Well-posedness of the2ptimal&ontrol3roblem5elated to'egenerate
&hemo-attraction0odels
SARAH SERHAL1,3, GEORGES CHAMOUN2, MAZEN SAAD1TONI SAYAH3
1Ecole Centrale de Nantes, Laboratoire de Mathématiques Jean Leray,
UMR CNRS 6629, 1 rue de la Noé, 44321 Nantes
FRANCE
2Faculty of engineering (ESIB), Saint Joseph University of Beirut
LEBANON
3Laboratoire de Mathématiques et Applications,
URMM, CAR, Faculté des Sciences,
Université Saint-Joseph de Beyrouth,
B.P 11-514 Riad El Solh, Beyrouth 1107 2050
LEBANON
Abstract: This paper delves into the mathematical analysis of optimal control for a nonlinear degenerate chemo-
taxis model with volume-filling effects. The control is applied in a bilinear form specifically within the chemical
equation. We establish the well-posedness (existence and uniqueness) of the weak solution for the direct prob-
lem using the Faedo Galerkin method (for existence), and the duality method (for uniqueness). Additionally, we
demonstrate the existence of minimizers and establish first-order necessary conditions for the adjoint problem.
The main novelty of this work concerns the degeneracy of the diffusive term and the presence of control over the
concentration in our nonlinear degenerate chemotaxis model. Furthermore, the state, consisting of cell density
and chemical concentration, remains in a weak setting, which is uncommon in the literature for solving optimal
control problems involving chemotaxis models.
Key-Words: Chemotaxis-model, degenerate parabolic problem, Existence, Optimal Control, Lagrange
multipliers
Received: May 19, 2023. Revised: May 17, 2024. Accepted: June 19, 2024. Published: July 16, 2024.
1 Introduction
Chemotaxis, the directed movement of organisms in
response to chemical gradients, is crucial in various
biological processes like embryogenesis, immunol-
ogy, cancer growth and wound healing. It allows
organisms to find nutrients, avoid predators or lo-
cate mates. For instance, cellular slime molds move
towards higher chemical concentrations secreted by
amoebae, while bacteria swim towards areas with
more oxygen. In pancreatic cancer therapy, iodine
acts as a chemoattractant to destroy cancer cells.
Mathematical models of chemotaxis are extensively
studied to understand and predict these biological
processes. Among these models, degenerate non-
linear chemotaxis systems have gained attention due
to their irregularity and the difficult proof of the
global existence and the uniqueness of weak solu-
tions. Various works, particularly those by [1], have
contributed to proving the global existence under cer-
tain assumptions, often employing methods like the
duality method used in [2]. We are particularly inter-
ested in a control on this problem, where we manipu-
late chemical concentrations to influence cell behav-
ior.
Bilinear control means that the control serves as a co-
efficient for a reaction term depends linearly on the
state. The control in this system governs the injec-
tion or extraction of a chemical substance within a
specified subdomain Ωc⊂Ω. Here, Ωrepresents a
bounded domain in R2with a smooth boundary ∂Ωof
class C2. Specifically, in this paper, we are concerned
with the study of an optimal control system arising in
chemotaxis as follows: Let (0, T )be a time interval,
where 0< T < +∞, and define QT= (0, T )×Ω,
the control problem can be described by the following
system in QT:
∂tN−∇·a(N)∇N+∇ · χ(N)∇C= 0,
∂tC−∇·(∇C) = αN −βC +fC1Ωc.
(1)
The homogeneous Neumann boundary conditions
over Σt= (0, T )×∂Ωare
a(N)∇N·η= 0,∇C·η= 0,(2)
where ηis the exterior unit normal to ∂Ω. The initial
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conditions on Ωare given by,
N(0, x) = N0(x), C(0, x) = C0(x).(3)
In the model above, f:Qc:= (0, T )×Ωc→Ris
the control with Ωc⊂Ω⊂Rdthe control domain,
where d= 2 or 3. The density of cells and the con-
centration of chemicals are denoted respectively by
Nand C. The function a(N)represents a non-linear
degenerate density-dependent diffusion coefficient.
The function χ(N)determines the sensitivity of
cells to be attracted or repulsed. Finally, the rates of
production and degradation are denoted respectively
by αand β.
Moreover, optimal theory provides well-designed
strategies to determine an optimal control over
chemoattractant concentration to achieve desired cell
density and concentration, particularly in cancer treat-
ment by example, decreasing cancer cell density
while minimizing effects on healthy tissues This is
achieved by minimization of a quadratic problem (de-
fined in section 3) where the cost function measures
the errors between the cell density, the concentration
and the desired given states; the cost of the control is
also minimized.
Many studies had significant attention to optimal
control problems governed by interconnected par-
tial differential equations. Notably, [3], discussed
the optimal control of a fully parabolic attraction-
repulsion chemotaxis model with a logistic source
in a two-dimensional setting, while [4], conducted a
thorough investigation into an optimal control prob-
lem for a haptotaxis model of solid tumor infiltra-
tion, exploring the use of multiple cancer treatments.
Recently, [5], analyzed a non-degenerate parabolic
chemo-repulsion model with nonlinear production in
two-dimensional domains.
The novelty of our study lies in extending the con-
cept of optimal control to our degenerate parabolic
chemo-attractant problem, where the challenge stems
from the strong degeneracy and non-linearity of the
diffusive term. This degeneracy not only complicates
the existence of weak solutions for the direct problem
but also poses challenges in constructing the adjoint
problem using Lagrange multipliers. In our work,
we consider that the set of constraints is given by the
concept of weak solutions with energy inequalities.
This set of admissible constraints is an unusual and
new concept in the literature for solving optimal
control problems involving chemotaxis models.
The structure of our paper is organized as follows: In
Section 2, we present the main results and establish
the well-posedness (existence and uniqueness) of our
nonlinear degenerate chemo-attractant problem using
the Faedo-Galerkin method. Section 3 is dedicated to
the optimal control problem. Here, we introduce our
optimal control approach, define a functional useful
for minimization, establish the existence of the con-
trol, and derive the adjoint-state problem. Finally,
employing the technique of regular points, we demon-
strate the existence of Lagrange multipliers, forming
a solution to this adjoint problem.
2 Setting of the Problem
First, let us introduce the notations that we will need
later. We assume that the chemotactic sensitivity
χ(N)vanishes when N≥Nmand that χ(0) = 0.
This condition, known as the volume-filling effect,
has a straightforward biological interpretation. Upon
normalization, we can assume that the threshold den-
sity is Nm= 1.
Moreover, the main data assumptions are:
a: [0,1] 7−→ R+, a ∈ C1([0,1]),(4)
where a(0) = a(1) = 0, a(s)>0for 0< s < 1,
χ: [0,1] 7−→ R, χ ∈ C1([0,1]) and χ(0) = χ(1) = 0
(5)
and
f∈L∞(QT).(6)
We will prove the existence and the uniqueness of a
weak solution of (1)-(3) along with its continuous de-
pendency on the control f.
This involves initially defining a weak solution for the
system (1)-(3).
Definition 2.1. Assume that 0≤N0≤1,C0≥0
and C0∈L∞(Ω). A pair (N, C)is said to be a weak
solution of (1)-(3) if
0≤N(t, x)≤1, C(t, x)≥0a.e. in QT,
N∈Cw(0, T ;L2(Ω)), ∂tN∈L2(0, T ; (H1(Ω))′),
A(N) := N
0
a(r)dr ∈L2(0, T ;H1(Ω)) ,
C∈L∞(QT)∩L2(0, T ;H1(Ω)) ∩C(0, T ;L2(Ω)),
∂tC∈L2(0, T ; (H1(Ω))′)
and (N,C) satisfy
T
0
< ∂tN, ψ1>(H1)′,H1dt
+QT
a(N)∇N· ∇ψ1dxdt
−QT
χ(N)∇C· ∇ψ1dxdt = 0,
(7)
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T
0
< ∂tC, ψ2>(H1)′,H1dt +QT∇C· ∇ψ2dxdt
=QT
[αN −βC +fC1Ωc]ψ2dxdt ,
(8)
for all ψ1and ψ2∈L2(0, T ;H1(Ω)), where
Cw(0, T ;L2(Ω)) denotes the space of continuous
functions with values in L2(Ω) endowed with the
weak topology, and <.,.> denotes the duality pairing
between H1(Ω) and (H1(Ω))′.
The next theorems state the existence and the unique-
ness of the weak solution.
Theorem 2.2. Under the assumptions (4) to (6), if
0≤N0≤1,C0≥0almost everywhere in Ω, and
C0∈L∞(Ω), then System (1)-(3) possesses a global
weak solution (N, C)as defined in Definition 2.1.
Theorem 2.3. The weak solution (N, C)of (1)-(3) is
unique under the following assumption: there exists
K0≥0such that ∀N1, N2∈[0,1],
(χ(N1)−χ(N2))2≤K0(N1−N2)(A(N1)−A(N2)).
Outline of the proofs of the theorems 2.2 and 2.3:
To prove Theorem 2.3, one can maintain the same
necessary estimates of ( [1], [2], [4], [5], [6], [7]) and
choose a positive δsuch that δ < min(1
K0,1
c2
χ+α2).
