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arXiv:2503.17049v1 [math.AP] 21 Mar 2025
Optimal control on a brain tumor growth model with
lactate metabolism, viscoelastic effects, and tissue damage
Giulia Cavalleri
Dipartimento di matematica “F. Casorati”, Universit`a degli Studi di Pavia,
via Ferrata 5, 27100 Pavia, Italy
E-mail: giulia.cavalleri01@universitadipavia.it
Alain Miranville
School of Mathematics and Statistics, Henan Normal University, Xinxiang, P. R. China
& Universit´e Le Havre Normandie, Laboratoire de Math´ematiques Appliqu´ees du
Havre (LMAH), 25, rue Philippe Lebon, BP 1123, 76063 Le Havre cedex, France
E-mail: alain.miranville@univ-lehavre.fr
Abstract
In this paper, we study an optimal control problem for a brain tumor growth model
that incorporates lactate metabolism, viscoelastic effects, and tissue damage. The
PDE system, introduced in [Cav+25], couples a Fisher–Kolmogorov-type equation
for tumor cell density with a reaction-diffusion equation for the lactate, a quasi-static
force balance governing the displacement, and a nonlinear differential inclusion for
tissue damage. The control variables, representing chemotherapy and a lactate-
targeting drug, influence tumor progression and treatment response. Starting from
well-posedness, regularity, and continuous dependence results proven in [Cav+25],
we define a suitable cost functional and prove the existence of optimal controls.
Then, we analyze the differentiability of the control-to-state operator and establish
a necessary first-order condition for treatment optimality.
Key words: tumor growth models, lactate kinetics, mechanical effects, damage,
optimal control, adjoint system, necessary optimality conditions.
AMS (MOS) Subject Classification: 35Q93, 49J20, 49K20, 35Q92, 92C50.
1 Introduction
Primary brain tumors, especially gliomas and glioblastomas, are characterized by an-
giogenesis, invasive growth, and necrosis. Despite advances in medical research and
treatment strategies, the brain tumor survival rate remains low (see, e.g., [DC16]), and
1
no significant improvement has been observed recently (see, e.g., [Szo+17]). For this
reason, it is crucial to explore new therapeutic approaches and to complement clinical
studies with mathematical modeling. The aim of the latter is not only to predict the
course of the disease, but also to optimize treatments, by adjusting the combination,
timing, and dosage of therapies to achieve the best possible outcome for the patient,
meaning maximum reduction of the tumor and minimum drug-related side effects. This
is where the present contribution comes into play. Specifically, we are interested in a pro-
tocol that combines chemotherapy with a drug targeting lactate, which seems promising
from an experimental point of view (cf. [Son+08], [Tan+21], [Guy+22]). From the math-
ematical perspective, only a few works go in this direction (see, e.g., [Aub+05], [Clo+09],
[Gui+18], [Che+21], [Che+22]) and even fewer regard optimal control (cf. [Che+24]).
This paper seeks to study an optimal control problem starting from the model recently
proposed and analyzed in [Cav+25], which describes the dynamics of a brain tumor tak-
ing into account the lactate metabolism, viscoelastic effects of the tissue, and possible
damage caused by surgery. The introduction of the damage variable in this context was
inspired by [Cav24], by one of the authors of the present contribution, where damage
was first incorporated into a tumor model. Among the huge literature regarding optimal
control for tumor growth models (see, e.g., [BAD15], [Col+21], [GL16], [GLR18], [EK19],
[Sig20], and the references cited therein) we would like to recall once again [Che+24],
where the authors deal with a brain tumor-specific model. The resulting PDE system
couples three parabolic equations, one for the tumor cell density, and the other two for
the intracellular lactate concentration and the capillary lactate concentration, respec-
tively. Another perspective related to the present work is the one from [GLS21], in which
the authors consider a phase-field model of Cahn–Hilliard type taking into account the
presence of a nutrient (such as oxygen or glucose) and mechanical effects.
The model. Let Ω be a bounded C2domain in Rnwith n= 2,3 and let νbe its
outward unit normal to ∂Ω. We are interested in the PDE system
∂tϕ−∆ϕ=U(ϕ, σ, z, χ1),(1.1a)
∂tσ−∆σ+K(ϕ, σ, z) = χ2S(ϕ, z),(1.1b)
−div Aε(∂tu) + B(ϕ, z)ε(u)=f,(1.1c)
∂tz−∆z+β(z) + π(z)∋ι−Ψ(ϕ, ε(u)),(1.1d)
posed in the parabolic cylinder Q:= Ω×(0, T ), where the domain Ω represents the brain
and the fixed time T > 0 is the medical treatment duration. In equations (1.1a) and
(1.1b) we introduced the notation
K(ϕ, σ, z):=k1(ϕ, z)σ
k2(ϕ, z) + σ, U (ϕ, σ, z, χ1):= (p(σ, z)−χ1)ϕ1−ϕ
N−ϕg(σ, z)
for the sake of brevity. As already pointed out, a slightly modified version of (1.1) was
proposed in [Cav+25]. We refer the reader to it for further modeling details and recall
only the most important aspects. The variable ϕrepresents the tumor cell density and
2
takes value in [0, N ], where the constant Nis the fixed carrying capacity of the tissue.
It is ruled by the Fischer–Kolmogorov type equation (1.1a). The parabolic reaction-
diffusion equation (1.1b) governs the evolution of σ, which is the intracellular lactate
concentration. The quasi-static balance of forces (1.1c) for a viscoelastic tissue describes
the dynamic of u, which is the small displacement field of each point with respect to a
reference undeformed configuration. The local tissue damage ztakes value in the interval
[0,1], where z= 0 means there is no damage while z= 1 means that the tissue is totally
damaged. Its evolution is ruled by the parabolic differential inclusion (1.1d). Finally,
χ1is the concentration of a cytotoxic drug, which decreases the tumor proliferation
p(σ, z) in the mass source U. Similarly, the term χ2represents a lactate targeting drug
and affects the lactate source term S. The system (1.1) is coupled with the boundary
conditions
∂νϕ=∂νz= 0,(1.2a)
∂νσ=σΓ−σ, (1.2b)
u=0,(1.2c)
and the initial conditions
ϕ(0) = ϕ0, σ(0) = σ0,u(0) = u0, z(0) = z0.(1.3)
The cost functional. Introducing the notation χ= (χ1, χ2), we define the following
cost functional
J((ϕ, σ, u, z),χ):=α1
2kϕ−ϕQk2
L2(Q)+α2
2kϕ(T)−ϕΩk2
H+α3ZΩ
ϕ(T) dx
+α4
2kσ−σQk2
L2(Q)+α5
2kσ(T)−σΩk2
H
+α6
2kpγ(ϕ)ε(u)k2
L2(Q)+α7
2kz−zQk2
L2(Q)
+α8ZΩ
z(T) dx+α9
2kχk2
L2(Q).
(1.4)
The non-negative constants α1,...,α9are weights that can not vanish all at the same
time, while ϕQ,ϕΩ,σQ,σΩ,zQare target functions. More explicitly, the term ϕQ(resp.
σQ,zQ) is a desired evolution for the tumor (resp. the lactate, the damage), while
ϕΩ(resp. σΩ) is a desired final configuration of the tumor (resp. concentration of the
lactate). The third and eighth addends measure the size of the tumor and the magnitude
of the damage at the end of the treatment. Since the presence of high mechanical stress,
especially in certain areas of the brain, can compromise its functionality, we are interested
in keeping it low. The sixth term serves this purpose, and γmay be, for instance, the
indicator function of a subdomain of Ω where the stress is intended to remain low.
Finally, the last addend is a L2regularization for the controls. We are interested in
studying the minimizers of the cost functional (1.4) subject to the PDE system (1.1)–
(1.3) and constrained to a suitable admissible control set. We define it as Uad =U1
ad ×U 2
ad
3
with
U1
ad :={χ1∈L2(V)∩L∞(Q):kχ1kL2(V)≤Cad, χ1≤χ1≤χ1a.e.},
U2
ad :={χ2∈L∞(Q):χ2≤χ2≤χ2a.e.},
where Cad >0 is a fixed constant and χ1,χ1,χ2,χ2∈L∞(Q) are given threshold
functions such that Uad is nonempty. The admissible control set Uad is a subset of
U=U1× U2:=hL2(V)∩L∞(Q)i×L∞(Q)
equipped with its natural norm that we denote with k·kU.
