On the approximate controllability of parabolic problems with non-smooth coefficients

Theorem 2.1.

Let Ω ⊂ R N be a bounded open set with smooth boundary, ω ⊂ Ω a non-empty open subset, and Y ⊂ R N a cell with the paving property in R N . Assume that { ε } is a parameter taking its values in a sequence of positive real numbers that converges to zero.

Let δ be a real number with 0 < δ < 1 and z ∈ L 2 ( Ω ) a given target function. For f ∈ L 2 ( ( 0 , T ) × Ω ) , u 0 ∈ L 2 ( Ω ) , and A ∈ M ( α , β , Ω ) a given symmetric N × N -matrix field as defined in ( 1.1 ), consider the following problem u ′ − div ( A ∇ u ) = f in Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 in Ω .

Suppose that, for some α B , β B ∈ R with 0 < α B < β B , there exists v 0 ∈ L 2 ( Ω ) and a Y-periodic matrix field B ∈ M ( α B , β B , Y ) , such that the sequence of matrices B ϵ ( x ) = B ( x ϵ ) has the property that for any f ∈ L 2 ( ( 0 , T ) × Ω ) , there exists { u ϵ 0 } ϵ ⊂ L 2 ( Ω ) (depending on f, B and v 0 ) with u ϵ 0 → v 0 strongly in L 2 ( Ω ) , such that the solution u ϵ of u ϵ ′ − div ( B ϵ ∇ u ϵ ) = f in Ω × ( 0 , T ) , u ϵ = 0 on ∂ Ω × ( 0 , T ) , u ϵ ( x , 0 ) = u ϵ 0 in Ω , verifies, for some ϵ 0 > 0 , (2.1) ‖ u ϵ ( · , T ) − u ( · , T ) ‖ L 2 ( Ω ) ⩽ δ 3 for all ϵ < ϵ 0 .

Then there exists a control v ∈ L 2 ( ( 0 , T ) × ω ) such that solution of the following problem p ′ − div ( A ∇ p ) = f + χ ω v in Ω × ( 0 , T ) , p = 0 on ∂ Ω × ( 0 , T ) , p ( x , 0 ) = u 0 in Ω , verifies ‖ p ( · , T ) − z ‖ L 2 ( Ω ) ⩽ δ .

Proof.

For every function ψ ∈ L 2 ( Y ) define its average M Y ( ψ ) = 1 | Y | ∫ Y ψ d y where | Y | denotes the measure of the set Y. Next consider the space of periodic functions with mean zero defined by W per ( Y ) = { ψ ∈ H per 1 ( Y ) , M Y ( ψ ) = 0 } . Let B 0 be the homogenized matrix corresponding to B ϵ (see [2,6]), i.e., (2.2) B 0 λ = M Y ( B ∇ q λ ) , for every λ ∈ R N , where w λ solves (2.3) ∫ Y B ∇ q λ ∇ ψ = 0 , for all ψ ∈ W per ( Y ) , ( q λ − λ · y ) ∈ W per ( Y ) .

Classical controllability results (see [25,26]) imply there exists φ 0 ∈ L 2 ( Ω ) such that the solution of (2.4) w 0 ′ − div ( B 0 ∇ w 0 ) = f + χ ω φ 0 in Ω × ( 0 , T ) , w 0 = 0 on ∂ Ω × ( 0 , T ) , w 0 ( x , 0 ) = v 0 in Ω , verifies (2.5) ‖ w 0 ( · , T ) − z ‖ L 2 ( Ω ) ⩽ δ 3 .

Consider now the problem p ′ − div ( A ∇ p ) = f + χ ω φ 0 in Ω × ( 0 , T ) , p = 0 on ∂ Ω × ( 0 , T ) , p ( x , 0 ) = u 0 in Ω . The hypothesis implies that there exists { ξ 0 ϵ } ϵ ⊂ L 2 ( Ω ) with ξ 0 ϵ → v 0 strongly in L 2 ( Ω ) such that the solutions of (2.6) w ϵ ′ − div ( B ϵ ∇ w ϵ ) = f + χ ω φ 0 in Ω × ( 0 , T ) , w ϵ = 0 on ∂ Ω × ( 0 , T ) , w ϵ ( x , 0 ) = ξ 0 ϵ in Ω , satisfy, for some ϵ 1 > 0 , the inequality (2.7) ‖ w ϵ ( · , T ) − p ( · , T ) ‖ L 2 ( Ω ) ⩽ δ 3 for all ϵ < ϵ 1 . Note that classical homogenization theory and correctors ([6]) implies that problem (2.4) is the homogenized limit of the sequence of problems (2.6) and one has the following corrector results ‖ w ϵ ( · , T ) − w 0 ( · , T ) ‖ C ( [ 0 , T ] ; L 2 ( Ω ) ) → 0 . This implies that there exists ϵ 2 > 0 such that (2.8) ‖ w ϵ ( · , T ) − w 0 ( · , T ) ‖ L 2 ( Ω ) ⩽ δ 3 for all ϵ < ϵ 2 . From (2.5), (2.7), (2.8) we conclude that ‖ p ( · , T ) − z ‖ L 2 ( Ω ) ⩽ δ , and so v = φ 0 satisfies the conclusion of the theorem. □

The next corollary shows that one can prove and compute approximate controls for parabolic problems associated with given micro-structures as long as their characteristic length is small enough, depending on the desired level of control accuracy. Indeed we have,

Corollary 2.2.

