Internal feedback stabilization for parabolic systems coupled in zero or first order terms

We consider systems of \begin{document}$ n $\end{document} parabolic equations coupled in zero or first order terms with \begin{document}$ m $\end{document} scalar controls acting through a control matrix \begin{document}$ B $\end{document} . We are interested in stabilization with a control in feedback form. Our approach relies on the approximate controllability of the linearized system, which in turn is related to unique continuation property for the adjoint system. For the unique continuation we establish algebraic Kalman type conditions.

1. Introduction. In this paper we study the local feedback stabilization for systems of parabolic equations in one dimension, i.e. on a bounded interval Ω ⊂ R. The equations are coupled in either first or zero order terms and we consider general boundary conditions, arbitrarily mixing Dirichlet, Neumann and Robin conditions. Under algebraic conditions of Kalman type concerning the coupling matrices of coefficients, we establish finite dimensional feedback stabilization with internal controls distributed in a subdomain ω ⊂ Ω and acting in part of the equations through a control matrix.
Since the case of general coupled systems, and especially those which are coupled only in first order terms, seems more delicate to treat, we consider systems in space dimension one. In our paper we are interested in fact in stabilization through a finite number of finite dimensional feedback controls and the strategy is as follows.
First we linearize the nonlinear system around the stationary state and we prove approximate controllability for it. For systems of two equations we treat the case of couplings in both zero and first order terms and homogeneous Dirichlet boundary conditions. For systems of n ≥ 3 equations, we will treat separately the cases of first or zero order couplings. The approximate controllability is obtained by proving the unique continuation property for the adjoint system under corresponding Kalman type conditions satisfied by the coupling matrix and the control matrix.
We consider an abstract formulation for the given problem as an evolution problem in a Hilbert space. With the result of approximate controllability for the linearized system at hand, we use a spectral decomposition of this Hilbert space with respect to the elliptic operator in a direct sum of closed and invariant subspaces for the semigroup. Moreover, one of these subspaces is finite dimensional, corresponding to the eigenvalues with positive real part (that is the unstable subspace) and the other one is infinite dimensional but stable. With this decomposition of the space we consider the controlled system projected onto these subspaces and we study the controllability of the finite dimensional system. The approximate controllability gives the exact controllability for the system in any time in the finite dimensional subspace and, consequently, complete stabilization for it. We stabilize by a feedback control the finite dimensional projection of the system and we prove, using the norm given by the solution to an appropriate Lyapunov equation, that this finite dimensional feedback control stabilizes the whole nonlinear system.
Concerning previous results on the stabilization of parabolic and parabolic like equations (Navier-Stokes for example) one may consult the monograph of V.Barbu [4]. We mention also for example the papers of V. Barbu and G.Wang [3,10] concerning stabilization of parabolic equations and the papers of V.Barbu, I.Lasiecka, R.Triggiani [9,5,6,7] for the stabilization of Navier-Stokes equations (with either internal or boundary controls). In these references the authors use a Riccati based aproach. In order to obtain finite dimensional stabilization they use a Kalman type condition for a finite dimensional projection; this in turn is obtained through a unique continuation property for systems of eigenfunctions of the elliptic part. There are also studies of stabilization of time dependent solutions, e.g. the papers [8,16], but the approach there is different when dealing with the open loop stabilization for the linearized system.
In our approach we use as an essential argument unique continuation for linear parabolic equations and this is in fact related to global Carleman estimates and the controllability problem. We mention here the controllability of linear parabolic equations with internally distributed controls, which was established by O.Yu.Imanuvilov, A.V.Fursikov (see [12] and references therein), through such observability inequalities for the adjoint equation.
The study of controlled systems of parabolic equations with fewer controls than equations need appropriate Carleman estimates, with partial observations for the adjoint to the linearized system and, in some cases, one may provide algebraic conditions concerning the couplings and control operators in order to obtain appropriate observability inequalities. Kalman type conditions for obtaining Carleman estimates and controllability for systems of parabolic equations coupled in zero order terms, with constant or time depending coupling coefficients, were established in [2,1]. The case of cascade like systems with one control and space depending couplings was studied in [13]. A more recent paper considering coupled linear systems (not only parabolic) and corresponding observation estimates is the paper of E.Zuazua and P.Lissy [17] where the equations in the system are linearly coupled with constant coefficients in the dominant part and/or in the zero order terms. Observability of the system is obtained in terms of a Kalman condition satisfied by the pair of coupling matrix and respectively the control matrix.
The boundary conditions are in general form, homogeneous, mixing Dirichlet, Neumann, Robin boundary conditions for each equation of the system. For a general formulation of these boundary conditions we chose diagonal matrices Γ 1 , Γ 2 , Γ 3 , Γ 4 ∈ M n×n (R) with the properties We consider a stationary solution of the uncontrolled system, denoted by y. The aim of the paper is to find a control in feedback form u = K(y − y), such that it stabilizes the controlled system (2.1) around the stationary state y with respect to a topology to be precised later. This will lead to the study of approximate controllability for a linear system following the unique continuation property for a linear adjoint system. Practically, the unique continuation property for the adjoint linear system will be the main point in proving the stability result. For this purpose we consider the following linear system obtained through linearization of the nonlinear system around the stationary state: In order to give the problem an abstract formulation, let us consider the Hilbert space H = [L 2 (Ω)] n and the operators By standard computations for adjoints to realizations of linear differential operators, one finds that and (2.6) We denoted by L * the formal adjoint L * p = D 2 x p − D x (η 1 (x)p) + η 0 (x)p and by A the transposed matrix of A. Note here that matricesΓ 2 ,Γ 4 may not be diagonal as are Γ 2 , Γ 4 . Remark 1. The operator A is generator of a strongly continuous semigroup but this may not be a contraction semigroup. Also, the operator A generates an analytic semigroup in H. Indeed, A is sum of a selfadjoint operator (y → D 2 x y with domain D(A)) and a lower order perturbation (see [19]). Compactness of the resolvent of A follows in a standard way by using Rellich embedding theorem.
The abstract formulation of the nonlinear system reads: and the linearized system (2.3) becomes The strategy to stabilize the nonlinear system (also found, for example, in [15]) is to construct a finite dimensional feedback control that stabilizes the linear system and then prove, by using an argument based on the Lyapunov equation, that the same control stabilizes the nonlinear system too. A first step towards proving stabilization of the linearized system is obtaining approximate controllability in arbitrary time T > 0 for the linearized system; this in turn is equivalent to the unique continuation property for the adjoint system. The adjoint to the linearized problem is the following: (2.9) The unique continuation property in time T for the above system is expressed in the following form: where B * p = B p| ω , with B denoting the transposed of matrix B.
Our paper is divided into two parts: The first part is dedicated to approximate controllability in given arbitrary time T > 0 and thus to unique continuation property for the adjoint system. This will be discussed in several settings: • systems of two equations and coupling matrices A 0 (x), A 1 (x), possibly nonconstant, and homogeneous Dirichlet boundary conditions (Γ 1 = Γ 3 = O n , Γ 2 = Γ 4 = I n ); • systems of n equations with couplings in zero order terms and constant coupling matrix (A 1 = 0, A 0 = cst); • systems of n equations with couplings in first order terms and constant coupling matrix (A 0 = 0, A 1 = cst).
Kalman type conditions for unique continuation are established.
The second part of the paper is devoted to feedback stabilization and the result is common to all situation when unique continuation property (UCP) is true.
We describe first the results concerning approximate controllability of coupled linear equations.