For the proof of Theorem 2.2, we note that a major
difficulty for the analysis of the first equation of (1) is
the strong degeneracy of the diffusion term. To solve
this problem, we modify the original diffusion term
a(N)by adding a small positive value εto create aε=
a(N) + ε. This adjustment allows us to consider the
following non-degenerate system:
∂tNε−∇·(aε(Nε)∇Nε) + ∇ · (χ(Nε)∇Cε) = 0,
∂tCε−∆Cε=αNε−βCε+fCε1Ωc,
(9)
with the following conditions:
a(Nε)∇Nε·η= 0,∇Cε·η= 0 on ΣT
Nε(0, x) = N0(x), Cε(0, x) = C0(x)in Ω.
(10)
To establish the existence of a weak solution to
the non-degenerate problem (9)-(10) in the sense
of the definition 2.1, we use the Faedo-Galerkin
method while the discrete maximum principle holds.
Convergence is achieved using a priori estimates and
compactness arguments.
The details of the Faedo-Galerkin method, the maxi-
mum principle and the convergence will be given in
the next sections.
2.1 The Faedo-Galerkin Solution
In this subsection, we will give the proof of the exis-
tence of a weak solution of the non-degenerate prob-
lem (9)-(10) by the Faedo Galerkin method. We con-
sider an appropriate spectral problem introduced in
[8], [9], where the eigenfunctions el(x)form an or-
thogonal basis in H1(Ω) and an orthonormal basis in
L2(Ω).
Our goal is to find finite-dimensional approximations
to the solutions of system (9)-(10) in the form of se-
quences {Nn,ε}n>1and {Cn,ε}n>1, defined for s≥0
and x∈¯
Ωas follows:
Nn,ε(t, x) =
n
l=1
qn,l(t)el(x)
Cn,ε(t, x) =
n
l=1
dn,l(t)el(x).
It’s important to note that this solution satisfies the
necessary boundary conditions.
Next, we need to determine the coefficients
{qn,l(t)}n
l=1 and {dn,l(t)}n
l=1 such that for
l= 1, . . . , n, the following equations hold:
< ∂tNn,ε, el>+Ω
aε(Nn,ε)∇Nn,ε · ∇eldx
−Ω
χ(Nn,ε)∇Cn,ε · ∇eldx = 0,
(11)
< ∂tCn,ε el>+Ω∇Cn,ε · ∇eldx
=Ω
(αNn,ε −βCn,ε)eldx +Ω
fCn,ε1Ωceldx,
(12)
along with the initial conditions:
Nn,ε(0, x) = N0,
Cn,ε(0, x) = C0.
Next, we use the orthonormality of the basis and
hence the above equations can be rewritten in the fol-
lowing form:
q′
n,l(t) = −Ω
aε(Nn,ε)∇Nn,ε · ∇eldx
+Ω
χ(Nn,ε)∇Cn,ε · ∇eldx
=: El
1(t, {qn,l}n
l=1,{dn,l}n
l=1)
(13)
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d′
n,l(t) = −Ω∇Cn,ε · ∇eldx
+Ω
(αNn,ε −βCn,ε)eldx
+Ω
fCn,ε1Ωceldx
=: El
2(t, {qn,l}n
l=1,{dn,l}n
l=1).
(14)
Let t′∈(0, T )and set X= [0, t′]. Choose R > 0
large enough so that the ball BR⊂RNcontains
{qn,l(0)}and {dn,l(0)}, and set Y=¯
BR. The com-
ponents El
i, i = 1,2, can be bounded on X×Y:
|El
i| ≤ K(ε, R, n), i = 1,2where K(ε, R, n)>
0depends only on ε, R, n. Due to the concept of
Carathéodory functions and the standard ODE theory,
we can conclude the existence of absolutely continu-
ous functions {qn,l}n
l=1 and {dn,l}n
l=1 for almost ev-
erywhere s∈[0, t′);t′>0. This proves that the
sequences (Nn,ε, Cn,ε)are well defined and the ap-
proximate solutions of (9)-(10) exist locally on [0, t′).
Furthermore, the local solution constructed above can
be extended to the entire time interval [0, T )(see,
[10]).
Next, we define ϕi,n(t, x) =
n
l=1
bi,n,l(s)el(x)for
i= 1,2, where the coefficients {bi,n,l}, for i= 1,2,
are absolutely continuous functions. Then, the ap-
proximate solution satisfies the weak formulation:
< ∂tNn,ε, ϕ1,n >+Ω
aε(N)∇Nn,ε · ∇ϕ1,n dx
+Ω
χ(Nn,ε)∇Cn,ε · ∇ϕ1,n dx = 0,
(15)
< ∂tCn,ε ϕ2,n >+Ω∇Cn,ε · ∇ϕ2,n dx
=Ω
(αNn,ε −βCn,ε)ϕ2,n dx
+Ω
fCn,ε1Ωcϕ2,n dx.
