Feedback stabilization of a boundary layer equation

DOI:10.1051/cocv/2010017 www.esaim-cocv.org

FEEDBACK STABILIZATION OF A BOUNDARY LAYER EQUATION PART 1: HOMOGENEOUS STATE EQUATIONS

Jean-Marie Buchot

and Jean-Pierre Raymond

Abstract. We are interested in the feedback stabilization of a fluid flow over a flat plate, around a stationary solution, in the presence of perturbations. More precisely, we want to stabilize the laminar- to-turbulent transition location of a fluid flow over a flat plate. For that we study the Algebraic Riccati Equation (A.R.E.) of a control problem in which the state equation is a doubly degenerate linear parabolic equation. Because of the degenerate character of the state equation, the classical existence results in the literature of solutions to algebraic Riccati equations do not apply to this class of problems. Here taking advantage of the fact that the semigroup of the state equation is exponentially stable and that the observation operator is a Hilbert-Schmidt operator, we are able to prove the existence and uniqueness of solution to the A.R.E. satisfied by the kernel of the operator which associates the ‘optimal adjoint state’ with the ‘optimal state’. In part 2 [Buchot and Raymond,Appl.

Math. Res. eXpress (2010) doi:10.1093/amrx/abp007], we study problems in which the feedback law is determined by the solution to the A.R.E. and another nonhomogeneous term satisfying an evolution equation involving nonhomogeneous perturbations of the state equation, and a nonhomogeneous term in the cost functional.

Mathematics Subject Classification.93B52, 93C20, 76D55, 35K65.

Received March 25, 2009. Revised November 2nd, 2009.

Published online April 23, 2010.

1. Introduction

We are interested in the feedback stabilization of a fluid flow over a flat plate, around a stationary solution, in the presence of perturbations. The control variable is a suction velocity through a small slot near the leading edge of the plate.

Keywords and phrases.Feedback control law, Crocco equation, degenerate parabolic equations, Riccati equation, boundary layer equations, unbounded control operator.

∗The two authors are supported by the project ANR CISIFS 09-BLAN-0213-03.

1Universit´e de Toulouse, UPS, Institut de Math´ematiques, 31062 Toulouse Cedex 9, France. [email protected];

[email protected]

2 CNRS, Institut de Math´ematiques, UMR 5219, 31062 Toulouse Cedex 9, France.

Article published by EDP Sciences c EDP Sciences, SMAI 2010

In the stationary case, the fluid flow in the boundary layer may be described by the Prandtl equations, or similarly by the Crocco equations [14]:

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

U_∞^s η ∂w

∂ξ −νw²∂²w

∂η² = 0 in (0, L)×(0,1),

ν

w∂w

∂η

(ξ,0) =vsw(ξ,0), lim

η→1w(ξ, η) = 0 for ξ∈(0, L),

w(0, η) =wb(η) for η∈(0,1).

(1.1)

Here (0, L) represents a part of the plate where the flow is laminar, (0,1) is thethicknessof the boundary layer in the Crocco variables, (U_∞^s,0) is the velocity of the incident flow,wb is thevelocityprofile in Crocco variables at ξ= 0,vs is a suction velocity throughout the plate, the positive constantν is the viscosity of the fluid. We set Ω = (0, L)×(0,1). The transformation used to rewrite the Prandtl equations into the Crocco equation is

ξ=x, η =us(x, y)

U_∞^s , w(ξ, η) = 1 U_∞^s

∂us

∂y (x, y), (1.2)

see [14], when (us, vs) is the stationary solution of the Prandtl system, and (x, y)∈(0, L)×(0,∞). Assuming that the regularity and compatibility conditions betweenwb andvsstated in [14], Theorem 3.3.2, are satisfied, the stationary equation (1.1) admits a unique solutionws in the class of functionswsatisfying

w∈Cb(Ω), K₁|1−η| ≤w(ξ, η)≤K₂|1−η|, ∂w

∂ξ

≤K₃|1−η|,

∂w

∂η ∈L^∞(Ω), w∂²w

∂η² ∈L^∞(Ω), ∂w

∂ξ ∈L^∞(Ω),

(1.3)

whereK₁,K₂, andK₃ are positive constants. This class of solution will be called the class of‘asymptotic type solutions’ because they may correspond to an asymptotic profile of some solutions to the Prandtl equations when xtends to infinity (see [7], Sect. 6, where we give an explicit example of such solutions). Another class of solutions important for applications is the class of ‘Blasius type solutions’ (the term comes from the fact that some solutions in that class can be obtained by solving the so-called Blasius differential equation) (see [7], Sect. 6, [14], p. 129).