Systems of two linear equations with first and zero order couplings. We consider systems of two coupled equations for w = (w 1 , w 2 ), with differential operator L with constant coefficients η 0 , η 1 ∈ R and possibly nonconstant couplings, under Dirichlet homogeneous boundary conditions: (2.11) If we consider the notations, for c(x) = 0 in ω, then we have the following results concerning the approximate controllability for the above linear systems: Theorem 2.1. For the linear system (2.11), with η 0 , η 1 ∈ R, if c(x) = 0 for x ∈ ω and one of the following hypotheses holds true (H1) the coefficients of the system are constants in the whole domain, (H2) the coupling coefficients are continuous in Ω, maybe nonconstant, and the function k = k(x) is not constant in ω; then the linear system (2.11) is approximately controllable in time T .
We mention here the paper of M.Duprez and P.Lissy [11] where similar systems of two parabolic equations are considered in higher space dimension with Dirichlet homogeneous boundary conditions and null controllability is proved. When null controllability occurs the approximate controllability is also obtained, but the conditions we ask for approximate controllability are easier to verify than those in [11] and our approach works also in the case of general boundary conditions as mentioned in Remark 5 below.
Systems of n equations with constant couplings. We consider now systems of n coupled parabolic equations in either zero order or first order terms, in space dimension one with constant coupling matrices A 0 or A 1 . (2.14) Parabolic systems with zero order couplings. We consider for the beginning the case where Remark 3. Controllability results and Carleman estimates for such controlled systems were established in [1] under homogeneous Dirichlet boundary conditions and in this case of zero order couplings our result is a consequence of the cited paper.
Here we use general homogeneous boundary conditions and we give a direct proof by reducing the question to the unique continuation result from [20].
Parabolic systems with first order couplings. Consider now the case where A 0 = 0, A 1 ∈ M n×n (R). Approximate controllability result in this case is: Theorem 2.3. Consider the linear system (2.3) with constant coefficients η 0 , η 1 and with constant couplings of order one, A 0 ≡ 0, A 1 ∈ M n×n (R). Suppose also that the following algebraic conditions concerning coupling matrix and matrices entering the boundary conditions are satisfied: Finite dimensional feedback stabilization of coupled parabolic systems.
Stabilization of the linearized system. In either of the cases when approximate controllability is verified, we prove the following feedback stabilization result for the linearized system: one has exponential stabilization: 3. Approximate controllability for linear systems.
3.1. Systems of two equations. Proof of theorem 2.1. We consider the homogeneous backward adjoint system (2.12), (2.9), and we study the unique continuation property for this system since this property is equivalent to the approximate controllability of the linear system (2.11). Now, suppose that p 1 ≡ 0 in (0, T ) × ω. We will treat each case corresponding to hypothesis (H1) and (H2) of Theorem 2.1.
Case 1. We are in the conditions of the hypothesis (H1) of Theorem 2.1, meaning that all coupling coefficients are constants. Since c = 0 in ω, c will not be zero in the entire Ω. Then, the system (2.12) becomes From the first equation of system (3.1) considered on (0, T ) × ω we get where h = γ c since the coefficients of the system are constants. Looking at the second equation of (3.1) on (0, T ) × ω, we obtain that giving that on the subdomain ω we have a separation of variables

Now, if we denote byp
Then the unique continuation property for homogeneous parabolic system verified by D tp1 and D tp2 gives that D tp1 ≡ 0 in (0, T ) × Ω and D tp2 ≡ 0 in (0, T ) × Ω, meaning thatp 1 andp 2 depend only on x in (0, T ) × Ω. Also, D xp1 and D xp2 verify the elliptic system with D xp1 ≡ 0, D xp2 = 0 in (0, T ) × ω, giving the only possibility, by unique continuation result in [20], that D x p 1 ≡ 0, D x p 2 ≡ 0 in (0, T ) × Ω and thus We see now that the boundary conditions forcesp 2 to be zero which implies p 1 ,p 2 ≡ 0 in (0, T )×Ω and thus p 1 , p 2 ≡ 0 which concludes the proof of approximate controllability.