(16)
2.2 Maximum Principle
In this section, we prove that the approximate solution
of the non-degenerate problem (9) -(10) satisfies the
following maximum principle.
Lemma 2.4. Assume that 0≤N0≤1and C0≥0,
then the approximate solution (Nn,ε, Cn,ε)satisfy
0≤Nn,ε ≤1and Cn,ε ≥0for a.e (t, x)∈QT.
Sketch of proof: To begin with, we denote by N−=
max(−N, 0) and N+=max(N, 0). To establish the
non-negativity of Nn,ε, we multiply (15) by −N−
n,ε
and then introduce the continuous and Lipschitz ex-
tension ˜χof χon Rsuch that
˜χ(s) =
0if s < 0
χ(s)if 0≤s≤1
0if s > 1,
to obtain χ(Nn,ε) = 0. This leads us to
1
2
d
dt Ω|N−
n,ε|2dx ≤0.
Finally, utilizing the fact that N0is non-negative, we
deduce that N−
n,ε = 0 for almost every (t, x)∈QT.
Therefore Nn,ε ≥0. Similarly to demonstrate that
Nn,ε ≤1, we follow the same steps by taking (Nn,ε −
1)+as a test function.
Similarly, to demonstrate Cn,ε ≥0, when fis nega-
tive, we proceed with the same approach. However,
when fis positive, for technical reasons, we extend
the function F(C, f ) := f C1Ωto ensure measurabil-
ity on ΩTand continuity with respect to C. This is
achieved by defining
˜
F(C, f ) = F(C, f )if C⩾0,
0else.
Therefore, it follows form (16) that Cn,ε satisfy
< ∂tCn,ε ϕ2,n >+Ω∇Cn,ε · ∇ϕ2,n dx
=Ω
αNn,εϕ2,n dx +Ω
˜
F(C+
n,ε, f)ϕ2,n dx.
(17)
To complete the proof, we proceed similarly as be-
fore, taking −C−
n,ε as a test function, and utilizing the
non-negativity of C0. This completes the proof of the
lemma 2.4. □
2.3 Convergence (Passing to the Limit with
n)
Finally, to ensure the existence of a weak solution for
the non-degenerate system (9)-(10), we still need to
pass to the limit with n. Thus, we take Nn,ε and Cn,ε
as a test function respectively in (15) and (16). Then
by integrating over (0, T )and employing Gronwall’s
inequalities, we establish that
Nn,ε ∈L∞(0, T ;L2(Ω)) ∩L2(0, T ;H1(Ω))
and
Cn,ε ∈L∞(0, T ;L2(Ω)) ∩L2(0, T ;H1(Ω)).
Additionally, we can easily demonstrate that
k(∂tNn,ε, ∂tCn,ε)kL2(0,T,H−1(Ω)) ≤A, where the
constant Ais independent of n.
Therefore, as n→+∞, we observe the following
weak convergences:
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•(Nn,ε, Cn,ε)⇀(Nε, Nε)weakly* in
L∞(0, T ;L2(Ω)),
•(Nn,ε, Cn,ε)⇀(Nε, Cε)weakly in
L2(0, T ;H1(Ω)),
•(∂tNn,ε, ∂tCn,ε)⇀(∂tNε, ∂tCε)weakly in
L2(0, T ;H−1(Ω)).
From these, we can deduce easily the weak conver-
gence of Nn,ε and Cn,ε to Nεand Cεrespectively,
satisfying the definition of a weak solution of the
non-degenerate system.
Next, we can prove that the sequence of approximate
solutions (Nε, Cε)converges to a weak solution
(N, C)of (1)-(3) in the sense of definition 2.1, when
εtends to zero by following the same guidelines of
[1].
In the absence of the control term f, numerical sim-
ulations were done in [7], [11], [12], using the finite
element method, and in [6], using the finite volume
method.
In our paper, we focus only on the theoretical study of
the optimal model.
3 Optimal Control Problem
This section focuses on establishing the existence
of optimal control. For that, we introduce the La-
grangian functions and construct the adjoint problem.
Additionally, we will prove the existence of a weak
solution to the adjoint problem.
Let us consider
f∈ F ={L∞(QT),such that ∀(t, x)∈QT,
fmin ≤f(t, x)≤fmax }.
We also have 0≤N0≤1and C0∈L∞(Ω) with
C0≥0in Ω. The function fdescribes the bilinear
control acting on the equation of the concentration.