We are interested in stabilizing a flow over a flat plate when the longitudinal incident velocity is of the form:

U_∞(t) =U_∞^s +u_∞(t). (1.4)

Using the Crocco transformation (see (1.2) and [14]) when the velocity of the external flow U_∞ is positive and only depends on t, the Prandtl system – describing the velocity field in the boundary layer over the flat plate – is transformed into a degenerate parabolic equation stated over Ω = (0, L)×(0,1), called the Crocco equation [3,4], System 4.7, p. 85, [14], p. 174, written down below:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

∂w

∂t +U_∞η∂w

∂ξ +U_∞

U_∞(1−η)∂w

∂η

−νw²∂²w

∂η² +U_∞

U_∞w= 0 in Ω×(0, T),

w(ξ, η,0) =w₀(ξ, η) in Ω,

ν w∂w

∂η

(ξ,0, t) = (vs+^½γu)w(ξ,0, t)−U_∞

U_∞(t) for (ξ, t)∈(0, L)×(0, T),

ηlim→1w(ξ, η, t) = 0 for (ξ, t)∈(0, L)×(0, T), w(0, η, t) =w₁(η, t) for (η, t)∈(0,1)×(0, T),

(1.5)

where ^½γ is the characteristic function of the slot γ = (x₀, x₁) ⊂(0, L), uis a control variable and vs is the function appearing in equation (1.1).

Due to the lack of existence result for the instationary Prandtl system whenU_∞(t) is of the form (1.4) (or to the corresponding instationary Crocco equation – see [14] for some results corresponding to particular profiles, and the more recent results in [20]), we have chosen to describe the velocity field in the boundary layer by solving the Crocco equation linearized about the stationary solutionws. Since the perturbationu_∞(t) and the control functionuare supposed to be small with respect toU_∞^s, the linearized model is an accurate approximation of the nonlinear one. This assertion, which is not proved, is actually confirmed by numerical experiments [4,7].

The Crocco equation (1.5) linearized aboutws with a boundary controluis the degenerate parabolic equation:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

∂z

∂t =Az+f (t, ξ, η)∈ (0,∞)×Ω,

z(0, ξ, η) =z₀(ξ, η) (ξ, η)∈ Ω,

√a z(t,0, η) =√

a zb(t, η) (t, η)∈(0,∞)×(0,1), (bz)(t, ξ,1) = 0, ∂z

∂η(t, ξ,0) = (^½γu+g)(t, ξ) (t, ξ)∈(0,∞)×(0, L),

(1.6)

where

Az=−a(η)∂z

∂ξ +b(ξ, η)∂²z

∂η² −c(ξ, η)z, f(t, ξ, η) =u_∞(t)d(ξ, η) +u_∞(t)

U_∞^s e(ξ, η), g(t, ξ) =− u_∞(t) νws(ξ,0)U_∞^s ·

(1.7)

The coefficients a,b,c,d, edepend on the stationary solutionws of the Crocco equation, and are defined by:

a=U_∞^sη, b=ν(ws)², c=−2ws

∂²ws

∂η² , d=−η∂ws

∂ξ , e=−ws−(1−η)∂ws

∂η ·

Assumptions on the coefficientsa,b,c,dandeare not the same ones ifwsbelongs to the class ofBlasius type solutionsor if it belongs to the class ofasymptotic type solutions.

In this paper we only consider the class ofasymptotic type solutionsbecause we have studied equation (1.6) in [6] whenwsbelongs to this class.

In the case of Blasius type solutions the so-called laminar-to-turbulent transition location – which is an important criterion in applications – is a nonlinear mapping depending on the state variablewand onU_∞. Its linearization about (ws, U_∞^s) – called the linearized transition location – is of the form _Ωψ(ξ, η)z(t, ξ, η) dξdη+ c₀u_∞(t), where the functionψbelongs toL²(Ω) and c₀ belongs toR(they can be determined numerically in a precise manner see [7], Sect. 6, Test 3).

Here, we consider observation operators of the more general form Cz(t,·) +yd(t,·) =

Ω

φ(·, ξ, η)z(t, ξ, η) dξdη+yd(t,·)∈L²(Ω), (1.8) whereφ∈L²(Ω×Ω) andy_d∈L²(0,∞;L²(Ω)) are given. ThusCis a Hilbert-Schmidt operator inL²(Ω). (For the linearized laminar-to-turbulent transition location the functionφ(x, y, ξ, η) =ψ(ξ, η) only depends on (ξ, η) and yd(t,·) = c₀u_∞(t) only depends on t.) It is obvious that the identity in L²(Ω) is not a Hilbert-Schmidt operator, however the identity operator from L²(Ω) into L²(Ω) equipped with a norm weaker than the usual one can also be written in the above form (see Prop.2.1).