Case 2. We argue now under hypothesis (H2) of Theorem 2.4, meaning that the function k is not constant in the subdomain ω, then, for p 1 ≡ 0 in (0, T ) × ω, the first equation of the system (2.12) gives Then p 2 verifies the second equation of (2.12) and since k is not constant there, then p 2 (t, x 0 ) must be zero in (0, T ), giving that p 2 ≡ 0 in (0, T ) × ω. Then, invoking again the unique continuation property for the parabolic system (2.12) we can conclude that p 1 ≡ 0, p 2 ≡ 0 in (0, T ) × Ω, thus obtaining the unique continuation property in this case.

Remark 5.
Observe that the conclusion of Theorem 2.1 under hypothesis (H2) remains valid under general boundary conditions described in (2.1): the proof makes no use of the imposed Dirichlet homogeneous conditions. We may also consider Theorem 2.1 under hypothesis (H1) with general boundary conditions in (2.1), (2.2). However, in this case, in order to have the conclusion on approximate controllability and correspondingly on unique continuation for the adjoint problem, one has to insure that the problem (3.4) has as unique solution with first componentp 1 ≡ 0 and second component constantp 2 ≡ cst the null solution. Considering that p satisfies boundary conditions described in (2.4), one sees that suchp satisfies Consequently, unique continuation is verified for the adjoint system if (0, 1) does not belong to at least one of the kernels of matrices (hΓ 1 +Γ 2 ) or (hΓ 3 +Γ 4 ).

Systems of n equations with constant coupling coefficients.
3.2.1. Zero order couplings. Proof of theorem 2.2. We consider system (2.3) with A 1 = 0. Approximate controllability in arbitrary time T > 0 is equivalent to unique continuation property for the the adjoint system (2.9): (3.6) From the adjoint equation above one obtains, after multiplying the adjoint system with B , that B A 0 p ≡ 0 in (0, T ) × ω. By induction, multiplying adjoint equation with matrix B(A 0 ) k , k = 1, · · · , n − 2 we obtain that B (A 0 ) k p ≡ 0 in (0, T ) × ω, k = 0, · · · , n − 1. (3.7) The Kalman rank condition (2.15): rank [A 0 |B] = n applied to (3.7) gives that which, by unique continuation for systems of parabolic equation (see, for example, the paper of J.-C.Saut and B.Scheurer [20]) gives that

3.2.2.
First order couplings. Proof of theorem 2.3. For simplifying notations we write the proof with η 1 = 0. This is not a loss of generality as we may replace A 1 by A 1 + η 1 I. We consider the controlled system (2.3) with A 0 = 0: The adjoint problem reads   (3.9) Approximate controllability in arbitrary time T > 0 for (3.8) is equivalent to unique continuation property for the the adjoint system: × Ω. We proceed as in the case of zero order couplings and we multiply the adjoint system (3.9) with B and we obtain Now, for k = 2, · · · , n − 2, we derivate k times in x the adjoint equation: (3.10) and we multiply by B (A 1 ) k to obtain in order to use the Kalman condition (2.16) we compute derivatives with respect to x in (3.11) and obtain 12) and find that D n−1 x p ≡ 0 in (0, T ) × ω, which by unique continuation for systems of parabolic equations applied to (3.10) with k = n − 1 gives D n−1 x p ≡ 0 in (0, T ) × Ω. This means that p is a polynomial of degree at most n − 2 of the form (3.13) Plugging the expression of p in the adjoint equation in (3.9) we obtain d k + η 0 d k + (k + 1)A 1 d k+1 + (k + 2)(k + 1)d k+2 = 0, (3.14) Now, we denote by c k := d k e η0t ,c k := c k (0). Observe that {c k } k satisfy where we denoted by a k := −(k + 1), k = 0, . . . , n − 3, b k = −(k + 2)(k + 1), k = 0, . . . , n − 4. Now, one may easily see that c k must be polynomials in t, with vector coefficients in R n and degree at most n − 2 − k. Denote by Dom(c k ) the coefficient of t n−2−k of the polynomial c k (this may be zero if the degree of c k is strictly smaller that n − 2 − k).
