EXISTENCE OF OPTIMAL CONTROL FOR WATER INFILTRATION INTO AN UNSATURATED POROUS MEDIUM

FOR WATER INFILTRATION INTO AN UNSATURATED POROUS MEDIUM

CORNELIA-ANDREEA CIUTUREANU

We study an identification problem for a mathematical model describing the rain- fall type infiltration into an unsaturated porous medium.

AMS 2000 Subject Classification: 93B30, 76S05, 93C20.

Key words: water infiltration in unsaturated soils, nonlinear boundary value prob- lems for parabolic PDE.

1. STATEMENT OF THE PROBLEM

Our purpose is to study an identification problem treated as an opti- mal control problem for a mathematical model describing the rainfall type infiltration into an unsaturated porous medium.

By Ω we designate an open bounded subset of R^N, N = 1,2,3, with boundary Γ := ∂Ω piecewise smooth, composed of two disjoint boundaries Γ_u and Γ_α, both sufficiently smooth. We consider that the soil boundary is horizontal and define Γu :={x∈Γ; x3 = 0},Γ = Γu∪Γα,where Γu∩Γα =∅ and Γ_α is a semipermeable underground boundary. Correspondingly, we set Q:= Ω×(0, T),Σu = Γu×(0, T) and Σα= Γα×(0, T).

Letx∈Ω be the vectorx= (x1, x2, x3). The time variabletruns in the interval (0, T),withT finite.

The mathematical model we consider has been introduced in [5]. It consists of Richards’ equation written for the soil moisture, θ(x, t), with an initial condition and boundary conditions expressing the input given by a vertical rainfall on the surface of the soil Γ_uand a flux across the boundary Γ_α. With our notation, we write the dimensionless mathematical model as

(1)















∂θ

∂t − 4D^∗(θ) +^∂K(θ)_∂x

3 =f inQ,

θ(x,0) =θ₀(x) in Ω,

(K(θ)·i3− ∇D^∗(θ))·ν =u on Σu,

(K(θ)·i₃− ∇D^∗(θ))·ν−αD^∗(θ) =f₀ on Σ_α.

MATH. REPORTS10(60),2 (2008), 143–154

By ν we denote the unit vector along the outward normal to Γ, i3 is the unit vector along theOx₃ axis, downwards orientated,f is a source or a sink within the flow domain, f0 given on Γα characterizes the properties of the domain exterior to Ω while αis related to the variable permeability on Γα.The vector q(x, t) :=K(θ)·i₃− ∇D^∗(θ) is the water flux. We consider that the Ox₃ axis is downwards orientated, so that u(x, t)≤0 and specify thatu=−u_R, where u_R≤0 is the physical rain rate.

For the unsaturated flow,D^∗ is related to the water diffusivity function, which is denoted by D. By K we denote the water hydraulic conductivity.

The hypotheses made with respect to the hydraulic functions in the un- saturated model are as follows:

(a) D : [0, θ_s) → [ρ,+∞) is a continuous and monotonically increasing function, blowing up at θ=θs;

(b)K : [0, θ_s]→[0, K_s] is monotonically increasing and Lipschitz;

(c)α: Γα→[αm, αM] is positive and continuous, withαm>0.

All constants further involved in the model, namely, θ_s (the saturation value of the moisture) and ρ =D(0), are considered to be positive such that the assumptions made above are consistent with specific hydraulic models well-known in soil sciences.

Even if in the physical domainθis positive, for the mathematical treat- ment we extend continuously the hydraulic functions to the negative axis, by setting D(r) =ρand K(r) = 0, forr ≤0.Thus, we have the following contin- uous extensions of the dimensionless diffusivity and hydraulic conductivity:

(2) D(θ) :=

ρ ifθ <0, D(θ) if 0≤θ < θ_s,

(3) K(θ) :=

0 ifθ <0, K(θ) if 0≤θ < θ_s.

In the unsaturated case the extended functionDsatisfies the hypotheses (i_D) D(θ)≥ρ >0, ∀θ∈(−∞, θ_s),

(iiD) lim

θ%θs

D(θ) = +∞, (iii_D) lim

θ%θs

Z θ 0

D(ξ)dξ= +∞.