We start first by proving the existence of a solution
for the following optimal control problem:
Find (N, C, f )∈ W × X × F minimizing
the functional
J(N, C, f ) := β1
2T
0kN(t)−Nd(t)k2
L2(Ω) dt
+β2
2T
0kC(t)−Cd(t)k2
L2(Ω) dt
+γf
2T
0kf(t)k2
L2(Ωc)dt,
subject to (N, C)be a weak solution
of the PDE system (1)-(3). in the sense of
Definition (2.1) and f∈ F ,
(18)
where
W={N∈Cw(0, T ;L2(Ω))
and ∂tN∈L2(0, T ; (H1(Ω))
′
)},(19)
X={C∈L∞(QT)∩L2(0, T ;H1(Ω))
∩C(0, T ;L2(Ω)) and ∂tC∈L2(0, T ; (H1(Ω))′)}.(20)
Here (Nd, Cd)∈L2(QT)2represent the target states,
and the non-negative numbers β2, β2and γfmeasure
the cost of the states and control, respectively.
We define the set of admissible solutions of (18) as
Sad ={s= (N, C, f )∈ W × X × F,where
sis a weak solution of (1) −(3) in QT}.
3.1 Existence of the Control
In this section, we prove the existence of a solution to
the control problem of (18).
Theorem 3.1. Assume that 0≤N0≤1,C0∈
L∞(Ω) , Nd∈L2(QT),Cd∈L2(QT)and f∈ F.
Then, there exists at least one solution of the optimal
control problem (18).
Proof. Theorem (2.2) ensures that Sad is non-empty.
With the non-negativity of the cost functional, J
has a greatest lower bound, leading to a minimiz-
ing sequence {sn}n∈N⊂ Sad with lim
n→+∞J(sn) =
inf
s∈Sad
J(s).
By the definition of J,(fn)nis bounded in L2(QT),
indicating the convergence of fnweakly to f∗. As
Fis closed, f∗∈ F. Let (Nn, Cn)be weak so-
lutions to (1)-(3) with fn. Now, as in section 2,
we can deduce the convergence of Nnand Cnto a
weak solution (N∗, C∗)of (1)-(3). This implies that
s∗= (N∗, C∗, f ∗)is a weak solution of the system
where s∗∈ Sad.
Finally, since Jis weakly lower semi-continuous,
then we conclude that
J(N∗, C∗, f ∗)⩽lim inf
n→∞ J(Nn, Cn, fn)
⩽inf
s∈Sad
J(s)⩽J(N∗, C∗, f ∗).
This establishes the existence of an optimal control
solution (18).
3.2 Optimality Condition and Dual Problem
We will now study the first-order optimality condi-
tions necessary for a local optimal solution (N, C, f )
of problem (18).
First, we consider the following (generic) optimiza-
tion problem:
min
s∈MJ(s)subject to G(s) = 0,(21)
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where J:X→Ris a functional, G:X→Y is
an operator, X and Y are Banach spaces, and M is a
nonempty closed and convex subset of X.
Definition 3.2. -([[13], chapter 6] (Lagrange multi-
plier) Let ¯s∈ S be a local optimal solution for prob-
lem (21). Suppose that Jand Gare Fréchet differ-
entiable in ¯s. Then, any ξ∈Y′is called a Lagrange
multiplier for (21) at the point ¯sif for all r∈ C(¯s),
we have
J′(¯s)[r]−ξ, G′(¯s)[r]≥0,
where C(¯s) = {θ(s−¯s) : s∈M, θ ≥0}is the
conical hull of ¯sin M.
Definition 3.3. Let ¯s∈ S be a local optimal solution
for problem (21). we say that ¯sis a regular point if
G′(¯s)[C(¯s)] = Y.
Theorem 3.4. ([14], Theorem 3.1) Let ¯s∈ S be a
local optimal solution for problem (21). Suppose that
Jis Fréchet differentiable in ¯s, and Gis continuous
Fréchet-differentiable in ¯s. If ¯sis a regular point, then
the set of Lagrange multipliers for (21) at ¯sis non-
empty.
Now, we will reformulate the optimal control problem
(18) in the abstract setting (21).
We define the following Banach spaces:
X:= W × X × L2(Qc),
Y:= L2(QT)×L20, T ; (H1(Ω))′×L2(Ω)
×L2(Ω).
The operator G= (G1, G2, G3, G4) : X→Y,
where
G1:X→L2(QT),
G2:X→L2(0, T ; (H1(Ω))′),
G3:X→L2(Ω),
G4:X→L2(Ω)
are defined at each point s= (N, C, f )∈Xby
< G1(s), φ1>=< ∂tN, φ1>L2(H1),L2((H1)′)
+ (a(N)∇N, ∇φ1)L2(QT)
−(χ(N)∇C· ∇φ1)L2(QT),
< G2(s), φ2>=< ∂tC, φ2>L2(H1),L2((H1)′)
+ (∇C, ∇φ2)L2(QT)
−(αN −βC +fC1Ωc, φ2)L2(QT),
and G3and G4satisfy respectively the constraints on
the initial conditions of N(t, x)and C(t, x)as fol-
lows:
G3(s) = N(0) −N0and G4(s) = C(0) −C0.