Our main objective is to determine a controlu, in feedback form, in order that the observationCz(t) +yd(t) decays to zero when t tends to infinity. For that we use the optimal control theory, and we consider the linear-quadratic control problem

(Pf,g,z_b,y_d,z₀) inf

J(z, u)|(z, u)∈L²(0,∞;Z)×L²(0,∞;U), (z, u) satisfies (1.6)

, whereZ =L²(Ω), U =L²(0, L), and

J(z, u) = 1 2

_∞

0

Cz(t) +yd(t)²Zdt+1 2

_∞

0

u(t)²Udt, whereC∈ L(Z) is the Hilbert-Schmidt operator of kernelφdefined above.

First of all we would like to explain in which aspects problem (Pf,g,z_b,y_d,z₀) is a classical matter of the optimal control theory, and what are the questions that the existing results in the literature cannot answer.

In Section 2 we give a precise definition of solution to equation (1.6), and we prove that it can be rewritten in the form

z =Az+B(^½γu) +F, z(0) =z₀. (1.9)

Moreover, the solution z to equation (1.9) belongs to Cb([0,∞);Z)∩L²(0,∞;Z), the mapping u → z is continuous from L²(0,∞;U) into Cb([0,∞);Z)∩L²(0,∞;Z), and the semigroup (e^tA)t≥0 is exponentially stable on Z. Thus it seems that we are in a very favorable position to characterize the optimal solution of (Pf,g,z_b,y_d,z₀) by means of a feedback law, and our control problem seems to enter into a classical setting.

Even if the analysis of the nonlinear model with the feedback law is not performed, let us explain why the results obtained for the LQ control problem (Pf,g,z_b,y_d,z₀) are quite new and interesting.

In Section 3, we are able to prove that (Pf,g,z_b,y_d,z₀) admits a unique solution (z, u), and that this solution is characterized by an optimality system of the form

⎧⎪

⎪⎨

⎪⎪

⎩

z=Az+B(^½γu) +F, z(0) =z₀,

−p =A^∗p+C^∗(Cz+yd), p(∞) = 0, u=−^½γB^∗p.

(1.10)

We want to prove that there exists an operator Π∈ L(Z) satisfying Π = Π^∗≥0, and a functionr∈L²(0,∞;Z) such that

p(t) = Πz(t) +r(t).

The main objective of the present paper is to obtain an algebraic Riccati equation characterizing Π. The equation satisfied byr, which involves the nonhomogeneous terms f,g,zb, andyd is studied in Part 2 [7]. To find an equation satisfied by Π, we study problem (Pf,g,z_b,y_d,z₀) in the case when f = 0, g = 0, zb = 0 and yd= 0. Denoting this problem by (Pz₀), we can easily show that

inf(Pz₀) = 1 2

Πz₀, z₀

L2(Ω).

SinceAis a degenerate parabolic operator, we explain at the beginning of Section 5 why the existing results in the literature are not sufficient to obtain a Riccati equation characterizing Π in the domain ofA. To overcome this difficulty we look for Π in the form of a Hilbert-Schmidt operator inL²(Ω), and we characterize the equation satisfied by its kernel π. The existence of a weak solution to the algebraic Riccati equation satisfied by π is studied in Section 5. In Section 6 we show that

inf(Pz₀) =1 2

Ω×Ωπ z₀⊗z₀,

for all solutionπto the algebraic Riccati equation. (z₀⊗z0denotes the function defined in Ω×Ω by (x, y, ξ, η) → z₀(x, y)z₀(ξ, η).) Thusπis unique and it is the kernel of Π. The analysis in the nonhomogeneous case, that is whenf,zb,g andyd are not necessarily zero, is performed in Part 2 [7]. Numerical results are also given in [7], showing the efficiency of the linear feedback law applied to the nonlinear Crocco equation in the presence of perturbations.

2. Assumptions and preliminary results

As in [6], we make the following assumptions on the coefficientsa,b, andc.

(H₁)a(η) =U_∞^s η forη∈[0,1], andb∈W^1,∞(Ω). There exist positive constantsCi,i= 1 to 4, such that C₁|1−η|²≤b(ξ, η)≤C₂|1−η|²,

∂b

∂η(ξ, η)≤C₃|1−η| and ∂b

∂ξ(ξ, η)

≤C₄|1−η|² for all (ξ, η)∈Ω.

(2.1)

(H₂) The functionc belongs toL^∞(Ω), and we denote byC₀ a positive constant such that

cL∞(Ω)≤C₀. (2.2)

The nonhomogeneous termsf,g,zb and the initial conditionz₀ and the functionφsatisfy (H₃)z₀∈L²(Ω),zb∈L²(0,∞;L²(0,1)) andg∈L²(0,∞;L²(0, L)).

(H₄)f ∈L²(0,∞;L²(Ω)), φ∈L²(Ω×Ω) andy_d∈L²(0,∞;L²(Ω)).

