Some recent results on controllability of coupled parabolic systems: towards a Kalman condition

Some recent results on controllability of coupled parabolic systems: Towards a Kalman condition Manuel González-Burgos, UNIVERSIDAD DE SEVILLA Thematic day on Control of Coupled Systems Institut Henri Poincaré, Paris, November 2010 M. González-Burgos Results on controllability of coupled parabolic systems GOAL: 1Show the important differences between scalar and non scalar problems. 2Give necessary and sufficient conditions (Kalman condition) which characterize the controllability properties of these systems. We will only deal with “Simple" Parabolic Systems: Coupling Matrices of Constant Coefficients. M. González-Burgos Results on controllability of coupled parabolic systems Contents 1The parabolic scalar case: The heat equation 2Finite-dimensional systems 3Two simple examples Distributed null controllability of a linear reaction-diffusion system Boundary null controllability of a linear reaction-diffusion system 4The Kalman condition for a class of parabolic systems. Distributed controls 5The Kalman condition for a class of parabolic systems. Boundary controls 6Comments and open problems M. González-Burgos Results on controllability of coupled parabolic systems 1. The parabolic scalar case: The heat equation Let Ω⊂RNbe a bounded domain, N≥1, with boundary ∂Ωof class C2. Let ω⊆Ωbe an open subset, γ⊆∂Ωa relative open subset and let us fix T>0. We consider the linear problems for the heat equation: (1)      ∂ty−∆y=v1ωin Q= Ω ×(0,T), y=0 on Σ = ∂Ω×(0,T), y(·,0) = y0in Ω, (2) (∂ty−∆y=0 in Q, y=v1γon Σ,y(·,0) = y0in Ω. In (1) and (2), 1ωand 1γrepresent resp. the characteristic function of the sets ωand γ,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- or in γ-boundary control-). M. González-Burgos Results on controllability of coupled parabolic systems 1. The parabolic scalar case: The heat equation Let Ω⊂RNbe a bounded domain, N≥1, with boundary ∂Ωof class C2. Let ω⊆Ωbe an open subset, γ⊆∂Ωa relative open subset and let us fix T>0. We consider the linear problems for the heat equation: (1)      ∂ty−∆y=v1ωin Q= Ω ×(0,T), y=0 on Σ = ∂Ω×(0,T), y(·,0) = y0in Ω, (2) (∂ty−∆y=0 in Q, y=v1γon Σ,y(·,0) = y0in Ω. In (1) and (2), 1ωand 1γrepresent resp. the characteristic function of the sets ωand γ,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- or in γ-boundary control-). M. González-Burgos Results on controllability of coupled parabolic systems 1. The parabolic scalar case: The heat equation Let Ω⊂RNbe a bounded domain, N≥1, with boundary ∂Ωof class C2. Let ω⊆Ωbe an open subset, γ⊆∂Ωa relative open subset and let us fix T>0. We consider the linear problems for the heat equation: (1)      ∂ty−∆y=v1ωin Q= Ω ×(0,T), y=0 on Σ = ∂Ω×(0,T), y(·,0) = y0in Ω, (2) (∂ty−∆y=0 in Q, y=v1γon Σ,y(·,0) = y0in Ω. In (1) and (2), 1ωand 1γrepresent resp. the characteristic function of the sets ωand γ,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- or in γ-boundary control-). M. González-Burgos Results on controllability of coupled parabolic systems 1. The parabolic scalar case: The heat equation Theorem (Distributed Controllability Results) Fix ω⊆Ωand T >0. Then, 1System (1) is approximately controllable at time T, i.e., for any ε>0 and y0,yd∈L2(Ω) there is v∈L2(Q)s.t. the solution y to (1) satisfies ky(·,T)−ydkL2(Ω) ≤ε. 2System (1) is null controllable at time T, i.e., for any y0∈L2(Ω) there is v∈L2(Q)s.t. the solution y to (1) satisfies y(·,T)≡0in Ω. Remark System (1) is null controllable at time T if and only if system (1) is exactly controllable to the trajectories