M. Duprez ∗ and P. Lissy †

Indirect controllability of some linear parabolic systems coupled by an operator of first order.

M. Duprez ^∗ and P. Lissy ^†

∗

Laboratoire de Mathématiques de Besançon UMR CNRS 6623, Université de Franche-Comté

†

Ceremade, UMR 7534, Université Paris-Dauphine

CEREMADE

UMR CNRS 7534

Abstract

This work is devoted to the study of the null and approximate controllability for some classes of linear coupled parabolic systems with less controls than equations. More precisely, for a given bounded domain Ω in R^N (N ∈ N^∗), we consider a system of two linear parabolic equations with coupling terms of first and zero order, and one controls localized in some arbitrary nonempty open subset ω of Ω. In the case of constant coupling coefficients, we provide a necessary and sufficient condition to obtain the null or approximate controllability in arbitrary small time. In the case N = 1, we also give a generic sufficient condition to obtain the null or approximate controllability in arbitrary small time for general coefficients depending on the space and times variables, provided that the supports of the coupling terms intersect the control domain ω. The results are obtained thanks to the fictitious control method together with an algebraic method and some appropriate Carleman estimates.

1. Setting

Let T > 0, Ω be a bounded domain in R^N (N ∈ N^∗) of class C² and ω be an nonempty open subset of Ω. Consider the system























∂_ty₁ = div(d₁∇y₁) + g₁₁ · ∇y₁ + g₁₂ · ∇y₂

+a₁₁y₁ + a₁₂y₂ + 1ωu in Q_T,

∂_ty₂ = div(d₂∇y₂) + g₂₁ · ∇y₁ + g₂₂ · ∇y₂

+a₂₁y₁ + a₂₂y₂ in Q_T,

y = 0 on Σ_T,

y(·, 0) = y⁰ in Ω,

(1)

where y⁰ ∈ L²(Ω; R²), u ∈ L²(Q_T), g_ij ∈ L^∞(Q_T; R^N), a_ij ∈ L^∞(Q_T) for all i, j ∈ {1, 2}, Q_T := Ω × (0, T), Σ_T := (0, T) × ∂Ω and

( d^ij_l ∈ W_∞¹ (Q_T), d^ij_l = d^ji_l in Q_T,

N

X

i,j=1

d^ij_l ξ_iξ_j > d₀|ξ|² in Q_T, ∀ξ ∈ R^N.

2. known results Cascade structure:

Let us suppose that

g₂₁ ≡ 0 in q_T and

(a₂₁ > a₀ in q_T or a₂₁ < −a₀ in q_T), for a constant a₀ > 0 and q_T := ω × (0, T).

Then System (1) is null controllable at time T.

Theorem 5 ([5], M. González-Burgos, L. De Teresa, 2010)

Condition on the dimension:

Let N = 1, d₁, d₂, a₁₁, g₁₁, a₂₂, g₂₂ be constant and sup- pose that

g₂₁ · ∇ + a₂₁ = P₁ ◦ θ in Ω × (0, T), where θ ∈ C²(Ω) with |θ| > C in ω₀ ⊂ ω and

P₁ := m₀ · ∇ + m₁,

for m₀, m₁ ∈ R. Moreover, assume that m₀ 6= 0. Then System (1) is null controllable at time T.

Theorem 5 ([7], S. Guerrero, 2007)

Boundary condition:

Assume that a_ij ∈ C⁴(Q_T), g_ij ∈ C³(Q_T)^N, d_i ∈ C³(Q_T)^N2 for all i, j ∈ {1, 2} and

∃γ 6= ∅ an open subset of ∂ω ∩ ∂Ω,

∃x₀ ∈ γ s.t. g₂₁(t, x₀) · ν(x₀) 6= 0 for all t ∈ [0, T], where ν is the exterior normal unit vector to ∂Ω.

Then System (1) is null controllable at time T. Theorem 5 ([2], A. Benabdallah et al, 2014)

3. Main results

Necessary and sufficient condition:

Let us assume that d_i, g_ij and a_ij are constant in space and time for all i, j ∈ {1, 2}.

Then System (1) is null controllable at time T if and only if g₂₁ 6= 0 or a₂₁ 6= 0.

Theorem 4 ([4], M. D., P. Lissy, 2015)

Additional result:

Let us assume that Ω := (0, L) with L > 0.