Let δ be a real number with 0 < δ < 1 , z ∈ L 2 ( Ω ) a given target function and f ∈ L 2 ( ( 0 , T ) × Ω ) . Suppose that { η } is a parameter taking its values in a sequence of positive real numbers that converges to zero, and { y η 0 } η ⊂ L 2 ( Ω ) a sequence such that y η 0 → y 0 strongly in L 2 ( Ω ) for some y 0 ∈ L 2 ( Ω ) .

Also consider a sequence of given matrices D η ( x ) = D ( x η ) , with D being Y-periodic, D ∈ M ( α D , β D , Y ) for some α D , β D ∈ R with 0 < α D < β D .

Then there exists a number 0 < η 0 < 1 and a function φ 0 independent of η < η 0 such that for any η < η 0 the solution of (2.9) y η ′ − div ( D η ∇ y η ) = f + χ ω φ 0 in Ω × ( 0 , T ) , y η = 0 on ∂ Ω × ( 0 , T ) , y η ( x , 0 ) = y η 0 in Ω , satisfies (2.10) ‖ y η ( · , T ) − z ‖ L 2 ( Ω ) ⩽ δ for all η < η 0 .

Proof.

The proof of the corollary follows as a consequence of Theorem 2.1, written for B = D and u = y η , but here we present a shorter more direct solution. Let D 0 be the homogenized matrix defined by (2.2) and (2.3) with matrix B replaced by matrix D. Classical control results imply that there exists a control function φ 0 ∈ L 2 ( Ω ) such that the solution of y ′ − div ( D 0 ∇ y ) = f + χ ω φ 0 in Ω × ( 0 , T ) , y = 0 on ∂ Ω × ( 0 , T ) , y ( x , 0 ) = y 0 in Ω , satisfies (2.11) ‖ y ( · , T ) − z ‖ L 2 ( Ω ) ⩽ δ 2 . On the other hand, the classical corrector results applied to the sequence of problems (2.9) imply that there exists 0 < η 0 < 1 such that (2.12) ‖ y η ( · , T ) − y ( · , T ) ‖ L 2 ( Ω ) ⩽ δ 2 for all η < η 0 . Finally, from (2.11) and (2.12), we obtain (2.10). □

Next we make the observation that our results presented above can be easily adapted to remain true (under certain assumptions to be described below) for the case when the periodicity assumption is replaced by the more general concept of H-convergence. Remark 2.3.

We recall (see [21,22]) that a sequence { A ϵ } ϵ ⊂ M ( α , β , Ω ) H-converges to A 0 ∈ M ( α ′ , β ′ , Ω ) (for some α ′ , β ′ with 0 < α ′ < β ′ ) iff for every function h ∈ H − 1 ( Ω ) the solution ξ ϵ of − div ( A ϵ ∇ ξ ϵ ) = h in Ω , ξ ϵ = 0 on ∂ Ω is such that i ) ξ ϵ ⇀ ξ 0 weakly in H 0 1 ( Ω ) ii ) A ϵ ∇ ξ ϵ ⇀ A 0 ∇ ξ 0 weakly in ( L 2 ( Ω ) ) N , where ξ 0 is the unique solution of the problem − div ( A 0 ∇ ξ 0 ) = h in Ω , ξ 0 = 0 on ∂ Ω .

By using the corrector results for the classical parabolic equations associated to an H-convergent matrix of coefficients (see [4]) and the unique continuation property for parabolic problems with C 1 coefficients (see [24]), the results of Theorem 2.1 and Corollary 2.2 remain true (with identical proofs) if, while still requiring that the approximation property (2.1) holds true, instead of periodicity of the sequences { B ϵ } ϵ and { D η } η one assumes that they are H-convergent with smooth (i.e., C 1 ) limits.

3. General strategy

In this section, we will state our strategy for the approximate control of general linear parabolic problems in heterogeneous media occupying possibly complex geometries (e.g. perforated domains, domains with inclusions or materials separated by interfaces). Thus consider parameter δ, Ω, ω and target function z ∈ L 2 ( Ω ) as above and denote by Γ ⊂ R N − 1 a possible connected interface separating Ω in two components or a disconnected set describing the boundary of perforations. In what follows, we will consider a generic parabolic flow { P } associated to a general source f ∈ L 2 ( ( 0 , T ) × Ω ) and initial condition u 0 . For this problem we assume a general heterogeneous media with given boundary conditions { BC } on ∂ Ω , for instance Dirichlet conditions and, if an interface Γ is considered as part of the geometry (i.e., as described above), with possibly interface conditions { IC } prescribed on Γ, e.g., flux-temperature proportionality conditions, (see [7–9,11,15]). We can also treat the case of a perforated domain, where one can assume for instance homogeneous or nonhomogeneus Robin conditions (in particular Neumann conditions, see [11]) on the boundaries of the holes. In this case, in the statement below, the fixed domain Ω has to be replaced by a varying one, with the obvious modifications, and for a nonperiodic setting one can use the H-convergence extension to perforated domains (see [5]).

We have: