Control of incomplete data problems. Application to an ecology problem

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Control of incomplete data problems. Application to an ecology problem

AbdennebiOmrane (joint work with L.Louison)

UMR 42 Ecologie des Forˆets de Guyane (EcoFoG) University of French Guiana, Campus of TrouBiran, Cayena

[email protected]

Benasque (Spain), August 27^th, 2019 8^thConference -PDE’s, Optimal Design and Numerics

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Biological aspect of nutrient uptake

Photosynthesis:

−carbon (C)

from carbon dioxide (CO2).

Root absorption:

- magnesium (Mg²⁺), - calcium (Ca²⁺), - potassium (K⁺), - nitrogen (NO₃⁻), - phosphorus (P), - water (H2O).

Absorption of nutrients by roots

small absorption zone need ofmore nutrients !

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Biological aspect of nutrient uptake

Photosynthesis:

−carbon (C)

Root absorption: - magnesium (Mg²⁺), - calcium (Ca²⁺), - potassium (K⁺), - nitrogen (NO₃⁻), - phosphorus (P), - water (H2O).

Absorption of nutrients by roots

small absorption zone need ofmore nutrients !

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Biological aspect of nutrient uptake

Photosynthesis:

−carbon (C)

Root absorption: - magnesium (Mg²⁺), - calcium (Ca²⁺), - potassium (K⁺), - nitrogen (NO₃⁻), - phosphorus (P), - water (H2O).

Absorption of nutrients by roots small absorption zone need ofmore nutrients !

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Description of the domain of study

The figure, above, shows the domain of study Ω, an open bounded set ofR²of boundary Γ.

Γ1: the root surface,

Γ2: the boundary between a piece of observed soil and the rest of soil, where Γ := Γ1∪Γ2et Γ1∩Γ2=∅.

For the study of uptake problem, we consider :

Q:=]0,T[×Ω, Σ1:=]0,T[×Γ₁,

Σ2:=]0,T[×Γ₂, complementary of Σ1.

Absorption mechanisms

root interception of nutrients, mass flow ..< 5%

diffusion :93%of phosphorus,80% of potassium.

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Description of the domain of study

Γ2: the boundary between a piece of observed soil and the rest of soil, where Γ := Γ1∪Γ2et Γ1∩Γ2=∅. For the study of uptake problem, we consider :

Q:=]0,T[×Ω, Σ1:=]0,T[×Γ₁,

diffusion :93%of phosphorus,80% of potassium.

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Description of the domain of study

Γ2: the boundary between a piece of observed soil and the rest of soil, where Γ := Γ1∪Γ2et Γ1∩Γ2=∅. For the study of uptake problem, we consider :

Q:=]0,T[×Ω, Σ1:=]0,T[×Γ₁,

diffusion :93%of phosphorus,80%

of potassium.

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The Nutrient uptake model

The Nye-Tinker-Barber (NTB) system (1980’s):

Let the functionc=c(t,x) represents the concentration of nutrient,









 α∂c

∂t +q∇c−D∆c = 0 in Q,

(D∇c−qc).−→n = h(c) on Σ1, (D∇c−qc).−→n = 0 on Σ2,

c(0,x) = c0(x) in Ω.

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Description of the NTB System:

α=b+θwithb: the buffer power andθ: the liquid saturation.

q∇represents the spatial convection withq: the Darcy flux, with divq= 0.

D∆the spatial diffusion withDthe diffusion coefficient.

h(c) = _K+c^{I c} the Michaelis-Menten function : nutrient absorption function at the root surface. The linear version ofhish(c) = ^{I c}_K whenK>>c.

I: the maximum uptake constant andK: the Michaelis-Menten constant.

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The Nutrient uptake model









 α∂c

∂t +q∇c−D∆c = 0 in Q,

c(0,x) = c0(x) in Ω.

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The Nutrient uptake model









 α∂c

∂t +q∇c−D∆c = 0 in Q,

c(0,x) = c0(x) in Ω.

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The existence of a unique solution for the NTB system

We introduce the Hilbert space : V =n

ψ∈H¹(Ω), ψ_|

Γ2 = 0o

, with kψk²_V =kψk²_L₂_(Ω)+k∇ψk²_L₂_(Ω).

Proposition

Suppose that the vector field q satisfies|q| ∈L^∞((0,T)×Ω). Then there is aunique solution c∈V (here c∈L²(0,T;V)) such that :

a(t;c, ψ) =L(t;ψ) ∀ψ∈V, (2) where

a(t;c, ψ) =1 2 Z

Ω

q.(ψ∇c−c∇ψ)dx+D Z

Ω

∇c∇ψdx ψ∈V, (3) and

L(t;ψ) = Z

Γ₁

h(c)ψ(x).ndγ ψ∈V. (4)

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Associated cultures (cropping)

Absorption of nutrients in polluted soils.

→ Banana needs important amount of : - Water,Nitrogen.

- Use ofchemicals: Nitrogen fertilizers !

→ Solution :Associated plants

Optimal control

→ Control of the nutrient

concentration, addition of nutrient (associated plant).

→ Control of systems of incomplete data (pollution).

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Associated cultures (cropping)

Optimal control

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Associated cultures (cropping)

Optimal control

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Nutrient uptake model with pollution & Optimal control

The nutrient uptake model with pollution is given by the following system :









 α∂c

∂t+q∇c−D∆c = g in Q,

(D∇c−qc).−→n = ^{I c}_K on Σ1, (D∇c−qc).−→n = −v on Σ2,

c(0,x) = 0 in Ω,

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withg∈G ⊂L²(Q) : unknown pollution function, andv ∈L²(Σ2) : control function.

Minimize : J(v,g) =kc(v,g)−cke²

L²(Σ₁)+Nkvk²

L²(Σ₂) ∀g ∈ G. (6)

A natural idea : inf

v∈L²(Σ2)

sup

g∈L²(Q)

J(v,g)

!

. But sup

g∈L²(Q)

J(v,g) =+∞! Indeed, we have :

c(v,g) =c(v,0) +c(0,g), and c(0,g) =A^∗g

→ No-regret control:J(v,g)≤J(0,g),∀g ∈G⊂L²(Q).

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Nutrient uptake model with pollution & Optimal control









 α∂c

c(0,x) = 0 in Ω,

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L²(Σ₁)+Nkvk²

L²(Σ₂) ∀g ∈ G. (6)

v∈L²(Σ2)

sup

g∈L²(Q)

J(v,g)

!

. But sup

g∈L²(Q)

c(v,g) =c(v,0) +c(0,g), and c(0,g) =A^∗g

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Nutrient uptake model with pollution & Optimal control









 α∂c

c(0,x) = 0 in Ω,

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L²(Σ₁)+Nkvk²

L²(Σ₂) ∀g ∈ G. (6)

v∈L²(Σ2)

sup

g∈L²(Q)

J(v,g)

!

. But sup

g∈L²(Q)

c(v,g) =c(v,0) +c(0,g), and c(0,g) =A^∗g

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Towards the low-regret . . . control

Definition

We say that the function u∈L²(Σ₂)is ano-regret control, if it is a solution of the following new MinMaxproblem :

v∈L2 (Σinf2 )

sup

g∈G

[J(v,g)−J(0,g)]

!

. (7)

J(v,g)−J(0,g) =J(v,0)−J(0,0) + 2hc(v,0),c(0,g)i

Remark : hc(v,0),c(0,g)i=hξ(v),gi, ξlinearsol. to the adjoint problem

The Low-regret control uγis a low-regret controliff:

inf

v∈L²(Σ2)

sup

g∈G

J(v,g)−J(0,g)−γkgk²_G

!

= inf

v∈L²(Σ₂)

J(v,0)−J(0,0)+sup

g∈G

2hξ(v),gi−γkgk²_G

!

(conjugate)

= inf

v∈L²(Σ₂)

J(v,0)−J(0,0)+1 γ ξ(v)

G0

= inf

v∈L²(Σ₂)

J^γ(v).

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Towards the low-regret . . . control

Definition

v∈L2 (Σinf2 )

sup

g∈G

[J(v,g)−J(0,g)]

!

. (7)

J(v,g)−J(0,g) =J(v,0)−J(0,0) + 2hc(v,0),c(0,g)i

Remark : hc(v,0),c(0,g)i=hξ(v),gi, ξlinearsol. to the adjoint problem The Low-regret control uγis a low-regret controliff:

inf

v∈L²(Σ₂)

sup

g∈G

!