Let us recall some notation introduced in [5,6]. LetH¹(0,1;d) be the closure ofC^∞([0,1]) in the norm:

zH¹(0,1;d)= 1

0

|z|²+|1−η|² ∂z

∂η ² dη

1/2

. (2.3)

To take the Dirichlet boundary conditionbz(ξ,1, t) = 0 into account, we denote byH_{1}¹ (0,1;d) the closure of C_c^∞([0,1)) in the norm · H1(0,1;d). According to Triebel [16], Theorem 2.9.2,

H¹(0,1;d) =H_{1}¹ (0,1;d).

Let us set

Γ₀=

[0, L)× {0}

∪

{0} ×(0,1)

, Γ₁=

{L} ×(0,1)

∪

(0, L]× {1}

. If the vectorfield

az,−b^∂z_∂η

belongs to (L²(Ω))², and its divergence belongs toL²(Ω), the normal trace on the boundary Γ of the vectorfield

az,−b_∂η^∂z

belongs toH^−1/2(Γ). We denote this normal trace byT

az,−b^∂z_∂η

. Let us recall the definitions of some trace spaces (see [13] or [8], Chap. 7, Sect. 2, Rem. 1)

H₀₀^1/2(Γ₀) =

ϕ∈L²(Γ₀)| ∃ψ∈H¹(Ω), ψ= 0 on Γ₁andψ=ϕon Γ₀

, H₀₀^1/2(Γ₁) =

ϕ∈L²(Γ₁)| ∃ψ∈H¹(Ω), ψ= 0 on Γ₀andψ=ϕon Γ₁

. We can defineT₀

az,−b^∂z_∂η

as an element in (H₀₀^1/2(Γ₀)) in the following way

T₀

az,−b∂z

∂η

, ϕ

(H^1/2₀₀ (Γ0)),H₀₀^1/2(Γ0)=

T

az,−b∂z

∂η

, γ₀ψ

H^−1/2(Γ),H^1/2(Γ)

for all ϕ∈ H₀₀^1/2(Γ₀), where γ₀ ∈ L(H¹(Ω), H^1/2(Γ)) is the trace operator and ψ∈ H¹(Ω) is a function such that ψ= 0 on Γ₁ andψ=ϕon Γ₀.

Similarly, if the vectorfield

−az,−_∂η^∂ (bz)

belongs to (L²(Ω))², and its divergence belongs toL²(Ω), the normal trace on the boundary Γ of the vectorfield

−az,−_∂η^∂ (bz)

, denoted byT

−az,−_∂η^∂ (bz)

, belongs to H^−1/2(Γ), and we can defineT₁

−az,−_∂η^∂ (bz)

by

T₁

−az,−∂

∂η(bz)

, ϕ

(H^1/2₀₀ (Γ1)),H₀₀^1/2(Γ1)=

T

−az,− ∂

∂η(bz)

, γ₀ψ

H^−1/2(Γ),H^1/2(Γ)

for allϕ∈H₀₀^1/2(Γ₁), whereψ∈H¹(Ω) is a function such thatψ= 0 on Γ₀ andψ=ϕon Γ₁. The differential operatorsAandA^∗ are defined by

Az=−a∂z

∂ξ +b∂²z

∂η² −cz, A^∗p=a∂p

∂ξ +∂²(bp)

∂η² −cp.

The unbounded operators inL²(Ω) associated with the above differential operators are given by:

D(A) =

z∈L²(0, L;H¹(0,1;d))|Az∈L²(Ω), T₀

az,−b∂z

∂η

= 0

, Az=Az for allz∈D(A),

D(A^∗) =

p∈L²(0, L;H¹(0,1;d))|A^∗p∈L²(Ω), T₁

−ap,−∂

∂η(bp)

= 0

, A^∗p=A^∗p for allp∈D(A^∗).

According to [6], Theorem 5.9, (A^∗, D(A^∗)) is the adjoint of (A, D(A)) and (A, D(A)) is the infinitesimal gen- erator of a strongly continuous semigroup onL²(Ω). As in [6], we also need to define the operators (Ak, D(Ak)) and (A^∗_k, D(A^∗_k)) by settingD(Ak) =D(A),D(A^∗_k) =D(A^∗),

Akζ=Aζ−k a ζ, for allζ∈D(A), and A^∗_kζ=A^∗ζ−k a ζ, for allζ∈D(A^∗).