As for K, it satisfies lim

θ%θs

K⁰(θ) <+∞. This property corresponds to a weaker nonlinear contribution of the hydraulic conductivity around the satu- ration value and implies that K is Lipschitz continuous on (−∞, θ_s], i.e.,

(ik) there exists M >0 such that

|K(θ₁)−K(θ2)| ≤M|θ₁−θ2|, ∀θ₁, θ2 ∈(−∞, θ_s].

We denote by D^∗ the primitive of the diffusivity D that vanishes at 0, i.e.,

(4) D^∗(θ) =

Z θ 0

D(ξ)dξ forθ < θ_s.

According to (iD)–(iiiD) the functionD^∗ is differentiable and monotoni- cally increasing in (−∞, θ_s), and satisfies

(i) (D^∗(θ₁)−D^∗(θ₂))(θ₁−θ₂)≥ρ(θ₁−θ₂)², ∀θ₁, θ₂ ∈(−∞, θ_s), (ii) lim

θ→−∞D^∗(θ) =−∞, (iii) lim

θ%θs

D^∗(θ) = +∞.

Condition (iii) can be found in the Broadbridge and White model (see [3]), a hydraulic model which covers a large variety of soils.

The identification problem for the time average of moisture observations available in the whole flow domain Ω has been studied in [6]. We want to get information about the water supply rate (or simply rain rate) knowing the time average of moisture observations in a subdomain Ω0 of Ω during the time period (0, T).Hence, we deal with an identification problem expressed in mathematical terms as

(5) min

u∈U_R

Z

Ω0

1 T

Z T 0

θ(x, t)dt−θ⁰(x) 2

dx,

where Ω₀ is a given subdomain of Ω andθ⁰(x) the time average of the moisture observations

θ⁰(x) = 1 T

Z T 0

θ^observed(x, t)dt, a function assumed to be in L²(Ω0).

It is a realistic assumption that rains are of finite rates, so that we are enabled to consider that uR = −u is bounded from above by a func- tion R ∈ L^∞(Σ_u). Hence we denote by U_R the admissible range of the rain rate, defined by

(6) UR:={u∈L^∞(Σu); −R(x, t)≤u(x, t)≤0 a.e. on Σu}.

Next, we introduce the functional framework and a Cauchy problem corre- sponding to the original boundary value problem (1) and prove existence and uniqueness results on the basis of the quasi m-accretivity of the associated operator.

Further, we will treat problem (5) and show that it has a solution.

2. EXISTENCE IN THE STATE SYSTEM

We shall use the same functional framework as in [6]. We denote by (·,·) and k·k the scalar product and, respectively, the norm on L²(Ω). The scalar products and norms defined in other spaces will be specified correspondingly.

For convenience, if confusion cannot arise, we shall indicate no longer the integration variable in the integrands.

Consider the spaceV =H¹(Ω) endowed with the norm

(7) kθk_V =

Z

Ω

|∇θ|²dx+ Z

Γα

α|θ|²dσ 1/2

,

which is equivalent to the standard Hilbertian norm. We shall denote by V⁰ the dual space of V, on which we introduce the scalar product

(8) (θ, θ)V⁰ =θ(ψ), ∀θ, θ∈V⁰, where ψ∈V satisfies the boundary value problem

(9) −4ψ=θ,

∂ψ

∂ν +αψ

Γα

= 0, ∂ψ

∂ν = 0 on Γu. Obviously, kψk_V =

θ V⁰.

We introduce the operatorA:D(A)⊂V⁰ →V⁰ by (10) hAθ, ψi_V⁰_,V :=

Z

Ω

∇D^∗(θ)· ∇ψ−K(θ)∂ψ

∂x3

dx+

Z

Γα

αD^∗(θ)ψdσ,

∀ψ∈ V, where D(A) :=

θ∈L²(Ω); D^∗(θ)∈V a.e. x∈Ω . Here, h·,·i_V0,V

is the pairing between V⁰ and V. Moreover, we define the operator B ∈ L(L²(Γ_u);V⁰) and the function f_Γ∈L²(0, T;V⁰) by

(11) Bu(ψ) :=− Z

Γu

uψdσ and f_Γ(t)(ψ) :=− Z

Γα

f0ψdσ, ∀ψ∈V.

With this notation we introduce the Cauchy problem

(12)

dθ

dt +Aθ=f+Bu+f_Γ, a.e. t∈(0, T).

