Feedback stabilization with one simultaneous control for systems of parabolic equations

In this work controlled systems of semilinear parabolic equations are considered. Only one control is acting in both equations and it is distributed in a subdomain. Local feedback stabilization is studied. The approach is based on approximate controllability for the linearized system and the use of an appropriate norm obtained from a Lyapunov equation. Applications to reaction-diffusion systems are discussed.

1. Introduction. In this paper we study the local feedback stabilization of systems of parabolic equations in a domain Ω ⊂ R n , with only one internally distributed control, supported in a bounded subdomain ω ⊂⊂ Ω. Because of the reduced number of controls, these systems may not be small time local controllable or, more precisely, the linearized system is not controllable. In fact, controllability for the linearized system is usually an argument, via appropriate algebraic Riccati or Lyapunov equations, to construct stabilizing feedbacks.
Controllability of linear parabolic equations with internally distributed controls supported in a subdomain was established by O.Yu.Imanuvilov using appropriate global Carleman estimates for the adjoint equation (see [9] for an introduction to the field). Local controllability of nonlinear equations or systems may be deduced from controllability of the linearized system. In such a situation the nonlinear system is small time local controllable.
Controllability of systems by a reduced number of controls is a challenging problem and positive results may be obtained under appropriate conditions on the coupling terms, as it is the case for example in the phase field models studied in [1]. When a good coupling is not verified for the linearized system, an issue to exploit the nonlinearity is the return method of J.-M.Coron, which is linearization along special solutions, constructed in such a way to fulfill coupling requirements for the linearized system (see [6,7,8]).
In this paper the strategy for stabilization is, in some sense, similar to the one in [11] and is based not on the controllability of the linearized system but on its approximate controllability. In fact, exact controllability for the linearized system or, equivalently, observability for the adjoint system, seems not to rely on standard Carleman estimates.
The first step in our approach is to linearize the system around the stationary state. Stabilization for the linear system uses a spectral decomposition of the space, with respect to the elliptic part, in a direct sum of two invariant closed subspaces. One of these subspaces is unstable, but finite dimensional, and the other is an exponentially stable infinite dimensional subspace. The system splits into two independent, controlled systems, one of which finite dimensional.
One key point in our analysis is to prove approximate controllability for the linearized system. This implies exact controllability for the finite dimensional system and, consequently, this has the property of complete stabilization. We may thus construct a feedback law stabilizing the finite dimensional part and then prove that this is stabilizing the full linearized system.
The fact that the feedback law constructed in the linear case is also stabilizing the nonlinear system is proved by using the solution of an appropriate Lyapunov equation.
The systems we want to approach, which are reaction-diffusion type systems, may not be small time local controllable if there is only one control acting in only one equation and we propose a strategy in which we still have one control but acting simultaneously in both equations. We obtain stabilization results if also the diffusion coefficients are different.
For other results concerning stabilization of parabolic equations and systems, with feedback supported in a subdomain, we refer to [3,2].

Preliminaries and main result.
Let Ω ⊂ R n , n ∈ {2, 3}, be a bounded connected domain with the boundary ∂Ω of class C 2 and let ω ⊂⊂ Ω be a nonempty open subset of Ω. We consider the following controlled reaction-diffusion type system: in Ω. (2.1) where d 1 , d 2 ∈ R + are the diffusion coefficients, f, g : R × R −→ R are C ∞ coupling nonlinearities, f 1 , g 1 ∈ L ∞ (Ω) , ψ ω ∈ C ∞ (Ω), supp ψ ω = ω, ψ ω > 0 in ω. u(t, ·) is the control which belongs to L 2 (ω) and by ψ ω u we denote the extension by 0 of u to Ω multiplied by ψ ω . Let (y, z) ∈ (L ∞ (Ω)) 2 be a stationary state of the system, that is in Ω y = 0, z = 0, on ∂Ω We want to find a finite dimensional feedback law u = K(y, z), such that system (2.1) becomes locally exponentially stable around Y = (y, z) in an appropriate space.