The interest of introducing the operator (Ak, D(Ak)) is explained right now. We can easily verify that a function z∈L²(0, T;L²(Ω)) is a weak solution to

z=Az in (0, T), z(0) =z₀, if and only if the functionζ= e^−kξz is a weak solution to

ζ =Akζ in (0, T), ζ(0) = e^−kξz₀. (2.4) We are able to prove estimates for ζ that can be translated in estimates for z. Actually, we have proved in [6], Theorem 6.2, that, for all z₀∈L²(Ω), the weak solutionζ∈L²(0, T;L²(Ω)) to equation (2.4) obeys the following inequality

1 2

₁

0

ξ 0

|ζ(x, η, t)|²dxdη+1 2

t 0

₁

0

a|ζ(ξ, η, τ)|²dηdτ +

t 0

₁

0

ξ 0

b

∂ζ

∂η

²+ ∂b

∂η

∂ζ

∂ηζ+ (c+ka)|ζ|²

dxdηdτ ≤1 2

₁

0

ξ 0

e^−2kx|z0(x, η)|²dxdη, (2.5)

for all t ∈ (0, T) and all ξ ∈ [0, L]. Formally estimate (2.5) could be obtained by multiplying equation (2.4) byζ and by making integrations in space and time. In that case we obtain an equality in (2.5) in place of an inequality. Due to the degenerate character of the operator Ak only an inequality has been proved in [6]. If we choosek >0 big enough, due to Lemma2.1below, inequality (2.5) can provide estimates forζ that can be translated in estimates forz. The existence ofk, for which we can establish a coercivity condition, is established in [6], Lemma 3.1. Due to the crucial role of this coercivity condition, we state and we give a complete proof of this lemma below.

Lemma 2.1. There existsk >0 such that ₁

0

b(ξ,·)

dz dη

²+∂b

∂η(ξ,·)dz

dηz+ (−C0+ka)z²

dη≥ C₁

2 z²H¹(0,1;d)+z²L²(0,1), (2.6) for allξ∈[0, L], allz inH¹(0,1;d).

Proof. Step 1. With the first inequality in (2.1) we can easily verify that α₁z²H¹(0,1;d)≤

₁

0

|z|²+|b(ξ,·)| dz

dη ²

dη≤α₂z²H¹(0,1;d), (2.7)

for allξ∈[0, L], and allz∈H¹(0,1;d), withα₁= min(C₁,1) and someα₂> α₁. Step 2. We set

βk(ξ;z, z) = ₁

0

b(ξ,·)

dz dη

²+∂b(ξ,·)

∂η dz

dηz+ (−c+ka)|z|² dη.

Using (2.7) and inequality (2.1), we have

βk(ξ;z, z) ≥ ₁

0

b

dz dη

²+∂b

∂η dz

dηz+ (−C0+ka)|z|²

dη

≥ ₁

0

C₁

2 |1−η|²dz dη

²+ ∂b

∂η dz dηz+

−C₀+ka−1 2

|z|²

dη+α₁

2 z²H1(0,1;d). From inequality (2.1), and Young’s inequality, it yields

₁

0

∂b

∂η dz

dηzdη≥ −C₃ε 2

₁

0

|1−η|²dz dη

²dη−C₃ 2ε

₁

0

|z|²dη, for allε >0. Consequently,βk(ξ;·,·) satisfies the estimate

βk(ξ;z, z)≥α₁

2 z²H¹(0,1;d)+

C₁ 2 −C₃ε

2

1 0

|1−η|² dz

dη

² dη+ ₁

0

−C₀+ka−1

2(1 + C₃ ε )

|z|²dη.

Now, we chooseεsuch that ^C₄¹ = ^C₂¹ −^C₂^3ε >0. We have

βk(ξ;z, z)≥α₁

2 z²H¹(0,1;d)+C₁ 4

₁

0

|1−η|² dz

dη

² dη+ ₁

0

−C₀+ka−1

2(1 +C₃ ε )

|z|²dη.

To establish the lemma, it is enough to prove that, there existsk >0 such that C₁

₁

0

|1−η|² dz

dη

² dy+ ₁

0

ka˜ |z|²dη≥ z²L²(0,1),

withC₁=C₁/(4˜r₀), ˜k=k/˜r₀, ˜c=c/˜r₀and ˜r₀=¹_2C

ε3 + 1

+C₀+ 1.

This can be shown by arguing by contradiction. We suppose that exists a sequence (zn)n⊂H¹(0,1;d) that satisfies

₁

0

|zn|²dy= 1 and C_{1 1}

0

|1−η|²dzn

dη

²dη+n ₁

0

a|zn|²dη <1. (2.8) Due to the second condition in (2.8), the sequence (zn)n(or at least a subsequence) tends to 0 almost everywhere in [0,1] and strongly inL²(,1) for all >0. Since the imbedding fromH¹(0,1) inL²(0,1) is compact and since ((1−η)zn)nis bounded inH¹(0,1), the sequence ((1−η)zn)_ntends to 0 inL²(0,1). We know that the sequence (zn)n converges to 0 in L²(1/2,1), and that the sequence ((1−η)zn)_n converges to 0 inL²(0,1/2). Thus, the sequence (zn)n converges to 0 inL²(0,1), which is in contradiction with the first condition in (2.8).