θ(0) =θ0.

The first equation of (12) is equivalent to

(13)

Z T 0

dθ

dt(t), φ(t)

V⁰,V

dt+ Z

Q

∇D^∗(θ)· ∇φ−K(θ) ∂φ

∂x₃

dxdt

= Z T

0

hf(t), φ(t)i_V0,V dt− Z

Σα

(αD^∗(θ) +f₀)φdσdt

− Z

Σu

uφdσdt, ∀φ∈L²(0, T;V).

Ifθis a strong solution to the Cauchy problem (12), then it satisfies (1) in the sense of distributions and the boundary conditions in the sense of the trace thsory (the proof can be found in [5]).

We shall prove the existence and uniqueness of the solution to the Cauchy problem (12) on the basis of fundamental results involving evolution equations with monotone operators in Banach spaces (see [1] and [2]). The existence and uniqueness result is based on the quasi m-accretivity of the operator A and we shall follow the ideas of [5], Chapter 5.

Proposition 1. Assume that hypothesis (i)–(iii)and (i_k) hold. Then A is quasi m-accretive on V⁰.

Proof. For λ >0 large enough we should prove that (λI+A)θ−(λI+A)θ, θ−θ

V⁰ ≥0 and

R(λI+A) =V⁰.

First, we have to show that Ais quasi-monotone. Ifλ >0 then (14) (λI+A)θ−(λI+A)θ, θ−θ

V⁰≥

λ−M² 2ρ

θ−θ

2 V⁰+ρ

2

θ−θ

2.

In conclusion, ifλis large enough, namelyλ≥ ^M_2ρ²,we get the quasi-monotony of A.

Next, we have to show thatR(λI+A) =V⁰,to mean that for eachg∈V⁰ there exists θ∈D(A) such that

(15) λθ+Aθ=g.

Denote ζ := D^∗(θ) ∈ V. Since D^∗(θ) is continuous and monotoni- cally increasing on (−∞,+∞) and R(D^∗) = (−∞,+∞), its inverse G(ζ) :=

(D^∗)⁻¹(ζ) is Lipschitz. Therefore, (15) can be written as

(16) λG(ζ) +AGζ =g,

with AG:V →V⁰ defined by hA_Gζ, ψi_V0,V :=

Z

G

∇ζ· ∇ψdx− Z

Ω

K(G(ζ))∂ψ

∂x₃dx+α Z

Γα

ζψdσ.

The strong monotony of λG+AG follows from (λG+A_G)ζ−(λG+A_G)ζ, ζ−ζ

V⁰,V ≥

≥

λρ−M² 2

G(ζ)−G(ζ)

2+1 2

ζ−ζ

2 V ≥0,

for λ large enough, namely, (λ > ^M_2ρ²). By Minty’s theorem, the operator is surjective, proving that (16) has a unique solution. This completes the proof of the quasi m-accretivity of A.

Theorem 2. Assume (i)–(iii)and (ik). Let

f∈W^1,1(0, T;V⁰), f₀∈W^1,1(0, T;L²(Γ_α)), u∈W^1,1(0, T;L²(Γ_u)), θ₀∈D(A) hold. Then there exists a unique solution θ ∈ C([0, T];V⁰) to problem (12) such that

(17) θ∈W^1,∞(0, T;V⁰)∩L^∞(0, T;D(A))∩L^∞(0, T;V),

(18) D^∗(θ)∈L^∞(0, T;V),

(19) j(θ)∈L^∞(0, T;L¹(Ω)).

If θλ and θµare two solutions to problem (12) corresponding to the data (20) f =f_λ, θ₀ =θ⁰_λ, f_Γ =f_λ^Γ, f₀ =f_λ⁰, u=u_λ, respectively,

f =f_µ, θ₀ =θ⁰_µ, f_Γ=f_µ^Γ, f₀=f_µ⁰, u=u_µ, then

(21)

kθ_λ(t)−θ_µ(t)k²_V0 ≤γ₁(α_m)

θ⁰_λ−θ⁰_µ

2 V⁰+

Z T 0

kf_λ(τ)−f_µ(τ)k²_V0dτ+ +

Z T 0

ku_λ(τ)−u_µ(τ)k²_L2(Γu)dτ + Z T

0

f_λ⁰(τ)−f_µ⁰(τ)