We make here the following convention: if a function y belongs to a space, let us say H 1 (Ω), the norm will be denoted, for simplicity, omitting to write the domain: y H 1 . In the same spirit, if a vectorial function Y = (y, z) ∈ L ∞ (Ω) × L ∞ (Ω), the norm will be denoted by Y L ∞ . If a function belongs to the intersection of some Banach spaces y ∈ X 1 ∩ X 2 , then the norm y X1∩X2 = max{ y X1 , y X2 }. The first step is to linearize the system around (y, z) and to construct a feedback that stabilizes the linearized system around zero.
Next step is to show that the same feedback stabilizes locally the nonlinear system. Stabilization for the nonlinear systems occurs in H 1 norm. For this step we use an equivalent norm given by the solution to an adapted Lyapunov equation. Stabilization in L ∞ norm is obtained by using the regularizing effect of parabolic systems.
The controlled linearized system is Let H be the Hilbert space L 2 (Ω) × L 2 (Ω) and consider the operators In operatorial form, with Y = (y, z) , F (Y ) = (f (y, z), g(y, z)) and F 1 = (f 1 , g 1 ) , the controlled system (2.1) is written: (2.4) . The stabilization result that will be proved in §3 for the linearized system is the following: Theorem 2.1. Suppose that the diffusion coefficients are distinct d 1 = d 2 and one of the following assumptions is true: • α is not identically constant in ω, or • α is a constant in Ω and 0 ∈ σ(L T ). Then the following conclusions hold: (i) The operator A = A + A 0 has compact resolvent and generates an analytic semigroup in H; (ii) The linear system (2.3) or (2.5) is approximately controllable in any time T ; (iii) For any δ > 0 there exist C = C(δ) > 0, a finite dimensional subspace U ⊂ L 2 (ω) and a linear continuous operator K ∈ L(H, U ) such that the operator A + BK generates an analytic semigroup of negative type satisfying The main result of this paper, proved in §4, and concerning the stability around the stationary state of the nonlinear system (2.1) respectively (2.4) is the following.
we have local exponential stabilization: 3. Feedback stabilization of linearized system. Proof of Theorem 2.1.
generates an analytic semigroup, by a standard argument on the lower order perturbations of selfadjoint operators. In fact, A 0 ∈ L(H) satisfies an estimate of the type A 0 y H ≤ ε Ay H + C(ε) y H for y ∈ D(A) (see [12]).
Moreover, A has compact resolvent as a consequence of Rellich compact embedding theorem. This fact implies that the spectrum σ(A) is discrete, with no finite accumulation point and is contained in an angular domain . Concerning the approximate controllability in time T for the linear problem (2.3) or (2.5), we know that this is equivalent to the unique continuation property for the backward adjoint problem. For this, consider the dual problem in Ω.
The unique continuation property we have to prove is: which, considering that B * (p, q) = [ψ ω · (p + q)]| ω , it means that we have to prove Suppose that p + q ≡ 0 in (0, T ) × ω, that is q = −p there. Since p satisfies both equations in (0, T ) × ω we get after adding the two equations multiplied by d 2 , respectively by d 1 , the identity By integration we find that for (t, x) ∈ (0, T ) × ω with α given in the hypothesis. Denote bỹ Suppose first that α is not identically constant in ω. Then, supp |∇α| ∩ ω has nonzero measure and in fact it has nonempty interior because of the C 1,θ regularity of α. It turns out that for x ∈ ω the equations in (3.6) are second order polynomials in t which are identically zero and thus all coefficients need to be zero. So, |∇α| 2p = |∇α| 2q ≡ 0 in ω ×(0, T ) and thusp =q ≡ 0 in (0, T )×[supp |∇α|∩ω]. By the above observation about the nonempty interior of supp |∇α|∩ω, from unique continuation, p =q ≡ 0 in Ω × (0, T ) and thus this occurs for p, q.