Remark 1.1. Notice that Uad is a nonempty, closed, and convex subset of U. Moreover,
there exists a positive constant Rsuch that
Uad ⊆ UR:={χ∈ U :kχkU< R}.
Aim and plan of the paper. The goal of this paper is to prove that the optimal
control problem has at least one minimizer and to derive first-order necessary conditions
for optimality. It is organized as follows. In Section 2, after introducing some notation
and preliminary results, we enlist the hypotheses we are going to use throughout the
paper. Then, we recall some known results about the state system (1.1)–(1.3) from the
previous work [Cav+25] and we prove a strict separation property for the damage. In
Section 3we properly state the optimal control problem and prove that it admits at least
one minimizer. In Section 4we analyze the linearized system, and in Section 5we use
it to prove the differentiability of the control-to-state operator. Finally, in Section 6, we
study the adjoint system and derive the first-order necessary conditions for optimality.
2 The state system
2.1 Notation and preliminaries
Notation. In what follows, for any real Banach space Xwith dual space X′, we
indicate its norm as k·kXand the dual pairing between X′and Xas h·,·iX. We denote
the Lebesgue and Sobolev spaces over Ω as Lp:=Lp(Ω), Wk,p :=Wk,p(Ω) and Hk:=
Wk,2(Ω), while for the Lebesgue spaces over Γ we use Lp
Γ:=Lp(Γ). For convenience, we
set
H:=L2
and we identify Hwith its dual space H′. We introduce
V:=H1, V0:=H1
0,
where H1
0represents the set of H1functions with zero trace at the boundary. Addition-
ally, we define
W:={v∈H2|∂νv= 0 on Γ}, W0:=H2∩V0.
4
In both cases, the natural norm induced by H2is denoted by k·kW. To simplify the no-
tation, we do not always distinguish between scalar, vector, and matrix-valued spaces.
For the sake of brevity, the norm of the Bochner space Wk,p(0, T ;X) is indicated as
k·kWk,p(X), omitting the time interval (0, T ). With the notation C0([0, T ]; X) we mean
the space of continuous X-valued functions. Finally, as is customary, Crepresents a
generic constant depending only on the problem’s data and whose value might change
from line to line or even within the same line. If we want to highlight a dependency on
a certain parameter, we put it as a subscript (e.g., Cτindicates a constant that depends
on τ,C0a constant that depends on the initial data, etc.).
Useful inequalities. We will make use of classical inequalities, such as H¨older, Young,
Gronwall, Poincar´e, and Poincar´e–Wirtinger. We will employ the following special cases
of Gagliardo–Nirenberg inequality (see [Nir59]).
Lemma 2.1. Let Ωbe a bounded Lipschitz domain in Rd. Then, it exists a constant C
such as, for every v∈V, it holds
kvkL4≤Ckvk
1
2
Hkvk
1
2
Vif n= 2,(2.1)
kvkL3≤Ckvk
1
2
Hkvk
1
2
V,kvkL4≤Ckvk
1
4
Hkvk
3
4
Vif n= 3.(2.2)
From this result, employing the Young inequality, it is easy to obtain an inequality that
we will extensively use throughout the paper and, thus, is worth mentioning. For every
δ > 0 there exists a positive constant Cδsuch that, for p= 3 and p= 4, the following
holds:
kvk2
Lp≤δkvk2
H+Cδk∇vk2
H,(2.3)
for every v∈V, both in dimension n= 2 and n= 3.
2.2 Hypotheses
In this section, we enlist the hypotheses under which we will work throughout the whole
paper. Most of them come from [Cav+25] and are necessary to establish solutions’ well-
posedness, additional regularity, and continuous dependence. However, we need stronger
assumptions to prove the results related to the optimal control problem. Specifically,
the higher regularity requested for the nonlinearities is because we need to handle the
corresponding terms in the linearized and adjoint systems.
(H1) We suppose that
p, g ∈C0,1(R2)∩C2(R2),(2.4)
0≤p≤p∗,0≤g≤g∗,(2.5)
where p∗,g∗denote positive constants, and that
Nis a positive constant. (2.6)
5
(H2) We assume that
k1, k2, S ∈C0,1(R2)∩C2(R2), and (2.7)
0≤k1≤k∗
1,0< k2∗≤k2≤k∗
2,0≤S≤S∗,(2.8)
where k∗
1,k2∗,k∗
2, and S∗are given constants.