Then System (1) is null controllable at time T if for an open subset (a, b) × O ⊆ q_T one of the following conditions is ver- ified:

(i) d_i, g_ij, a_ij ∈ C¹((a, b), C²(O)) for i, j = 1, 2 and g₂₁ = 0 and a₂₁ 6= 0 in (a, b) × O,

1/a₂₁ ∈ L^∞(O) in (a, b).

(ii) d_i, g_ij, a_ij ∈ C³((a, b), C⁷(O)) for i = 1, 2 and

|det(H(t, x))| > C for every (t, x) ∈ (a, b) × O, where

H :=



















−a₂₁ +∂_xg₂₁ g₂₁ 0 0 0 0

−∂_xa₂₁ +∂_xxg₂₁ −a₂₁ + 2∂_xg₂₁ 0 g₂₁ 0 0

−∂_ta₂₁ +∂_txg₂₁ ∂_tg₂₁ −a₂₁ +∂_xg₂₁ 0 g₂₁ 0

−∂_xxa₂₁ +∂_xxxg₂₁ −2∂_xa₂₁ + 3∂_xxg₂₁ 0 −a₂₁ + 3∂_xg₂₁ 0 g₂₁

−a₂₂ +∂_xg₂₂ g₂₂ − ∂_xd₂ −1 −d₂ 0 0

−∂_xa₂₂ +∂_xxg₂₂ −a₂₂ + 2∂_xg₂₂ −∂_xxd₂ 0 g₂₂ − 2∂_xd₂ −1 −d₂

















 .

Theorem 5 ([4], M. D., P. Lissy, 2015)

Remarks: Even though the last condition seems compli- cated, it can be simplified in some cases :

1. System (1) is null controllable at time T if there exists an open subset (a, b) × O ⊆ q_T such that







g₂₁ ≡ κ ∈ R^∗ in (a, b) × O, a₂₁ ≡ 0 in (a, b) × O,

∂_xa₂₂ 6= ∂_xxg₂₂ in (a, b) × O.

2. If the coefficients depend only on the time variable, the condition becomes :

∃ (a, b) ⊂ (0, T) s.t.:

g₂₁(t)∂_ta₂₁(t) 6= a₂₁(t)∂_tg₂₁(t) in (a, b).

4. Proof of Theorem 4 System (1) can be written as







∂_ty = div(D∇y) + G · ∇y + Ay + e₁1ωu in Q_T,

y = 0 on Σ_T,

y(·, 0) = y⁰ in Ω,

(1)

where y⁰ ∈ L²(Ω; R²), D := (d₁, d₂) ∈ L(R^2N, R^2N), G := (g_ij) ∈ L(R^2N, R²) and A := (a_ij) ∈ L(R²).

Strategy:

Analytic problem:

Find (z, v) with Supp(v) ⊂ ω × (ε, T − ε) s.t. :







∂_tz = div(D∇z) + G · ∇z + Az + N(1ωv) in Q_T,

z = 0 on Σ_T,

z(0, ·) = y⁰, z(T, ·) = 0 in Ω, where the operator N is well chosen.

Difficulties: • the control is in the range of the operator N,

• v has to be regular enough.

Algebraic problem:

Find (bz,v)b with Supp(z,b bv) ⊂ ω × (ε, T − ε) s.t. :

∂_tzb = div(D∇zb) + G · ∇zb+ Azb+ Bbv + Nf in Q_T, Conclusion:

The couple (y, u) := (z − z,b −bv) will be a solution to system (1) and will satisfy y(T) ≡ 0 in Ω

Resolution of the algebraic problem

We use the notion of the algebraic resolvability (see [6]) used for the first time in the control theory of pde’s in [3].

The algebraic problem can be rewritten as :

For f := 1ωv, find (z,b bv) with Supp(z,b bv) ⊂ ω×(ε, T −ε) such that

L(z,b bv) = Nf,

where L(z,b bv) := ∂_tzb− div(D∇zb) − G · ∇zb− Azb− Bbv. .

This problem is solved if we can find a differential operator M such that

L ◦ M = N.