at time T: for every trajectory y∗of (1) (a solution to (1) associated to y∗ 0∈L2(Ω)) there exists v∈L2(Q)such that y(·,T)≡y∗(·,T)in Ω. M. González-Burgos Results on controllability of coupled parabolic systems 1. The parabolic scalar case: The heat equation Theorem (Distributed Controllability Results) Fix ω⊆Ωand T >0. Then, 1System (1) is approximately controllable at time T, i.e., for any ε>0 and y0,yd∈L2(Ω) there is v∈L2(Q)s.t. the solution y to (1) satisfies ky(·,T)−ydkL2(Ω) ≤ε. 2System (1) is null controllable at time T, i.e., for any y0∈L2(Ω) there is v∈L2(Q)s.t. the solution y to (1) satisfies y(·,T)≡0in Ω. Remark System (1) is null controllable at time T if and only if system (1) is exactly controllable to the trajectories at time T: for every trajectory y∗of (1) (a solution to (1) associated to y∗ 0∈L2(Ω)) there exists v∈L2(Q)such that y(·,T)≡y∗(·,T)in Ω. M. González-Burgos Results on controllability of coupled parabolic systems 1. The parabolic scalar case: The heat equation Adjoint Problem: Let us fix ϕ0∈L2(Ω) and consider the adjoint problem (3) (∂tϕ+ ∆ϕ=0 in Q, ϕ=0 on Σ, ϕ(T) = ϕ0in Ω. It is well known: Theorem System (1) is exactly controllable to trajectories at time T if and only if there 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 Results on controllability of coupled parabolic systems 2. Finite-dimensional systems Let us consider the autonomous linear system (4) y0=Ay+Buin [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∈Rn, is there a control u∈L2(0,T;Rm)such that the solution yto the problem satisfies y(T) = yd???? Let us define (controllability matrix) [A|B]=[B|AB |A2B| · · · | An−1B]∈ L(Cnm;Cn). M. González-Burgos Results on controllability of coupled parabolic systems 2. Finite-dimensional systems Let us consider the autonomous linear system (4) y0=Ay+Buin [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∈Rn, is there a control u∈L2(0,T;Rm)such that the solution yto the problem satisfies y(T) = yd???? Let us define (controllability matrix) [A|B]=[B|AB |A2B| · · · | An−1B]∈ L(Cnm;Cn). M. González-Burgos Results on controllability of coupled parabolic systems 2. 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 (4) is exactly controllable at time T, for every T >0. 2There exists T >0such that system (4) is exactly controllable at time T. 3rank [A|B] = n (Kalman rank condition). 4ker[A|B]∗={0}. M. González-Burgos Results on controllability of coupled parabolic systems 2. Finite-dimensional systems Goal We have a complete characterization of the controllability results for finite-dimensional linear differential systems (a Kalman condition). Is it possible to obtain similar results for PDE systems? We will focus on coupled linear parabolic systems. M. González-Burgos Results on controllability of coupled parabolic systems 2. Finite-dimensional systems What are the possible generalizations to Systems of Parabolic Equations? M. González-Burgos Results on controllability of coupled parabolic systems 3. Two simple examples 3.1 Distributed null controllability of a linear reaction-diffusion system Let us consider the 2 ×2 linear reaction-diffusion system (5)    yt−D∆y=a1,1a1,2 a2,1a2,2y+1 0v1ωin Q, y=0 on Σ,y(·,0) = y0in Ω. Here Ω,ωand Tare as before, y0∈L2(Ω; R2)and D=d10 0d2,d1,d2>0(A=a1,1a1,2 a2,1a2,2,B=1 0). One has Theorem System (5) is exactly controllable to trajectories at time T if and only if det [A|B]6=0⇐⇒ a2,16=0. M. González-Burgos Results on controllability of coupled parabolic systems 3. Two simple examples 3.1 Distributed null controllability of a linear reaction-diffusion system Let us consider