Now, we take M=W × X × F (a closed convex
subset of X). Thus, our optimal control problem (18)
is reformulated as follows:
min
s∈MJ(s)subject to G(s) = 0.(22)
Regarding the differentiability of the functional Jand
the constraint operator G, we use a direct calculation
similar to [15], [16], to obtain the following results.
Lemma 3.5. The functional J:X→Ris Fréchet
differentiable and its derivative at s= (N, C, f)∈X
in the direction r= (U, V, F )∈Xis given by
J′(s)[r] =β1T
0Ω
(N−Nd)U dxdt
+β2T
0Ω
(C−Cd)V dxdt
+γfT
0Ωc
fF dxdt.
Lemma 3.6. The operator G:X→Yis continu-
ous Fréchet differentiable and its derivative at s=
(N, C, f )∈Xin the direction r= (U, V, F )∈Xis
the linear operator
G′(¯s)[r] = G′
1(¯s)[r], G′
2(¯s)[r], G′
3(¯s)[r], G′
4(s)[r]
defined by
< G′
1(s)[r], φ1>=< ∂tU, φ1>+(a(N)∇U, ∇φ1)
+ (a′(N)U∇N, ∇φ1)−(χ(N)∇V, ∇φ1)
−(χ′(N)U∇C, ∇φ1),
< G′
2(s)[r], φ2>=< ∂tV, φ2>+(∇V, ∇φ2)
−(αU, φ2)+(βV, φ2)
−(fV +F C, φ2),
and
G′
3(s)[r] = U(0), G′
4(s)[r] = V(0).
Next, we aim to prove the existence of Lagrange mul-
tipliers.
3.2.1 Existence of Lagrange Multipliers
In this section, we prove the existence of Lagrange
Multipliers under the assumptions:
∇N∈L∞(QT),(23)
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a(N)vanishes at a finite number of points
in QT.(24)
Furthermore, the existence is guaranteed if a local op-
timal solution of problem (22) is a regular point of
operator G. Therefore, we first need to prove the fol-
lowing Lemma:
Lemma 3.7. Let a′(N)⩽ca(N)and χ′(N)≤
Ka(N)for a positive constant cand K, and suppose
that (23) and (24) hold. If s= (N, C, f )∈ Sad =
{s= (N, C, f )∈M:G(s)=0}, then s is a regular
point.
Proof. For a given (N, C, f )∈ Sad, let
(gN, gC, U0, V0)∈Y. As 0is in C(¯
f), we only
need to prove the existence of (U, V )that solves the
following problem:
∀φ1, φ2∈L2(H1)
< ∂tU, φ1>+(a(N)∇U, ∇φ1)
+ (a′(N)U∇N, ∇φ1)−(χ(N)∇V, ∇φ1)
−(χ′(N)U∇C, ∇φ1) =< gN, φ1>,
< ∂tV, φ2>+(∇V, ∇φ2)−(αU, φ2)
+ (βV, φ2)−(fV, φ2) =< gC, φ2>,
U(0) = U0, V (0) = V0.
(25)
First, we note that despite the linearity of the problem,
the degeneracy due to the term a(N)adds a layer of
difficulty to the analysis. Therefore, as discussed in
Section 2 of this article, we replace a(N)with a(N)+
εand define the following non-degenerate problem:
< ∂tUε, φ1>+(aε(N)∇Uε,∇φ1)
+ (a′(N)Uε∇N, ∇φ1)−(χ(N)∇Vε,∇φ1)
−(χ′(N)Uε∇C, ∇φ1) =< gN, φ1>,
(26)
< ∂tVε, φ2>+(∇Vε,∇φ2)−(αUε, φ2)
+ (βVε, φ2)−(fVε, φ2) =< gC, φ2>, (27)
Uε(0) = U0, Vε(0) = V(0).(28)
We can establish the existence of a solution to the non-
degenerate problem by following the same steps as
in Section 2, employing the Faedo-Galerkin method.
Now, we still need to tend εto zero.
Taking Vεas a test function in (27), multiplying (26)
by δand then taking Uεas a test function in (26),
we apply Young’s inequality to each equation, using
the assumptions (23)-(24), and a′(N)⩽ca(N)and
χ(N)⩽ka(N), for positive constants cand K. After
that, we sum these two equations together and choose
δ≥0such that
1−2δK2ka(N)kL∞≥0.
Finally by applying Gronwall’s inequality, we con-
clude that (Uε, Vε)satisfies:
Uε∈L∞0, T ;L2(Ω),(29)
Vε∈L2(0, T ;H1(Ω)),(30)
√ε∇Uε∈L20, T ;L2(Ω),(31)
a(N)∇Uε∈L20, T ;L2(Ω),(32)
and are bounded in these spaces independently of ε.