2

L²(Γα)dτ

. The estimate

(22)

kθ(t)k² ≤ Z

Ω

j(θ(x, t))dx+

Z t 0

dθ dτ(τ)

2 V⁰

dτ+ Z t

0

kD^∗(θ(τ))k²_V dτ ≤

≤γ2(αm) Z

Ω

j(θ0(x))dx+ Z T

0

kf(τ)k²_V0dτ + +

Z _T

0

ku(τ)k²_L2(Γu)dτ + Z _T

0

kf₀(τ)k²_L2(Γα)dτ

also holds, where j:R→(−∞,∞) is defined by j(r) :=Rr

0 D^∗(ξ)dξ, ∀r∈R.

We note that j is a proper convex lower-semicontinuous function onR and

(23) ∂j(r) =

( D^∗(r), r∈(−∞, θ_s),

∅, r≥θs

(see [5]).

Proof. The existence of a unique solution to (12) follows from the general results concerning evolution equations withm-accretive operators (see [1], [2], [5]). It belongs to the spaces specified in (17). Suppose now that θλ and θµ

are two solutions to problem (12) corresponding to the data (20). Multiply the equation

d

dt(θ_λ−θ_µ) +Aθ_λ−Aθ_µ=f_λ−f_µ+f_λ^Γ−f_µ^Γ+Bu_λ−Bu_µ

byθλ−θ_µscalarly inV⁰, and integrate it over (0, t) witht∈(0, T).After some computations we get

1

2kθ_λ(t)−θ_µ(t)k²_V0+ρ 2

Z _t

0

kθ_λ(τ)−θ_µ(τ)k²dτ ≤

≤ 1 2

θ_λ⁰−θ⁰_µ

2 V⁰+1

2

1 +c²_u+c²_α+M² ρ

Z t 0

kθ_λ(τ)−θ_µ(τ)k²_V0dτ+

+1 2

Z t 0

kf_λ(τ)−fµ(τ)k²_V0dτ+1 2

Z t 0

ku_λ(τ)−uµ(τ)k²_L2(Γu)dτ+ +1

2 Z t

0

f_λ⁰(τ)−f_µ⁰(τ)

2

L²(Γα)dτ, wherecu,cαare constants depending on _α¹

m, the dimensionN and the domain Ω (occuring from the trace theorem and the Poincar´e inequality). Next, we apply Gronwall’s lemma for g(t) =kθ_λ(t)−θ_µ(t)k²_V0 and deduce (21).

For the second estimate, we multiply the first equation of (12) by _dτ^dθ scalarly inV⁰ and integrate the result over (0, t)×Ω witht∈(0, T).We obtain

Z t 0

dθ dτ(τ)

2 V⁰

dτ+ Z t

0

Z

Ω

∇D^∗(θ(τ))· ∇ψdxdτ+ Z t

0

Z

Γα

αD^∗(θ(τ))ψdσdτ =

= Z t

0

Z

Ω

f(τ)ψdxdτ − Z t

0

Z

Γu

u(τ)ψdσdτ − Z t

0

Z

Γα

f₀(τ)ψdσdτ+ +

Z t 0

Z

Ω

K(θ(τ))∂ψ

∂x3

dxdτ,

with ψ∈V satisfying the boundary problem

−4ψ= dθ

dτ(τ), ∂ψ

∂ν Γu

= 0 and

∂ψ

∂ν +αψ

Γα

= 0.

Taking into account that Z

Ω

D^∗(θ(τ))dθ

dτ(τ)dx= Z

Ω

dj(θ)

dτ (τ)dx= d dτ

Z

Ω

j(θ)dx, after integration with respect to τ,we get

Z

Ω

j(θ(x, t))dx+ Z t

0

dθ dτ(τ)

2 V⁰

dτ ≤

≤ Z

Ω

j(θ0)dx+1 2

Z t 0

dθ dτ(τ)

2 V⁰

dτ + 2 Z t

0

kf(τ)k²_V0dτ+ +2

Z t 0

c²_uku(τ)k²_L2(Γu)dτ + 2 α_m

Z t 0

kf₀(τ)k²_L2(Γα)dτ+ 2M² Z t

0

kθ(τ)k²dτ.