the 2 ×2 linear reaction-diffusion system (5)    yt−D∆y=a1,1a1,2 a2,1a2,2y+1 0v1ωin Q, y=0 on Σ,y(·,0) = y0in Ω. Here Ω,ωand Tare as before, y0∈L2(Ω; R2)and D=d10 0d2,d1,d2>0(A=a1,1a1,2 a2,1a2,2,B=1 0). One has Theorem System (5) is exactly controllable to trajectories at time T if and only if det [A|B]6=0⇐⇒ a2,16=0. M. González-Burgos Results on controllability of coupled parabolic systems 3. Two simple examples 3.1 Distributed null controllability of a linear reaction-diffusion system Proof: =⇒: If a2,1=0, then y2is independent of v. ⇐=: The controllability result for system (5) is equivalent to the observability inequality:∃C>0 such that kϕ1(·,0)k2 L2+kϕ2(·,0)k2 L2≤CZZω×(0,T) |ϕ1(x,t)|2dx dt, where ϕis the solution associated to ϕ0∈L2(Ω; R2)of the adjoint problem: (6) −ϕt−D∆ϕ=A∗ϕin Q, ϕ=0 on Σ, ϕ(·,T) = ϕ0in Ω. It is a consequence of well known global Carleman estimates for parabolic equations. M. González-Burgos Results on controllability of coupled parabolic systems 3. Two simple examples 3.1 Distributed null controllability of a linear reaction-diffusion system Proof: =⇒: If a2,1=0, then y2is independent of v. ⇐=: The controllability result for system (5) is equivalent to the observability inequality:∃C>0 such that kϕ1(·,0)k2 L2+kϕ2(·,0)k2 L2≤CZZω×(0,T) |ϕ1(x,t)|2dx dt, where ϕis the solution associated to ϕ0∈L2(Ω; R2)of the adjoint problem: (6) −ϕt−D∆ϕ=A∗ϕin Q, ϕ=0 on Σ, ϕ(·,T) = ϕ0in Ω. It is a consequence of well known global Carleman estimates for parabolic equations. M. González-Burgos Results on controllability of coupled parabolic systems 3. Two simple examples 3.1 Distributed null controllability of a linear reaction-diffusion system Lemma There exist a positive regular function, α0, and two positive constants C0and σ0(only depending on Ωand ω) s.t.                    I(φ)≡ZZQ e−2sα[sρ(t)]−1|φt|2+|∆φ|2 +ZZQ e−2sα[sρ(t)] |∇φ|2+ZZQ e−2sα[sρ(t)]3|φ|2 ≤C0 ZZω×(0,T) e−2sα[sρ(t)]3|φ|2+ZZQ e−2sα|φt±∆φ|2!, ∀s≥s0=σ0(Ω,ω)(T+T2)and φ∈L2(0,T;H1 0(Ω)) s.t. φt±∆φ∈L2(Q). The functions ρ(t)and α=α(x,t)are given by ρ(t)=[t(T−t)]−1,α(x,t) = α0(x)/t(T−t). M. González-Burgos Results on controllability of coupled parabolic systems 3. Two simple examples 3.2 Boundary null controllability of a linear reaction-diffusion system As before, system (7) 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)≤CZT 0 |ϕ1,x(0,t)|2dt, holds. Again ϕis the solution associated to ϕ0∈H1 0(0, π;R2)of the adjoint problem: (8)    −ϕt−Dϕxx =A∗ϕin Q, ϕ|x=0=ϕ|x=1=0 on (0,T), ϕ(·,T) = ϕ0in (0, π). Let us see that, in general, this inequality fails (even if a2,1=16=0!!!!!). M. González-Burgos Results on controllability of coupled parabolic systems 3. Two simple examples 3.2 Boundary null controllability of a linear reaction-diffusion system A necessary condition: Proposition Assume that system (7) is null controllable at time T. 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 (8) into an o.d. system. Let us consider the sequence of 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 Results on controllability of coupled parabolic systems 3. Two simple examples 3.2 Boundary null controllability of a linear reaction-diffusion system A necessary condition: Proposition Assume that system (7) is null controllable at time T. 