Moreover, one can show that
k(∂tUε, ∂tVε)kL2(0,T,H−1(Ω)) ≤A,
where the constant Ais independent of ε.
Thus, there exist a solution (U, V )and a subsequence
of (Uε, Vε)still denotes as the sequence such that, as
εgoes to 0,
Uε⇀ U weakly* in L∞0, T ;L2(Ω),(33)
Vε⇀ V weakly in L2(0, T ;H1(Ω)) ,(34)
ε∇Uε⇀0weakly in L2(QT),(35)
a(N)∇Uε⇀ ξ weakly in L2(QT),(36)
∂tUε⇀ ∂tUεweakly in L20, T ; (H1(Ω))′,(37)
∂tVε⇀ ∂tVεweakly in L20, T ; (H1(Ω))′.(38)
We still have to show that ξ=a(N)∇Na.e. on
QT. To do this, we use the assumption (24) and follow
the guidelines presented in [17].
With the above convergence, we can easily show that
(U, V )is a solution to the problem (25).
Now, using Lemma 3.7 and Definition 3.3, we can
conclude the following result about the existence of
Lagrange multipliers.
Theorem 3.8. Let s= (N, C, f )∈ Sad be a
local optimal solution for the control problem (22)
and suppose that there exist positive constants c
and Ksuch that a′(N)⩽ca(N)and χ(N)⩽
Ka(N). If the assumptions (23) and (24) hold, then
all for all (U, V, F )∈W×X× C(f), there ex-
ists a Lagrange multiplier (λ, η, ϕ, ψ)∈L2(QT)×
L2(0, T, H1(Ω)) ×L2(QT)×(H1(Ω))′such that
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β1T
0Ω
(N−Nd)U dxdt
+β2T
0Ω
(C−Cd)V dxdt
+γfT
0Ωc
fF dxdt −T
0h∂tU, λidt
+T
0Ω
(−a(N)∇U+χ(N)∇V)· ∇λ dxdt
+T
0Ω
(−a′(N)U∇N+χ′(N)U∇C)· ∇λ dxdt
−T
0
< ∂tC, η > dt −T
0Ω∇C· ∇η dxdt
+T
0Ω
(αU −βV +fV 1Ωc)η dxdt
+T
0Ω
F C1Ωcη dxdt −Ω
U(0)ϕ dx
−Ω
V(0)ψ dx ≥0.
Remark 3.9. From the previous theorem 3.8, we ob-
tain an optimality system, for which we can consider
the following spaces:
Wu0:= {u∈ W :u(0) = 0}and
Xv0:= {v∈ X :v(0) = 0}.
Corollary 3.10. Let s= (N, C, f )represent a lo-
cal optimal solution for the optimal control problem
(22) and suppose that there exist positive constants
c and K such that a′(N)⩽ca(N)and χ(N)⩽
Ka(N). If the assumptions (23) and (24) hold,
then any Lagrange multiplier (λ, η)∈L2(QT)×
L2(0, T ;H1(Ω)) provided by Theorem 3.8 , satisfies
the following system :
∀U∈ Wu0and ∀V∈ Xv0,
T
0h∂tU, λidt +T
0Ω
a(N)∇U· ∇λ dxdt
+T
0Ω
(a′(N)U∇N−χ′(N)U∇C)· ∇λ dxdt
−T
0Ω
αUη dxdt (39)
=β1T
0Ωd
(N−Nd)U dxdt,
T
0h∂tC, λidt +T
0Ω∇C· ∇λ dxdt
−T
0Ω
χ(N)∇V· ∇λ dxdt +T
0Ω
βUη dxdt
−T
0ΩC
fV η dxdt. (40)
=β2T
0Ω
(C−Cd)V dxdt,
and the optimality condition: ∀¯
f∈ F,
T
0Ωε
(γff+Cη) ( ¯
f−f)dxdt ≥0.(41)
Outline of proof: To get the equations (39) and (40)
satisfied by λand η, we set (V, F ) = (0,0), and since
Wu0is a vector space, we obtain the first equation
(39). Similarly, by setting (U, F ) = (0,0) and con-
sidering that XV0is a vector space, we derive the sec-
ond equation (40).
Now, to obtain the optimality condition, we take
(U, V ) = (0,0) and choose F=¯
f−f∈ C(¯
f)for all
˜
f∈ F.
Remark 3.11. A pair (λ, η)satisfying first equation
(39) and second equation(40) corresponds to the con-
cept of a weak solution of the problem
−∂tλ−∇·(a(N)∇λ) + a′(N)∇N· ∇λ
−χ′(N)∇C· ∇λ−αη =β1(N−Nd),in QT
−∂tη−∆η+βη +∇ · (χ(N)∇λ)−fη1Ωc,
=β2(C−Cd),in QT
λ(T) = 0, η(T) = 0 in Ω,
a(N)∇λ·n = 0,(∇η−χ(N)∇λ)·n = 0,
on ΣT.