Thanks to this lemma we can prove the following theorem.

Theorem 2.1. The operator(A, D(A))is the infinitesimal generator of a strongly continuous semigroup expo- nentially stable on L²(Ω).

Proof. The complete proof of this result is given in [6], Proof of Theorem 6.1. We only explain how the exponential stability of the semigroup (e^At)t≥0, can be obtained. By using Lemma2.1and inequality (2.5), we can show that, for allz₀∈L²(Ω), the functionz(t) = e^Atz₀obeys

zL²(0,∞;L²(Ω)) ≤Cz₀L²(Ω).

The exponential stability follows from Datko’s Theorem (seee.g.[21], Thm. 3.1(i)).

In the following we shall denote byω >0 an exponent andC(ω)≥1 a constant depending onω such that e^At_L(L²(Ω))≤C(ω) e^−ωt and e^A∗t_L(L²(Ω)) ≤C(ω) e^−ωt for allt >0.

As in [6], it is useful to introduce a parameterkto obtain estimates of solutions of different equations related to the operatorA.

Now we show that there is a norm inL²(Ω), weaker than the usual one, which is associated with a Hilbert- Schmidt operator. More precisely, we have the following:

Proposition 2.1. For1≤i <∞and1≤j <∞, let us set ψi,j(x, y) =

2 Lsin

iπx L

√

2 sin (jπy), and

φα(x, y, ξ, η) = ∞ i,j=1

1

(i^2α+j^2α)^1/2ψi,j(x, y)ψi,j(ξ, η) with α >1.

Then φα belongs toL²(Ω×Ω). Let Cαbe the Hilbert-Schmidt operator defined by Cαz=

Ω

φα(·, ξ, η)z(ξ, η) dξdη.

The mapping

z −→ CαzL²(Ω)=

⎛

⎝^∞

i,j=1

1 i^2α+j^2α

Ω

ψi,jz ₂⎞

⎠

1/2

, is a norm inL²(Ω) weaker than the usual one.

Proof. The family (ψi,j)_1≤i,j≤∞is a Hilbertian basis ofL²(Ω), and the family (ψi,j⊗ψi,j)_1≤i,j≤∞is a Hilbertian basis ofL²(Ω×Ω). Thus it is easy to see that

φα²L²(Ω×Ω)= ∞ i,j=1

1

i^2α+j^2α <∞.

The end of proof is obvious.

3. Control system

In this section, we want to prove that equation (1.6) can be rewritten as a control evolution equation of the form

z =Az+B(^½γu) +F, z(0) =z₀. (3.1)

In particular we want to define the operatorsAandB, and the functionF.

3.1. Existence and uniqueness results for the state equation

To define solutions to equation (1.6) by the transposition method, we introduce the adjoint system:

−p=A^∗p+ψ in (0,∞), p(∞) = 0. (3.2)

Due to Theorem2.1, and with results in [6], we can prove the following theorem.

Theorem 3.1. Let ψ∈L²(0,∞;L²(Ω)). The system(3.2)admits a unique weak solutionpsuch that p∈Cb([0,∞);L²(Ω))∩L²(0,∞;L²(0, L;H¹(0,1;d))),

√ap∈Cw([0, L];L²(0,∞;L²(0,1))),

whereCw([0, L];L²(0,∞;L²(0,1)))is the space of continuous functions from[0, L]intoL²(0,∞;L²(0,1))equipped with its weak topology and Cb([0,∞);L²(Ω)) is the space of bounded and continuous functions from [0,∞) intoL²(Ω). It satisfies the estimate

pL∞(0,∞;L2(Ω))+√

apL∞(0,L;L2(0,∞;L2(0,1)))+pL2(0,∞;L2(0,L;H1(0,1;d)))≤CψL2(0,∞;L2(Ω)). (3.3) We define weak solutions to equation (1.6) by the transposition method.

Definition 3.1. A functionz∈L²

0,∞;L²(Ω)

is a weak solution to equation (1.6) if and only if we have

Q

zψdτdξdη =

Q

f pdτdξdη+

Ω

p(0, ξ, η)z₀(ξ, η) dξdη

− _∞

0

L 0

b(ξ,0) (g+^½γu)(τ, ξ)p(τ, ξ,0) dτdξ+ _∞

0

₁

0

a(η)zb(τ, ξ)p(τ,0, η) dτdη, (3.4) for allψ∈L²

0,∞;L²(Ω)

, where pis the solution to equation (3.2), andQ= Ω×(0,∞).