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 (8) into an o.d. system. Let us consider the sequence of 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 Results on controllability of coupled parabolic systems 3. Two simple examples 3.2 Boundary null controllability of a linear reaction-diffusion system Consider also BK=   B . . . B   ∈R2K,(B=1 0)and L∗ K=diag (−λ1D+A∗,−λ2D+A∗,· · · ,−λKD+A∗)∈ L(R2K). Taking in (8) arbitrary initial data ϕ0,K=PK `=1a`φ`∈H1 0(0, π;R2)where a`∈R2,it is not difficult to see that system (8) is equivalent to the o.d. system (9) −Z0=L∗ KZon [0,T],Z(0) = Z0∈R2K. From the observability inequality for system (8) we deduce the unique continuation property for the solutions to (9): B∗ KZ(·) = 0 in (0,T)=⇒Z≡0. M. González-Burgos Results on controllability of coupled parabolic systems 3. Two simple examples 3.2 Boundary null controllability of a linear reaction-diffusion system In particular system Y0=LKY+BKvon [0,T],Y(0) = Y0∈R2K. is exactly controllable at time T. Then rank [LK|BK] = 2K. We deduce that L∗ Kcannot have eigenvalues with geometric multiplicity 2 or greater. But θ=−d1λk=−d2λjis an eigenvalue of L∗ Kwith two linearly independent eigenvectors V1,V2∈R2Kgiven by:          V1= (V1,`)1≤`≤K,V1,k=1 0and V1,` =0∀`6=k, V2= (V2,`)1≤`≤K,V2,j= 1 λj(d1−d2) 0!and V2,` =0∀`6=j. M. González-Burgos Results on controllability of coupled parabolic systems 3. Two simple examples 3.2 Boundary null controllability of a linear reaction-diffusion system The result has been proved in E. FERNÁNDEZ-CARA, M. G.-B., L. DE TERESA,Boundary controllability of parabolic coupled equations, J. Funct. Anal. 259 (2010), no. 7, 1720–1758. Remark Again, the system is always null controllable at time T if we exert two controls. In fact, system (7) is approximately controllable at time T ⇐⇒ pd1/d26∈ Q. M. González-Burgos Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls Let Ω⊂RNbe a bounded domain, N≥1, with boundary ∂Ωof class C2. Let ω⊆Ωbe an open subset and let us fix T>0. For n,m∈Nwe consider the following n×nparabolic system (10) (∂ty−D∆y=Ay+Bv1ωin Q, y=0 on Σ,y(·,0) = y0(·)in Ω, with A∈ L(Rn)and B∈ L(Rm;Rn)are constant matrices y0∈L2(Ω; Rn)and D=diag (d1,d2,· · · ,dn)∈ L(Rn),(di>0,∀i). v∈L2(Q;Rm)is the control (mcomponents). Remark This problem is well posed: For any y0∈L2(Ω; Rn)and v∈L2(Q;Rm), problem (10) has a unique solution y∈L2(0,T;H1 0)∩C0([0,T]; L2). M. González-Burgos Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls (10) (∂ty−D∆y=Ay+Bv1ωin Q, y=0 on Σ,y(·,0) = y0(·)in Ω. Remark We want to control the whole system (n equations) with m controls. The most interesting case is m <n or even m =1. Difficulties: 1In general m <n. 2Dis not the identity matrix. M. González-Burgos Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls The adjoint problem: (11) −∂tϕ= (D∆ + A∗)ϕin Q, ϕ=0 on Σ, ϕ(·,T) = ϕ0in Ω, where ϕ0∈L2(Ω; Rn). Then, the exact controllability to the trajectories of system (10) is equivalent to the existence of C>0 such that, for every ϕ0∈L2(Ω; Rn), the solution ϕ∈C0([0,T]; L2(Ω; Rn)) to the adjoint system (11) satisfies the observability inequality: (12) ||ϕ(·,0)||2 L2(Ω) ≤CZZω×(0,T) |B∗ϕ(x,t)|2, M. González-Burgos Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls Let us consider {λk}k≥1the sequence of eigenvalues for −∆with homogenuous Dirichlet boundary conditions and {φk}k≥0the corresponding normalized eigenfunctions. Theorem (A Necessary Condition) If system (10) is null controllable at time T then (13) 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 Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls Idea of the proof: The objective is to prove the observability inequality (12): (12) ||ϕ(·,0)||2 L2(Ω) ≤CZZω×(0,T) |B∗ϕ(x,t)|2. To this end we use two arguments: Prove a Carleman type observability estimate for a scalar equation of order nin time, Prove a coercivity property for the Kalman operator K. M. González-Burgos Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls Let us consider ϕa regular solution of the adjoint system (11) and take Φ = n X i=1 αiϕi,with αi∈R∀i:1≤i≤n. Then, Φis a regular solution to the linear scalar equation of order nin time det (Id∂t+D∆ + A∗) Φ = 0 in Q, ∆jΦ = 0 on Σ,∀j≥1. M. González-Burgos Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls The key point is to prove a Carleman inequality for the solutions to the previous problem: Theorem Let n,k1,k2∈N. There exist two constants r0and C(only depending on Ω,ω, n, D,A, k1and k2) such that k1 X i=0 k2 X j=0 J(3−4(i+j),∆i∂j tΦ) ≤CZZω×(0,T) e−2sα[sρ(t)]3+r0|Φ|2, , ∀s≥s0=σ0(Ω,ω)(T+T2)(see Lemma 6) and Φsolution to the previous problem. M. González-Burgos Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls From this result and after some operations, one deduces ZT 0 e −2sM0 t(T−t)[sρ(t)]3k∆kK∗ϕk2 L2(Ω)nm ≤CZZω×(0,T) e−2sα[sρ(t)]3+r|B∗ϕ|2 for every s≥σ0T+T2. In this inequality, ρand αare as in Lemma 6, M0=maxΩα0and r≥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 ker K∗6={0}. M. González-Burgos Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls 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(∆k)nm, one has kzk2 L2(Ω)n≤Ck∆kK∗zk2 L2(Ω)nm . So, from the previous inequality we get ZT 0 e −2sM0 t(T−t)[sρ(t)]3kϕk2 L2(Ω)nm ≤CZZω×(0,T) e−2sα[sρ(t)]3+r|B∗ϕ|2 and the observability inequality (12): (12) ||ϕ(·,0)||2 L2(Ω) ≤CZZω×(0,T) |B∗ϕ(x,t)|2. M. González-Burgos Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls Summarizing 1We have established a Kalman condition ker K∗={0} which characterizes the controllability properties of system (10). 2The Kalman condition for system (10) 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 (10) at time T. Again, approximate and null controllability are equivalent concepts for system (10). M. González-Burgos Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls A special case:D=Id. It is possible to get better results when D=Id. In this case system (10) is given by (14) (∂ty−∆y=Ay+Bv1ωin Q, y=0 on Σ,y(·,0) = y0(·)in Ω, where again A∈ L(Rn)and B∈ L(Rm;Rn)are constant matrices and y0∈L2(Ω; Rn)is given. In this case, kerK∗={0}is equivalent to the algebraic Kalman condition rank [A|B] = rank [B|AB |A2B| · · · | An−1B] = n. In this case we can obtain a better Carleman inequality for the adjoint system (15) (−∂tϕ−∆ϕ=A∗ϕin Q, ϕ=0 on Σ, ϕ(·,T) = ϕ0(·)in Ω. M. González-Burgos Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls Theorem There exist a positive function α0∈C2(Ω) (only depending on Ωand ω), positive constants C0and σ0(only depending on Ω,ω,n,m,Aand B) and a positive integer `≥3(only depending on n and m) such that, if rank [A|B] = n, for every ϕ0∈L2(Q;Rn), the solution ϕto (15) satisfies I(ϕ)≤C0s`ZZω×(0,T) e−2sαρ(t)`|B∗ϕ|2, ∀s≥s0=σ0T+T2. In this inequality, α(x,t),ρ(t)and I(z)are as in Lemma 6. M. González-Burgos Results on controllability of coupled parabolic systems 4. The Kalman condition for a class of parabolic systems. Distributed controls 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. M. González-Burgos Results on controllability of coupled parabolic systems 5. The Kalman condition for a class of parabolic systems. Boundary controls Let us consider the boundary controllability problem: (16)      yt=yxx +Ayin Q= (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 (16), 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 Results on controllability of coupled parabolic systems 5. 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: 1We reformulate the null controllability problem for system (16) as a vector moment problem. 