Suppose now that α is a constant in Ω and 0 ∈ σ(L T ). System (3.6) becomes By the unique continuation property for systems of parabolic equations, we obtain that p 1 = q 1 ≡ 0 in (0, T )×Ω. This implies thatp,q are independent of t in (0, T )×Ω and it means that in (0, T ) × Ω and, by (3.7), (p,q) is a solution to the elliptic system (3.8) By computation we have that and by adding the equations in (3.8) we get: Using the unique continuation hypotesis p + q = 0 in (0, T ) × ω, we deduce that p +q = 0 in ω and we conclude, by the unique continuation property for elliptic equations, thatp +q = 0 in Ω. In what follows the strategy for stabilization is very similar to what was done in [10,11] but we present it for the sake of completeness. Since A generates an analytic semigroup in H and has compact resolvent, σ(A) lies in a cone V λ,φ := {z ∈ C, |arg(z − λ)| ∈ (π − φ, π]}, for some λ ∈ R, φ ∈ (0, π 2 ). Consider some δ > 0 and choose δ 2 > δ such that σ(A) ∩ {Re λ = −δ 2 } = ∅, (3.12) Observe that σ 1 has a finite number of elements and, moreover, it contains also all the eigenvalues λ with Re λ > 0, coresponding to the unstable states. Observe also that, for some φ ∈ (0, π 2 ), σ 2 ⊂ V −δ2,φ . Coresponding to this decomposition of the spectrum we can split the complexified space H c into a direct sum H c = H 1 ⊕ H 2 , where the subspaces H 1 , H 2 are invariat under the complexified operator denoted also by A and σ(A| H1 ) = σ 1 , σ(A| H2 ) = σ 2 . Let P be the projector onto the space H 1 corresponding to this decomposition, Q = I − P , A 1 = P A, A 2 = QA. Remark 1. Observe that, since H 1 is finite dimensional and is generated by eigenfunctions or generalized eigenfunctions of A, by elliptic regularity H 1 ⊂ (C 1,θ (Ω)) 2 ∩ (H s (Ω)) 2 , θ ∈ (0, 1), s ∈ [0, 2].
We project the linear equation (2.5) on H 1 , H 2 and denoting by W 1 := P W, W 2 := (I − P )W , we have The first one is a finite dimensional linear system in the space H 1 . From the approximate controllability in time T established in (ii) we have that the reachable set {W u (T, ·), u ∈ L 2 (0, T, L 2 (ω))}, with W u solution to (2.5), is dense in H. So, the projection of the reachable set on the finite dimensional space H 1 is the entire H 1 . It means that the first equation is exactly controllable in any time T , and so it is completely stabilizable: such that we have the following exponential decay of the finite dimensional semigroup in H 1 e t(A1+P BK1) L(H1) ≤ ce −δ1t . (3.14) We defineK := K 1 • P . Denoting by WK 1 the solution of the finite dimensional equation stabilized byK, then We consider the feedback control u :=KW 1 =KP W so the solution of the second equation is, by variation of constants formula, Now, if we pass to norms in variation of constants formula, using the estimates obtained on W 1 and e tA2 , If we choose, in the stabilization of W 1 , δ 1 > δ 2 then there exists a constant C = C(δ 1 , δ 2 ) such that Together with the estimate on WK 1 and looking only to the real part of the system, that is taking K := ReK, we find that K stabilizes the linear system (3.13): (3.20) which completes the proof of Theorem 2.1.

4.
Local stabilization of the nonlinear system. Proof of Theorem 2.2.