From (i) we deduce that R

Ωj(θ)dx ≥ ^ρ₂kθk² and, after applying Gron- wall’s lemma to g(t) =kθ(t)k², we get

kθ(t)k² ≤π₀(α_m) exp 4M²

ρ t

×

× Z

Ω

j(θ0)dx+ Z T

0

kf(τ)k²_V0+ku(τ)k²_L2(Γu)+kf₀(τ)k²_L2(Γα)

dτ

. Thus, we have

(24) Z

Ω

j(θ(x, t))dx+ Z t

0

dθ dτ(τ)

2 V⁰

dτ ≤π0(αm)Sexp 4M²

ρ t

, where

S = Z

Ω

j(θ0)dx+ Z T

0

kf(τ)k²_V0 +ku(τ)k²_L2(Γu)+kf₀(τ)k²_L2(Γα)

dτ.

Finally, we multiply equation (12) byD^∗(θ) scalarly inV and integrate it over (0, t)×Ω with t∈(0, T).Similarly to the above, we obtain

(25)

Z

Ω

j(θ(x, t))dx+ Z T

0

kD^∗(θ(τ))k²_V dτ ≤4Sexp 4M²

ρ t

.

Adding (24) and (25), we obtain (22), as claimed. By (22) we also get (19).

Theorem 3. Let

(26) f ∈L²(0, T;V⁰), u∈L²(0, T;L²(Γu)), f0∈L²(0, T;L²(Γα)),

(27) θ0 ∈L²(Ω) such that j(θ)∈L¹(Ω), θ0 < θs a.e. in Ω.

Then there exists a unique solution θ∈C([0, T];L²(Ω)) to problem (12) that satisfies

(28) θ∈W^1,2(0, T;V⁰)∩L²(0, T;V), D^∗(θ)∈L²(0, T;V), (29) j(θ)∈L^∞(0, T;L¹(Ω))and θ < θs a.e. in Ω.

The solution also satisfies estimates (21)–(22).

The proof is similar to that in [5] of Theorem 3.10, page 89.

3. IDENTIFICATION PROBLEM IN THE CASE OF A TIME AVESAGE OBSERVATION

In this section we study problem (5). For that, let us set

(30) MT(θ) = 1

T Z T

0

θ(x, t)dt.

Theorem 4. Let f ∈L²(0, T;V⁰), f0∈L²(Σα), θ0∈L²(Ω)and θ0 < θs

a.e. in Ω.Then problem (5) has at least one solution.

Proof. Letd= min

u∈UR

R

Ω0

1 T

RT

0 θ(x, t)dt−θ⁰(x)2

dx and let {u_n} ⊂U_R be a minimizing sequence, that is,

(31) d≤ Z

Ω0

1 T

Z T 0

θ_n(x, t)dt−θ⁰(x) ²

dx≤d+ 1

n, n≥1,

where θn is the solution to (12) with u = un. According to Theorem 3, θn

satisfies (28), (29), (21) and (22) for each n. Because{u_n} ⊂U_R,selecting a subsequence, if necessary, we have

u_n→uweak-star in L^∞(Σ_u) and u∈U_R.

We approximatef,uandf0by sequencesfn∈W^1,1(0, T;V⁰), un∈W^1,1(0, T; L²(Γ_u)), f_n⁰∈W^1,1(0, T;L²(Γ_α)) and take a sequence

θ⁰_n ∈D(A) such that θ_n⁰ → θ0 strongly in V⁰,

fn → f strongly in L²(0, T;V), f_n⁰ → f0 strongly inL²(0, T;L²(Γα)), un → u strongly inL²(0, T;L²(Γu)).

By Theorem 2, the Cauchy problem (32)

dθn

dt +Aθ_n=f_n+Bu_n+f_n^Γ, a.e. in (0, T), θ_n(0) =θ_n⁰

has a unique solution θ_n∈C([0, T];V⁰) such that

θn∈W^1,∞(0, T;V⁰)∩L^∞(0, T;D(A))∩L^∞(0, T;V), D^∗(θ_n)∈L^∞(0, T;V), j(θ_n)∈L^∞(0, T;L¹(Ω))

and θn also satisfies estimates (21)–(22) with constants independent ofn.

By estimate (21), the sequence{θ_n} is Cauchy inC([0, T];V⁰)∩L²(0, T; L²(Ω)).Hence

θ_n→θ strongly inC([0, T];V⁰), (33)

θn→θ strongly inL²(0, T;L²(Ω)).

By (22) we have dθn

dt → dθ

dt weakly inL²(0, T;V⁰), D^∗(θn)→η weakly in L²(0, T;V),

and the latter implies thatθ_n→θweakly inL²(0, T;V⁰).Moreover,K(θ_n)→ K(θ) strongly inL²(0, T;L²(Ω)).

Since θ_n → θ in L²(0, T;L²(Ω)), we have θ_n(x, t) → θ(x, t) a.e. on Q.

By Egorov’s theorem, for any ε > 0 there exists a measurable subset Qε ⊂ Q such that meas(Q_ε) < ε and θ_n → θ uniformly on Q\Q_ε. Since D^∗ is a continuous function on (−∞, θ_s) we have D^∗(θ_n(x, t)) → D^∗(x, t) on Q\Q_ε, hence D^∗(θn) → D^∗(θ) strongly in L²(Q\Q_ε). Therefore, η =D^∗(θ) a.e. on Q, soD^∗(θ_n)→D^∗(θ) weakly inL²(0, T;V).

SinceA is quasim-accretive onV⁰,its realization Aeon L²(0, T;V⁰) also is quasi m-accretive. Thus,Aeis demiclosed, i.e.,

Aθe _n→Aθe weakly in L²(0, T;V⁰).

Passing to the limit in the problem dθ_n

dt +Aθe _n=f_n+Bu_n+f_n^Γ, θ_n(0) =θ⁰_n, we deduce that θsatisfies problem (12).

Since D^∗(θ) ∈ L²(0, T;V), we have R

Ω(D^∗(θ))²dx ≤ const. a.e. t ∈ (0, T),which implies that θ(x, t)< θs a.e. in Q.

Using (33) we deduce that

kM_T(θn)−MT(θ)k_L2(Ω0)≤ kM_T(θn)−MT(θ)k_L2(Ω)≤

≤ 1 T

Z T 0

kθ_n(t)−θ(t)kdt≤ 1

√

T kθ_n−θk_L2(Q), which implies that M_T(θ_n)→M_T(θ) strongly inL²(Ω₀).

Passing to the limit in (31) and taking into account the weak lower semicontinuity property, we obtain

d≤ Z

Ω0

1 T

Z T 0

θ(x, t)dt−θ⁰(x) ²

dx≤

≤lim inf

n→∞

Z

Ω0

1 T

Z T 0

θn(x, t)dt−θ⁰(x) 2

dx≤d.

Thusu is a solution to problem (5).

Remark 5. The identification problem (5) says that we can retrace the rain history even if the observations are available only in a limited part of the flow domain.

Remark 6. Sometimes, the observations are available only in a small interval of time around a moment t₀. The above result is still true for the identification problem

(34) min

u∈U_R

Z

Ω

Mt0(θ)−θ⁰(x)2

dx

, where

Mt0(θ) = 1 2η

Z t0+η t0−η

θ(x, t)dt, t0 ∈(0, T),

and θ⁰(x) is the time average of the observations over the time interval (t0−η, t0+η) in Ω.

The approach is similar to the previous one. We shall only present the differences. The norm on L²(Ω) of the difference of Mt0(θn) and Mt0(θ) is equal to

Z

Ω

1 2η

Z t0+η t0−η

(θ_n(t)−θ(t)) dt 2

dx

!1/2

≤

≤ Z

Ω

1 2η

Z T 0

(θn(t)−θ(t)) dt 2

dx

!1/2

≤ T

2ηkM_T(θn)−MT(θ)k_L2(Ω)≤

≤ 1 2η

Z T 0

kθ_n(t)−θ(t)kdt≤ 1

√2ηkθ_n−θk_L2(Q).

Hence Mt0(θn) → Mt0(θ) strongly inL²(Ω) and so, according to Theorem 4, there exists at least a controlu satisfying (34).

REFERENCES

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Received 4 February 2008 “Caius Iacob–Gheorghe Mihoc”

Institute of Mathematical Statistics and Applied Mathematics Calea 13 Septembrie nr. 13 050711 Bucharest 5, Romania [email protected]