In [6], Theorem 6.6, it is shown that if z ∈ L²

0,∞;L²(Ω)

is a weak solution to equation (1.6), in the sense of semigroup theory, then it is also a solution in the sense of transposition, that is to say in the sense of Definition 3.1. By taking in (3.4) functions ψ of the form ψ(t, ξ, η) = −θ(t)ζ(ξ, η)−θ(t)A^∗ζ(ξ, η), where ζ∈D(A^∗) andθ∈ D(R⁺), we recover the weak formulation of the definition in the sense of semigroup theory.

The initial condition can also be recovered by choosing a particular sequence of functionsψ.

Theorem 3.2. Letf be inL²(0,∞;L²(Ω)),g∈L²(0,∞;L²(0, L)),u∈L²(0,∞;L²(0, L)),zb∈L²(0,∞;L²(0,1)), andz₀∈L²(Ω), then equation(1.6)admits a unique weak solutionz∈L²(0,∞;L²(Ω)). Moreover

z∈L²(0,∞;L²(0, L;H¹(0,1;d)))∩Cb([0,∞);L²(Ω)),

√a z∈Cw([0, L];L²(0,∞;L²(0,1))), and the solution obeys:

zL^∞(0,∞;L²(Ω))+√

azL^∞(0,L;L²(0,∞;L²(0,1)))+zL²(0,∞;L²(0,L;H¹(0,1;d)))

≤C₅

fL²(Q)+uL²(0,∞;L²(0,L))+gL²(0,∞;L²(0,L))+zbL²(0,∞;L²(0,1))+z₀L²(Ω)

. (3.5) Proof. Theorem 3.2 is proved in [6], Theorem 6.6. Its proof relies on inequality (2.5), on Lemma 2.1, and on an approximation procedure (the boundary terms u, g and zb are approximated by a sequence of distributed

terms).

3.2. Dirichlet and Neumann operators

Letv belong toL²(0, L) andzb ∈L²(0,1). We define the solution to the Neumann problem Aw= 0 in Ω, √

a w(0,·) = 0 in (0,1), (bw)(·,1) = 0 and ∂w

∂η(·,0) =v in (0, L), (3.6) and to the Dirichlet problem

Aζ= 0 in Ω, √

aζ(0,·) =√

azb in (0,1), (bζ)(·,1) = 0 and ∂ζ

∂η(·,0) = 0 in (0, L), (3.7) by the transposition method as follows.

Definition 3.2. A functionw∈L²(Ω) is a weak solution to equation (3.6) if and only if we have

Ω

wA^∗pdξdη=− L

0

b(ξ,0)v(ξ)p(ξ,0) dξ for allp∈D(A^∗). (3.8) Similarly, a function ζ∈L²(Ω) is a weak solution to equation (3.7) if and only if we have

Ω

ζA^∗pdξdη=− ₁

0

a(η)zb(η)p(0, η) dξ for allp∈D(A^∗). (3.9) Using the method in [6], Proof of Theorem 6.6, we can establish the following theorem.

Theorem 3.3. Let v∈L²(0, L), then equation (3.6)admits a unique weak solutionw∈L²(Ω). Moreover w∈L²(0, L;H¹(0,1;d)), √

aw∈Cw([0, L];L²(0,1)), and

√

awL^∞(0,L;L²(0,1))+wL²(0,L;H¹(0,1;d))≤CvL²(0,L). (3.10)

Let zb∈L²(0,1), then equation (3.7)admits a unique weak solutionζ∈L²(Ω). Moreover ζ∈L²(0, L;H¹(0,1;d)), √

aζ ∈Cw([0, L];L²(0,1)), and the solution obeys:

√

aζL^∞(0,L;L²(0,1))+ζL²(0,L;H¹(0,1;d))≤CzbL²(0,1). (3.11) Proof. We briefly give the proof of (3.10). The second statement can be proved in the same way. The uniqueness of solution to equation (3.6) is obvious. The only difficult point is the existence of a solution and estimate (3.10).

We proceed by approximation. We set vn(ξ, η) = nv(ξ)χn(η), where χn is the characteristic function of the interval (0,_n¹). Let wn be the solution to equation

Awn=b vn. (3.12)

It can be shown thatζn = e^−kξwn satisfies an inequality similar to (2.5). More precisely, we have 1

2 ₁

0

a ζn(x, η)²dη+ ₁

0

x 0

b

∂ζn

∂η

²+∂b

∂η

∂ζn

∂η ζn+ (c+ka)ζ_n²

dξdη≤ ₁

0

x 0

e^−kξb vnζndξdη, (3.13) for allx∈[0, L]. With Lemma2.1and classical majorizations we arrive at

√

aζnL^∞(0,L;L²(0,1))+ζnL²(0,L;H¹(0,1;d))≤CvnL²(0,L),

where the constantC is independent ofn. Therefore, there exists a subsequence, still indexed bynto simplify the notation, such that

ζn w weakly inL²(0, L;H¹(0,1;d)),

√aζn √

aw weakly-star inL^∞(0, L;L²(Ω)), (3.14) for some function w ∈ L^∞(0, L;L²(0,1))∩L²(0, L;H¹(0,1;d)). By passing to the limit in the variational formulation satisfied byζn, we can show thatwis a weak solution to equation (3.6).