As B p(t, x) = 0, x ∈ ω one finds that B c k (t) ≡ 0, t ∈ (0, T ), k = 0, . . . , n − 2, (3.16) and thus, letting t = 0 we obtain B c k ≡ 0, k = 0, . . . , n − 2. (3.17) Observe now that Dom(c n−2 ) =c n−2 and knowing that the degree of c k is at most n − 2 − k we may write with P (l) k a polynomial of degree at most l. Identifying the coefficients of t n−3−k in left and right sides of (3.15) we obtain (3.18) Consequently, and thus c n−2 =c n−2 . Now we use the boundary condition from (3.9) in x = 0, which is satisfied also by p(t, x)e η0t , and we find: This is a polynomial of degree at most n − 2 and thus all coefficients are zero, in particular the coefficient of t n−2 which is (3.20) Now we look at the other end, x = l for the corresponding boundary conditions and find that: Observe that this is again a polynomial in t of degree at most n − 2 and thus all coefficients are null and in particular the coefficient of t n−2 is zero; this latter coefficient appears only in c 0 , and thus From (3.17), (3.20), (3.22) and hypothesis (2.17) (with η 1 = 0) we find thatc n−2 = 0. and, consequently, the degree of polynomial c k (t), k = 0, . . . , n − 3 is at most n − 3 − k.
Observe that H 1 is finite dimensional and is generated by eigenfunctions and generalized eigenfunctions of the elliptic operator corresponding to eigenvalues from σ 1 (A).
We consider P to be the projection on H 1 corresponding to the direct sum H 1 ⊕ H 2 , Q := I − P and A 1 = P A, A 2 = QA. Project now the equation (2.3) on H 1 , H 2 and, for some w solution of (2.3), we denote by W 1 = P w, W 2 = Qw. Then we have the two problems: Since we have approximate controllability in time T for the linear system in H, we have that the reachable set {W u (T, ·), u ∈ L 2 (0, T ; L 2 (ω))}, Now we define the operatorK := K 1 • P and we denote by WK 1 a solution for the equation in H 1 stabilized byK 1 . Then we have the estimate for the solution, We construct the feedback control We pass to norms in the above formula, using that A 2 is the generator of a stable semigroup on H 2 , where we have used the estimate obtained on W 1 and e tA2 . Now, if we choose δ >δ when stabilizing W 1 , then there exists C = C(δ, δ) such that We consider the real part of the system and we take K := ReK to find that K stabilizes the linear system (2.3): 5. Local stabilization of the nonlinear system. In order to prove stabilization of the nonlinear system (2.1) around the stationary state y with the feedback control u = K(y − y), we will study an equivalent property, the stability in zero for the system satisfied by z := y − y: Since we are interested in local stabilization and in order to avoid blow-up phenomena, we will study first a system obtained through truncation and then show that for initial data small enough in appropriate norm the solution to the truncated feedback controlled system is solution to the initial nonlinear controlled system. For this purpose we consider the cutoff function where |·| 2 is the Euclidean norm in R 2n , R > 2 y L ∞ and we will consider F R = ρ R F instead of F in the above system (5.1). Then, the system we will work with takes the form where for w ∈ H 1 (Ω), [Rw](x) = R Dy,y (Dw(x), w(x)), x ∈ Ω, and R Dy,y (ζ, y) is the remainder of a first order Taylor development of F R (ζ, y): If we also consider the Taylor expansion of order two for F R (ξ, y) and taking into account that F R is compactly supported, we find further the following for the remainder R Dy,y (ζ, y): Regarding the operator Rz = R Dy,y (Dz, z), using the linear growth we obtain the following estimates and using the quadratic growth we get with C a constant depending only on R. In order to estimate the solution to nonlinear problem (5.2) we use norms given by a selfadjoint operator P which is solution to a Lyapunov equation and in this sense we need the following Lemma whose proof was communicated to us by C. Lefter. This result appears also in [14](see also [15]) but under the more restrictive assumption D(Ã) = D(Ã * ) which is no more needed in this proof.