Stabilization of system (2.4) to the stationary state Y with feedback u = K(Y − Y ) is equivalent to proving stability in 0 for the system satisfied by Z := Y − Y : Since our result has as consequence local stabilization in L ∞ , it is natural to study a truncated system with F replaced by F R = ρ R F with R > 2 Y L ∞ and a cutoff function ρ R satisfying: where we denoted by | · | 2 the Euclidean norm in R 2 . The truncated system is Remark 3. Observe that F R is a C 2 function with compact support and so it has bounded derivatives in R 2 . Considering a Taylor expansion for F R , the following estimates are both true uniformly for w belonging to bounded subsets of R 2 , say |w| 2 ≤ R: |R w (ζ)| 2 ≤ C min{|ζ| 2 , |ζ| 2 2 }, for all ζ ∈ R 2 , with a constant C depending only on R.
We will use the quadratic growth estimate in the local stabilization arguments. The linear growth estimate will be useful in questions related to bounds in stronger norms and, what is essential, to prove that solutions to the truncated system remain solutions to the nonlinear initial system if the initial data is small enough.
Using the linear growth, one has immediately that By a simple computation using the C 1,θ regularity of Y and the properties of the functions F R , R w , one also obtains the following estimate: with a constant depending only on R.
Consider in the above proposition β = 1 2 and the corresponding operator P. Multiply then (4.2) with PZ in H and obtain Using the quadratic growth estimate for R Y and the Lyapunov equation we find: Observe that in space dimension n ≤ 3, H 1 (Ω) ⊂ L 4 (Ω) and thus We denoted by C, C 1 , C 2 various constants which do not depend on Z. It turns out that for some δ 1 > 0 small enough, the neighborhood {Z, (PZ, Z) H < δ 1 } of 0 is invariant for the flow and satisfies.
for some δ > 0. Exponential decay for the norm H 1 follows immediately: which is partly the conclusion of the theorem. Last step of the proof is to show that we may stabilize the truncated system in L ∞ -norm and a consequence of this fact is the local L ∞ stabilization of the original nonlinear system. Rewrite (4.2) as where T(Z) = (A 0 + BK)Z + R Y (Z), by (4.3),(4.4), satisfies in either of the norms X ∈ {L 2 , L ∞ , H 1 }. First, by parabolic maximum principle (see [4]) applied independently to the equations in (4.7), we obtain that It turns out that choosing a τ < 1 2C there exists (4.9) The following estimates for solutions to (4.7) is classic for parabolic-like nonhomogeneous equations or systems: Inserting (4.4) into (4.10), using (4.6) and (4.8), we obtain that It is now clear that, since H 2 (Ω) ⊂ L ∞ (Ω) with continuous embedding, for some δ 3 > 0, if Z 0 H 1 < δ 3 and considering also (4.9) we have Z(t) L ∞ ≤ R, t > 0. and exponential stabilization occurs locally, in H 2 norm for (4.1) and thus we proved the local stabilization around a stationary state for the initial nonlinear system. The proof of Theorem 2.2 is completed.
The result there is that given T > 0 the system with m = 3 is null controllable in time T , with initial data from a small neighborhood in L ∞ . The return method is used and linearization is done along a particular nonstationsary solution. This kind of linearization is not compatible with our stabilization strategy because spectral decomposition is not working anymore. For m = 2 the system is not controllable in any time (see [7]). Their result remains true, with unchanged proof, for the case The result in this paper applies to the feedback stabilization of system (4.12) but introducing the same control also in the second equation. Local feedback stabilization at (0, 0) is obtained in this way for      y t − d 1 ∆y = g(y, z) + ψ ω u, t > 0, x ∈ Ω z t − d 2 ∆z = y m + Rz + ψ ω u, t > 0, x ∈ Ω y = 0, z = 0, t > 0, x ∈ ∂Ω, (4.13) with arbitrary m ≥ 1, if the diffusion coefficients are different d 1 = d 2 and hypotheses of Theorem 2.1 for the linearized system are verified. More precisely, when verifying the hypotheses of Theorem 2.1 in this particular case, we see that α(x) is a constant and there is only one value for R, say R, for which 0 ∈ σ(L T ). The stabilization results stated in Theorems 2.1, 2.2 occur if R = R.