 in QT, y(0,·) = Bv,y(π, ·) = 0 on (0,T), y(·,0) = y0in (0, π), Let ϕbe a solution of the adjoint problem: −ϕt−Dϕxx +A∗ 0ϕ=0 in QT, ϕ(0,·) = ϕ(π, ·) = 0 on (0,T), ϕ(·,T) = ϕ0∈H1 0(0, π)2in (0, π). If yis a solution of the direct problem, then hy(T), ϕ0i−hy0, ϕ(0)i=ZT 0 v(t)B∗Dϕx(0,t)dt Thus y(T) = 0⇐⇒ ∃v∈L2(0,T)such that ZT 0 v(t)B∗Dϕx(0,t)dt =−hy0, ϕ(0)i,∀ϕ0∈H1 0(0, π;R2) M. González-Burgos Controllability of non-scalar parabolic systems
9. New phenomena: Minimal time of controllability Fattorini-Russell Method M. González-Burgos Controllability of non-scalar parabolic systems
9. New phenomena: Minimal time of controllability Fattorini-Russell Method σ(−D∂2 xx +A∗ 0) = Sk≥1k2,dk2:= Sk≥1{λk,1,λk,2}. {Φk,i}a (Riesz) basis of H1 0(0, π)2, where Φk,i=Vk,isin kx,i=1,2 are eigenfunctions of the operator −D∂2 xx +A∗ 0. Vk,1and Vk,2: eigenvectors of the matrix k2D+A∗ 0associated to the eigenvalues k2,dk2. M. González-Burgos Controllability of non-scalar parabolic systems
9. New phenomena: Minimal time of controllability (18) yt−Dyxx +A0y=0 in QT, y(0,·) = Bv,y(π, ·) = 0 on (0,T), y(·,0) = y0in (0, π), Objective: Existence of v∈L2(0,T)s.t. ZT 0 v(t)B∗Dϕx(0,t)dt =−hy0, ϕ(0)i,∀ϕ0∈H1 0(0, π;R2) Choosing ϕ0=Φk,i,we have ϕ(·,t) = e−λk,i(T−t)Φk,iand ϕ(x,0) = e−λk,iTΦk,i(x), ϕx(0,t) = ke−λk,i(T−t)Vk,i The identity connecting yand ϕwrites (moment problem) kB∗DVk,iZT 0 v(T−t)e−λk,itdt =−e−λk,iThy0,Φk,ii,∀(k,i) M. González-Burgos Controllability of non-scalar parabolic systems
9. New phenomena: Minimal time of controllability (18) yt−Dyxx +A0y=0 in QT, y(0,·) = Bv,y(π, ·) = 0 on (0,T), y(·,0) = y0in (0, π), Objective: Existence of v∈L2(0,T)s.t. ZT 0 v(t)B∗Dϕx(0,t)dt =−hy0, ϕ(0)i,∀ϕ0∈H1 0(0, π;R2) Choosing ϕ0=Φk,i,we have ϕ(·,t) = e−λk,i(T−t)Φk,iand ϕ(x,0) = e−λk,iTΦk,i(x), ϕx(0,t) = ke−λk,i(T−t)Vk,i The identity connecting yand ϕwrites (moment problem) kB∗DVk,iZT 0 v(T−t)e−λk,itdt =−e−λk,iThy0,Φk,ii,∀(k,i) M. González-Burgos Controllability of non-scalar parabolic systems
9. New phenomena: Minimal time of controllability (18) yt−Dyxx +A0y=0 in QT, y(0,·) = Bv,y(π, ·) = 0 on (0,T), y(·,0) = y0in (0, π), Approximate controllability: a necessary condition (I) kB∗DVk,iZT 0 v(T−t)e−λk,itdt =−e−λk,iThy0,Φk,ii,∀(k,i) A necessary condition: B∗DVk,i6=0 for all k≥1,i=1,2 Recall d6=1 , B∗= (0,1),Vk,1= 1 1 (d−1)k2!,Vk,2=0 1,∀k≥1. So, here B∗DVk,i6=0,∀k≥1,i=1,2 (algebraic Kalman condition) M. González-Burgos Controllability of non-scalar parabolic systems