Proof. Let Q(z) = ∞ 0 (−Ã) β+ 1 2 e tÃ z 2 H dt with D(Q) = {z|Q(z) < +∞}. Observe that Q is densely defined in H as it is well defined on n≥1 D(−Ã) n , which is a dense subspace of H. Moreover, Q is a quadratic form as it is easy to see that Denote also by Q the bilinear form In fact one has There exists a selfadjoint operator P in H with domain and Q(z) = Pz, z , ∀z ∈ D(P).
One also knows that, asÃ is a negative operator generator of an analytic semigroup in a Hilbert space one has equalities between spaces of real and complex interpolation and domains of powers of the operator −Ã (see [18]): Suppose now that β = 1 2 and denote by Observe that for z ∈ D((−Ã) β ) and the first statement of the lemma is proved, which is the fact that Q defines an equivalent norm in D((−Ã) β ) which coincides with D((−A) β ) for β ∈ [0, 1].
We want now to prove that D((−Ã) 2β ) ⊂ D(P) with continuous embedding and for this we need to prove that there exists C > 0 such that Now, for someδ > 0 small, if we place the system in the neighborhood of zero with z H 1 <δ, the solution remains there and satisfies, for some δ > 0 giving exponential decay for the norm H 1 : Now since the dimension of the space we work in is one, we have H 1 (Ω) ⊂ L ∞ (Ω) and the decay in the L ∞ -norm follows. The truncated system stabilizes in L ∞ norm.
We now prove that for initial data small enough in D((−A) ν ) for given ν ∈ ( 3 4 , 1), the solution to the truncated problem is solution to the initial problem. To do this we need estimates of z in W 1,∞ (Ω); for this purpose we estimate z in D((−Ã) ν ) and we take into account the continuous embeddings D((−Ã) ν ) ⊂ W 2ν,2 (Ω) ⊂ W 1,∞ (Ω). One has for fixed > 0 and τ ∈ R + z(τ + ) = e Ã z(τ ) + It is clear now that for z(0) D((−Ã) ν ) small enough z is solution to the nonlinear system (5.1) for time t ≥ . Regarding the estimates for t ∈ [0, ], using classical arguments concerning continuous dependence of the solution on initial data, we may say that if one chooses z(0) D((−Ã) ν ) small enough then z L ∞ (0, ;D((−Ã) ν ) is small enough. Correspondingly, one has z L ∞ (0, ;W 1,∞ (Ω) < R and thus z is solution to (5.1) on [0, ∞); stabilization for truncated equation is in fact stabilization for initial nonlinear equation when initial data is small. Stabilization in H 2 or in D(Ã) also follows as above using again the regularizing effect of the analytic semigroup.
We mention here that since the initial data y 0 satisfies the boundary conditions and belongs to H 2ν , by a classic result of R.Seeley (see [21]), y 0 ∈ D((−Ã) ν ) and the stability estimates obtained in terms of norms in fractional spaces reduce to the estimates using Sobolev norms.

Remark 7.
Observe that if we are in the framework of Theorem 2.5, the feedback stabilizing (2.1) (or equivalently (2.7)) stabilizes also the system with A perturbed by a small lower order operator: A + T where T is for example a first order linear differential operator with coefficients having small enough L ∞ norm. As a consequence, if we have to stabilize nonconstant solutions then the linearized system has nonconstant coupling matrices A 0 (x), A 1 (x). We may then consider a modified linear controlled system with constant coupling matrices A 0 = 1 l l 0 A 0 (x)dx and A 1 = 1 l l 0 A 1 (x)dx. Following the procedure in this paper one may prove that if the modified controlled system is feedback stabilizable and A 0 (·) − A 0 L ∞ and A 1 (·) − A 1 L ∞ are small enough, then the same feedback stabilizes (2.1).