3.3. Control system

We denote byN andD the operators defined by

N v=w, Dzb=ζ

wherewis the solution to equation (3.6), andζ is the solution to equation (3.7).

Observe that N belongs to L(L²(0, L), L²(0, L;H¹(0,1;d))), and that D belongs to L(L²(0,1), L²(0, L;H¹(0,1;d))). Moreover according to Definition3.2, we have

N^∗A^∗p=−b(ξ,0)p(ξ,0) and D^∗A^∗p=−a(η)p(0, η) for allp∈D(A^∗).

ThusN^∗A^∗pis the trace of−bpon (0, L)× {0}.

Using the extrapolation method the semigroup (e^tA)t∈R⁺ can be extended to (D(A^∗)). Denoting the cor- responding semigroup by (e^tA)t∈R⁺, the generator (A, D(A)) of this semigroup is an unbounded operator in (D(A^∗)) with domainD(A) = Z.

First assume that g∈C_c¹(0,∞, L²(0, L)),u∈C_c¹(0,∞;L²(0, L)), andzb∈C_c¹(0,∞;L²(0,1)), and set w(t) =N(^½γu(t) +g(t)), ζ(t) =Dzb(t).

Letz be the unique weak solution to equation (1.6), and setZ =z−w−ζ. We can check thatZ is the weak solution to the equation

Z =AZ−w−ζ+f, Z(0) =z₀, that is

Z(t) = e^tAz₀+ t

0

e^(t−τ)Af(τ)dτ− t

0

e^(t−τ)Aw(τ)dτ− t

0

e^(t−τ)Aζ(τ)dτ.

Making integration by parts, we can show that (seee.g.[2]) equation (1.6) can be rewritten in the form z=Az +f + (−A)N g + (−A)N( ^½γu) + (−A)Dz b, z(0) =z₀. (3.15) This equation is still meaningful if g ∈ L²(0,∞;L²(0, L)), u ∈ L²(0,∞;L²(0, L)), and zb ∈ L²(0,∞;

L²(0,1)). We set

F =f+ (−A)N g + (−A)Dz b and B= (−A)N, (3.16) and we obtain equation (3.1) if, by abuse of notation, we replaceAbyA.

4. Optimal control

Let us recall the definition of (Pf,g,z_b,y_d,z₀) inf

J(z, u)|(z, u)∈L²(0,∞;Z)×L²(0,∞;U), (z, u) satisfies (4.2)

, where

J(z, u) = 1 2

_∞

0

Cz(t) +yd(t)²Zdt+1 2

_∞

0

u(t)²Udt, (4.1)

with

z =Az+B(^½γu) +F, z(0) =z₀, (4.2)

and F is defined in (3.16). Let us recall that Z =L²(Ω), U =L²(0, L), C∈ L(Z), andy_d∈L²(0,∞;Z) are defined in the introduction. In the above setting · Z and · U denote respectively the norm inZ and inU, and the associated inner products will be denoted by (·,·)Z and (·,·)U.

Theorem 4.1. Assume that (H₁)−(H₄) are fulfilled. Then problem (Pf,g,z_b,y_d,z₀) admits a unique solution (¯z,u).¯

Proof. The proof is classical. We briefly introduce the main ingredients for the convenience of the reader. Let us denote by z(u) the solution to equation (4.2) corresponding to u. Due to Theorem 2.1, J(z(0),0) < ∞.

Thus (Pf,g,z_b,y_d,z₀) admits minimizing sequences, and minimizing sequences are bounded inL²(0,∞;U). Due to Theorem3.2, if a sequence (un)n converges weakly inL²(0,∞, U) to someu, then (z(un))nconverges weakly in L²(0,∞;L²(0, L;H¹(0,1;d))) to z(u). Thus, by standard arguments, if (un)n is a minimizing sequence, converging to ufor the weak topology ofL²(0,∞;U), then

J(z(u), u)≤lim inf

n→∞ J(z(un), un) = inf(Pf,g,z_b,y_d,z₀).

Thus, (z(u), u) is a solution of (Pf,g,z_b,y_d,z₀). The uniqueness follows from the strict convexity of the mapping

u →J(z(u), u).

Theorem 4.2. If (¯z,u)¯ is the solution to(Pf,g,z_b,y_d,z₀)then

¯

u(t) =^½γbp|¯γ×{0}=−^½γB^∗p(t),¯ (4.3) wherep¯is the solution to equation(3.2)with

ψ=C^∗(C¯z+yd).