= inf

v∈L²(Σ₂)

J(v,0)−J(0,0)+sup

g∈G

!

(conjugate)

= inf

v∈L²(Σ₂)

J(v,0)−J(0,0)+1 γ ξ(v)

G0

= inf

v∈L²(Σ₂)

J^γ(v).

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Towards the low-regret . . . control

Definition

v∈L2 (Σinf2 )

sup

g∈G

[J(v,g)−J(0,g)]

!

. (7)

J(v,g)−J(0,g) =J(v,0)−J(0,0) + 2hc(v,0),c(0,g)i

Remark : hc(v,0),c(0,g)i=hξ(v),gi, ξlinearsol. to the adjoint problem The Low-regret control uγis a low-regret controliff:

inf

v∈L²(Σ₂)

sup

g∈G

!

= inf

v∈L²(Σ₂)

J(v,0)−J(0,0)+sup

g∈G

!

(conjugate)

= inf

v∈L²(Σ₂)

J(v,0)−J(0,0)+1 γ ξ(v)

G0

= inf

v∈L²(Σ₂)

J^γ(v).

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Existence of a unique low-regret control

Proposition

The optimal problem inf

v∈L²(Σ₂)

J^γ(v)admits a unique solution uγ calledlow-regret controlfor the NTB system with pollution.

Proof - We recall thatJ^γ(v)=J(v,0)−J(0,0)+1 γkξ(v)k²

L2 (Q) and that J(v,0) =kc(v,0)−ck˜ ²

L²(Σ₁)+Nkvk²

L²(Σ₂) and J(0,0) =k˜ck²

L²(Σ₁). The cost functionJ^γ(v) satisfiesJ^γ(v)≥ −J(0,0), for anyv∈L²(Σ2). Therefore, it existskγ= inf

v∈L²(Σ₂)

J^γ(v). Consider a minimizing sequence{vn(γ)}={vn}, then :

kc(vn,0)−ck˜ ²_L₂_(Σ

1)+Nkvnk²_L₂_(Σ

2)+1

γkξ(vn)k²_L₂_(Q)≤kγ+ 1 +kck˜ ²

L2 (Σ1 )

.

→ Notations :A:=α∂

∂t+q∇ −D∆ inQ, andB:=D∇ −qon Σ.

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Existence of a unique low-regret control

Proposition

v∈L²(Σ₂)

L²(Σ₁)+Nkvk²

L²(Σ₂) and J(0,0) =k˜ck²

L²(Σ₁).

The cost functionJ^γ(v) satisfiesJ^γ(v)≥ −J(0,0), for anyv∈L²(Σ2). Therefore, it existskγ= inf

v∈L²(Σ₂)

1)+Nkvnk²_L₂_(Σ

2)+1

γkξ(vn)k²_L₂_(Q)≤kγ+ 1 +kck˜ ²

L2 (Σ1 )

.

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Existence of a unique low-regret control

Proposition

v∈L²(Σ₂)

L²(Σ₁)+Nkvk²

L²(Σ₂) and J(0,0) =k˜ck²

v∈L²(Σ₂)

1)+Nkvnk²_L₂_(Σ

2)+1

γkξ(vn)k²_L₂_(Q)≤kγ+ 1 +k˜ck²

L2 (Σ1 )

.

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Existence of a unique low-regret control

Proposition

v∈L²(Σ₂)

L²(Σ₁)+Nkvk²

L²(Σ₂) and J(0,0) =k˜ck²

v∈L²(Σ₂)

1)+Nkvnk²_L₂_(Σ

2)+1

γkξ(vn)k²_L₂_(Q)≤kγ+ 1 +k˜ck²

L2 (Σ1 )

.

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Characterization of the low-regret control to the NTB system

Proposition

The low-regret control uγsatisfies to :

hc(uγ,0)−c,c(w,˜ 0)i_{L2 (Σ}

1 )+Nhuγ,wi_{L2 (Σ}

2 )+h1

γξ(u_γ), ξ(w)i_{L2 (Q)}= 0, ∀w∈L²(Σ₂).