4 Conclusion
In this paper, we addressed an optimal control prob-
lem for the concentration in a non-linear parabolic
system with a two-sidedly degenerate equation and
volume-filling effects equation. We provided a rig-
orous analysis of the mathematical model, proposing
an optimal control strategy to reduce cancer cell den-
sity while minimizing the impact on healthy cells in
the body, and this model can also help to reduce pat-
tern formation. We demonstrated that the direct con-
trol problem is well-posed. Additionally, we estab-
lished the characterization of the global optimal solu-
tion. By imposing regularity conditions and utilizing
the technique of Fréchet derivatives, we give a rigor-
ous mathematical justification of the existence of the
adjoint problem and demonstrate the existence of La-
grange multipliers within the admissible constraint set
as weak solutions.
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References:
[1] G. Chamoun, M. Saad, R. Talhouk, A coupled
anisotropic chemotaxis–fluid model: the case of
two-sidedly degenerate diffusion, Comput. Math.
Appl. 68(9), 2014, 1052–1070.
[2] P. Laurencot and D. Wrzosek, A Chemotaxis
model with threshold density and degenerate dif-
fusion, Nonlinear differential equations and their
applications, vol. 64, Birkhauser, Boston, 2005,
pp. 273-290,
[3] C. Liu and Y. Yuan, Optimal Control of a
Fully Parabolic Attraction-Repulsion Chemo-
taxis Model with Logistic Source in 2D, Ap-
plied Mathematics and Optimization; New York,
Vol.85, Iss. 1, 2022.
[4] F. Dai and B. Liu, Optimal control and pattern for-
mation for a haptotaxis model of solid tumor inva-
sion, J. Frankl. Inst. 356(16), 9364-9406, 2019.
[5] F. Guillén-González, E. Mallea-Zepeda, É.J.
Villamizar-Roa, On a bi-dimensional chemo-
repulsion model with nonlinear production and
a related optimal control problem, Acta Appl.
Math,2020, 170, 963–979.
[6] F. Filbet, A finite volume scheme for the Patlak–
Keller–Segel chemotaxis model, Numerische
Mathematik 104, 2006: 457-488.
[7] Hassan, Sattar M., and Akil J. Harfash, Finite
element approximation of a Keller–Segel model
with additional self-and cross-diffusion terms and
a logistic source.Communications in Nonlinear
Science and Numerical Simulation, 104, 2022.
[8] M. Bendahmane and K.H Karlsen, Analysis of
a class of degenerate reaction-diffusion systems
and the bidomain model of cardiac tissue. Netw
Heterog Media, 2006, 1:185–218.
[9] P.T. Sowndarrajanand L. Shangerganesh, Opti-
mal control problem for cancer invasion parabolic
system with nonlinear diffusion. Optimization,
67(10), 2018, 1819–1836.
[10] M. Bendahmane, F. Mroue, M. Saad and R.
Talhouk, Mathematical analysis of cardiac elec-
tromechanics with physiological ionic model,
DCDS-B, 24, 2019.
[11] Saito, Norikazu, Conservative upwind finite-
element method for a simplified Keller–Segel
system modeling chemotaxis. IMA journal of nu-
merical analysis 27.2 (2007): 332-365.
[12] S. Badia, J. Bonilla and J. V. Gutiérrez-
Santacreu, Bound-preserving finite element ap-
proximations of the Keller–Segel equations.
Mathematical Models and Methods in Applied
Sciences, 33(03), 609-642, 2023.
[13] F.Tröltzsch, Optimal control of partial differ-
ential equations. Theory, methods, and applica-
tions, AMS Providence, Rhode Island, 2010.
[14] J. Zowe, and S. Kurcyusz, Regularity and stabil-
ity for the mathematical programming problem in
Banach spaces. Appl. Math. Optim, , Vol.5, 1979,
49-62.
[15] P. Braz e Silva, F. Guillén-González, C. Pe-
rusato and M.A Rodriguez-Bellido, Bilinear Op-
timal Control of the Keller–Segel Logistic Model
in 2D-Domains, Appl Math Optim 87, 2023, 55.
[16] H. Tang and Y. Yuan, Optimal control for a
chemotaxis–haptotaxis model in two space di-
mensions, Bound Value Probl, 2022, 79.
[17] G. Marinoschi, R. M. Mininni and S. Romanelli,
An Optimal Control Problem in Coefficients for a
Strongly Degenerate Parabolic Equation with In-
terior Degeneracy, J Optim Theory Appl, 2017,
173:56–77.
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