The last equality is equivalent to

M^∗ ◦ L^∗ = N^∗,

where L^∗ is given for all ϕ ∈ C_c^∞(Q_T)² by L^∗ϕ :=





L^∗₁ϕ L^∗₂ϕ L^∗₃ϕ



 =







−∂_tϕ₁ − div(d₁∇ϕ₁) + P₂

j=1{g_j1 · ∇ϕ_j − a_j1ϕ_j}

−∂_tϕ₂ − div(d₁∇ϕ₂) + P₂

j=1{g_j2 · ∇ϕ_j − a_j2ϕ_j} ϕ₁





 . We remark that

(g₂₁ · ∇ − a₂₁)L^∗₁ϕ

L^∗₁ϕ + (∂_t + div(d₁∇) − g₁₁ · ∇ + a₁₁)L^∗₃ϕ

= N^∗ϕ, where

N^∗ := g₂₁ · ∇ + a₂₁.

Thus the algebraic problem is solved by taking M^∗



 ψ₁ ψ₂ ψ₃



 :=

(g₂₁ · ∇ − a₂₁)ψ₃

ψ₁ + (∂_t + div(d₁∇) − g₁₁ · ∇ + a₁₁)ψ₃

.

Resolution of the analytic problem The system







∂_tz = div(D∇z) + G · ∇z + Az + N(1ωv) in Q_T,

z = 0 on Σ_T,

z(0, ·) = y⁰, z(T, ·) = 0 in Ω,

is null controllable if for all ψ⁰ ∈ L²(Ω; R²) the solution of the adjoint system







−∂_tψ = div(D^∗∇ψ) − G^∗ · ∇ψ + A^∗ψ in Q_T,

ψ = 0 on Σ_T,

ψ(T, ·) = ψ⁰ in Ω.

satisfies the following inequality of observability Z

Ω

|ψ(0, x)|²dx 6 C_obs

Z _T

0

Z

ω

ρ₀|N ^∗ψ(t, x)|² dxdt,

where ρ₀ can be chosen to be exponentially decreasing at times t = 0 and t = T .

This inequality is obtain by applying the differential operator

∇∇N ^∗ = ∇∇(−a₂₁ + g₂₁ · ∇)

to adjoint system. More precisely we study the solution φ_ij := ∂_i∂_jN^∗ψ of the system







−∂_tφ_ij = D∆φ_ij − G^∗ · ∇φ_ij + A^∗φ_ij in Q_T,

∂φ

∂n = ^∂(∇∇N_∂n ^∗ψ) on Σ_T,

φ(T, ·) = ∇∇N ^∗ψ⁰ in Ω.

Remark : We need that ∇∇N^∗ commutes with the other operators of the system, what is possible with some con- stant coefficients.

Following the ideas of Barbu developed in [1], the control is chosen as the solution of the problem







minimize J_k(v) := 1 2

Z

Q_T

ρ⁻¹₀ |v|²dxdt + k 2

Z

Ω

|z(T )|²dx, v ∈ L²(Q_T, ρ^−1/2)².

(2) We obtain the regularity of the control with the help of the below Carleman inequality

P3 k=1

Z Z

Q_T

ρ_k|∇^kψ|²dxdt 6 C_T

Z _T

0

Z

ω

ρ₀|N ^∗ψ(t, x)|² dxdt, where ρ_k are some appropriate weight functions.

5. References

[1 ] V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Op- tim., 42 (2014), 73–89.

[2 ] A. Benabdallah, M. Cristofol, P. Gaitan and L. De Teresa, Controllability to trajec- tories for some parabolic systems of three and two equations by one control force, Math. Control Relat. Fields, 4 (2014), 17–44.

[3 ] J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier- Stokes system with a distributed control having two vanishing components, Invent.

Math., 198 (2014), 833–880.

[4 ] M. Duprez and P. Lissy, Indirect controllability of some linear parabolic systems of m equations with m - 1 controls involving coupling terms of zero or first order, Submitted.

[5 ] M. González-Burgos and L. de Teresa, Controllability results for cascade systems ofm coupled parabolic PDEs by one control force, Port. Math., 67 (2010), 91–113.

[6 ] M. Gromov, Partial differential relations, Springer-Verlag, 9 1986.

[7 ] S. Guerrero, Null controllability of some systems of two parabolic equations with one control force, SIAM J. Control Optim., 46 (2007), 379–394.

Contrôlabilité des EDP et applications, CIRM 2015