Proof - We use the Euler-Lagrange formula : lim

λ→0

J^γ(u_γ+λw)− J^γ(u_γ) λ

= 0.

Theorem

The low-regret control is characterized by the unique quadruplet{cγ, ρ_γ, ξ_γ,p_γ}s.t. :











Acγ= 0, A^∗ξ_γ= 0, Aργ=_γ¹ξ_γ, A^∗p_γ= 0 in Q, (Bcγ).n=h(c_γ), (B^∗ξ_γ).n=r_γ, (Bργ).n=h(ρ_γ), −(B^∗p_γ).n=k_γ onΣ₁, (Bcγ).n=−uγ, (B^∗ξγ).n= 0, (Bργ).n= 0, (B^∗pγ).n= 0 onΣ2, c_γ(0) = 0, ξ_γ(T) = 0, ρ_γ(0) = 0, p_γ(T) = 0 inΩ,

r_γ=c(u_γ,0) +_K^Iξ(u_γ), m_γ=c_γ+ ˜c+ρ_γ, and k_γ=−mγ+_K^Ip_γ, with the adjoint equation :

p_γ+Nu_γ= 0 in L²(Q).

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Characterization of the low-regret control to the NTB system

Proposition

The low-regret control uγsatisfies to :

hc(uγ,0)−c,c(w,˜ 0)i_{L2 (Σ}

1 )+Nhuγ,wi_{L2 (Σ}

2 )+h1

γξ(u_γ), ξ(w)i_{L2 (Q)}= 0, ∀w∈L²(Σ₂).

Proof - We use the Euler-Lagrange formula : lim

λ→0

J^γ(u_γ+λw)− J^γ(u_γ) λ

= 0.

Theorem

The low-regret control is characterized by the unique quadruplet{cγ, ρ_γ, ξ_γ,p_γ}s.t. :











Acγ= 0, A^∗ξ_γ= 0, Aργ=_γ¹ξ_γ, A^∗p_γ= 0 in Q, (Bcγ).n=h(c_γ), (B^∗ξ_γ).n=r_γ, (Bργ).n=h(ρ_γ), −(B^∗p_γ).n=k_γ onΣ₁, (Bcγ).n=−uγ, (B^∗ξγ).n= 0, (Bργ).n= 0, (B^∗pγ).n= 0 onΣ2, c_γ(0) = 0, ξ_γ(T) = 0, ρ_γ(0) = 0, p_γ(T) = 0 inΩ,

r_γ=c(u_γ,0) +_K^Iξ(u_γ), m_γ=c_γ+ ˜c+ρ_γ, and k_γ=−mγ+_K^Ip_γ, with the adjoint equation :

p_γ+Nu_γ= 0 in L²(Q).

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Some remarks

Work in progress

→ No-regret control

→ Numerical simulation

→ Comparison : Numerical curves to the measurements ?

Nutrient transfert : Plant-fungus association

→ Mycorhizes ?

→ Problem of scale ?

→ Other applications in biology..

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Some remarks

Work in progress

→ No-regret control

→ Numerical simulation

→ Comparison : Numerical curves to the measurements ?

Nutrient transfert : Plant-fungus association

→ Mycorhizes ?

→ Problem of scale ?

→ Other applications in biology..

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References

Lions J.-L. (1992),Contrôle à moindres regrets des systèmes distribués, C. R.

Acad. Sciences de Paris, Ser I Maths, 315 pp 1253-1257.

Louison L., Omrane A., Ozier-Lafontaine H., Picart D. (2015),Modeling plant nutrient uptake : Mathematical analysis and optimal control, Evol. Equations and Control Theory, 4 (2) pp 193-203.

Louison L., Omrane A.,Modeling plant nutrient uptake under pollution : Mathematical analysis and optimal control(to appear).

Ptashnyk M. (2010),Derivation of a macroscopic model for nutrient uptake by hairy-roots, Nonlinear Analysis : Real World Applications, 11, pp 4586–4596.

Roose T. (2001),A mathematical model of plant nutrient uptake, J. Math. Biol 42, pp 347–360.

Tinker P. B., Nye P. H. (2000),Solute movement in the rhizosphere, Oxford Press University.

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Control of incomplete data problems. Application to an ecology problem