2Existence and bounds of a family biorthogonal to appropriate complex matrix exponentials. Let us fix η≥1, an integer, T∈(0,∞]and {Λk}k≥1⊂Ca sequence. Let us recall that the family {ϕk,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−Λktϕ∗ l,i(t)dt =δklδij,∀(k,j),(l,i) : k,l≥1,0≤i,j≤η−1. M. González-Burgos Results on controllability of coupled parabolic systems 5. 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: 1We reformulate the null controllability problem for system (16) as a vector moment problem. 2Existence and bounds of a family biorthogonal to appropriate complex matrix exponentials. Let us fix η≥1, an integer, T∈(0,∞]and {Λk}k≥1⊂Ca sequence. Let us recall that the family {ϕk,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−Λktϕ∗ l,i(t)dt =δklδij,∀(k,j),(l,i) : k,l≥1,0≤i,j≤η−1. M. González-Burgos Results on controllability of coupled parabolic systems 5. The Kalman condition for a class of parabolic systems. Boundary controls Theorem 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, ∃ {ϕk,j}k≥1,0≤j≤η−1biorthogonal to tje−Λktk≥1,0≤j≤η−1such that, for every ε>0, there exists C(ε,T)>0satisfying kϕk,jkL2(0,T;C)≤C(ε,T)eε<Λk,∀(k,j) : k≥1,0≤j≤η−1. M. González-Burgos Results on controllability of coupled parabolic systems 5. The Kalman condition for a class of parabolic systems. Boundary controls Reference F. AMMAR-KHODJA, A. BENABDALLAH, M. G.-B., L. DE TERESA,The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, submitted. M. González-Burgos Results on controllability of coupled parabolic systems 6. Comments and open problems Most of the controllability results for parabolic systems are open. Two “simple” open problems A.- Let us consider the distributed controllability problem (10) (∂ty−D∆y=Ay+Idv1ωin Q, y=0 on Σ,y(·,0) = y0(·)in Ω. with A∈ L(Rn)(as before), B=Idand with D∈ L(Rn)a non-symmetric matrix such that the Jordan canonical form Jis real and positive definite, i.e., J∈ L(Rn)and ξJξ∗>0,∀ξ∈Rn, ξ 6=0. Some partial results by E. FERNÁNDEZ-CARA, M. G.-B., L. DE TERESA, in preparation. M. González-Burgos Results on controllability of coupled parabolic systems 6. Comments and open problems Most of the controllability results for parabolic systems are open. Two “simple” open problems A.- Let us consider the distributed controllability problem (10) (∂ty−D∆y=Ay+Idv1ωin Q, y=0 on Σ,y(·,0) = y0(·)in Ω. with A∈ L(Rn)(as before), B=Idand with D∈ L(Rn)a non-symmetric matrix such that the Jordan canonical form Jis real and positive definite, i.e., J∈ L(Rn)and ξJξ∗>0,∀ξ∈Rn, ξ 6=0. Some partial results by E. FERNÁNDEZ-CARA, M. G.-B., L. DE TERESA, in preparation. M. González-Burgos Results on controllability of coupled parabolic systems 6. Comments and open problems B.- Consider again the boundary controllability problem    yt−Dyxx =Ayin Q= (0, π)×(0,T), y|x=0=Bv,y|x=1=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,B=1 0and A=0 0 1 0 . We know: d1=d2:Approximate and null controllability at time T>0. Kalman condition for general A∈ L(R2)and B∈ L(Rm;R2)(the only interesting case is m=1). d16=d2:Approximate controllability at time T>0⇐⇒ pd1/d26∈ Q. d16=d2: There exist d1,d2such that the null controllability property fails at any time T:F. LUCA, L. DE TERESA, 2011. M. González-Burgos Results on controllability of coupled parabolic systems 6. Comments and open problems C.- Kalman condition: Only in the cases presented here. Other situations ? M. González-Burgos Results on controllability of coupled parabolic systems Thanks for your attention ! ¡ Gracias por vuestra atención ! M. González-Burgos Results on controllability of coupled parabolic systems