(H3) The fourth-order tensors A= (aijkh), B= (bij kh):R2→Rn×n×n×nsatisfy:
Ais constant, symmetric, and strictly positive definite, and (2.9)
B ∈ C0,1(R)∩C2(R2) is bounded, symmetric, and positive definite.(2.10)
Moreover, we assume
f∈L∞(H).
In the following, being able to handle the maximal monotone operator βand its higher
derivatives will be essential. To this end, the key point is proving a strict separation
property for the damage variable z. This leads us to adopt a more restrictive form of
the convex potential ˆ
β, moving beyond the quite general hypotheses employed in the
previous analysis [Cav+25].
(H4) We consider a b
β:R→[0,+∞] such that
b
βis lower semicontinuous, convex, and (2.11)
b
β∈C3(0,1).(2.12)
Its derivative β:=b
β′satisfies the growth conditions:
lim
r→0+β(r) = −∞,lim
r→1−β(r) = +∞.(2.13)
(H5) We consider a function bπ∈C1(R)∩C3(0,1) and we denote by π:=bπ′its derivative,
requiring that
bπis concave, (2.14)
πis Lipschitz continuous. (2.15)
Example 2.2. The prototypical example to keep in mind is the logarithmic potential
ˆ
β(r) + ˆπ(r) = C1rln r+ (1 −r) ln (1 −r)−C2r2
for some given and positive constants C1, C2, which clearly fulfills the hypotheses (H4)
and (H5).
(H6) We suppose that
ι∈L∞(Q)∩H1(0, T ;H).(2.16)
6
(H7) We assume that Ψ ∈W2,∞(Ω ×R×Rn×n) and that
Ψ(x, ·,·):R×Rn×n→Ris Lipschitz continuous, i.e.,
∃CΨ>0 s.t. |Ψ(x, ϕ1, ǫ1)−Ψ(x, ϕ2, ǫ2)| ≤ CΨ|ϕ1−ϕ2|+|ǫ1−ǫ2|(2.17)
for a.e. x∈Ω, for all ǫ1, ǫ2∈Rn×n,ϕ1, ϕ2∈R.
In the following, for the sake of brevity, we will omit in the notation the dependence of
Ψ on the point x, using Ψ(ϕ, ǫ) instead of Ψ(x, ϕ, ǫ).
Remark 2.3. We require the boundedness of Ψ(ϕ, ε(u)) to establish a separation prop-
erty for the damage variable z. It is worth noting that this assumption is not as restrictive
as it might initially appear. Indeed, we will prove the boundedness of ϕ, and since we
are working within the framework of linear elasticity, ε(u) is expected to remain small.
Consequently, under hypothesis (2.17), the boundedness of Ψ(ϕ, ε(u)) would follow ask-
ing that Ψ(0,0)∈L∞(Ω). However, as we are unable to prove the boundedness of ε(u)
mathematically, it becomes necessary to explicitly impose Ψ the boundedness of as an
additional condition. The boundedness of the higher derivatives is required to handle
the linearized coefficients in the linearized and adjoint systems.
(H8) Regarding the boundary conditions, we suppose that
σΓ∈L2(0, T ;L2
Γ),0≤σΓ≤M0,(2.18)
where M0is a fixed positive constant.
(H9) Regarding the initial conditions, we assume that
ϕ0∈W, 0≤ϕ0≤N, (2.19)
σ0∈H, 0≤σ0≤M0,(2.20)
u0∈W0,(2.21)
z0∈W, 0<ess inf(z0),ess sup(z0)<1.(2.22)
Remark 2.4. Since the domain of such ˆ
βis the physically meaningful interval [0,1], the
initial datum z0must be chosen such that its values lie within this interval. Recalling
again that our goal is to prove a separation property for the damage, we request that z0
stays bounded away from 0 and 1.
Finally, regarding the cost functional J, we make the following assumptions.
(H10) The coefficients α1,...,α9are nonnegative constants that can not vanish all at the
same time.
(H11) The target functions satisfy
ϕQ, σQ, zQ∈L2(Q), ϕΩ, σΩ∈H. (2.23)
7
(H12) The coefficient γ:Ω×R→[0,+∞) is a Carath´eodory function such that
γ(x, ·)∈C1(R) (2.24)
for a.e. x∈Ω. Moreover, it exists a constant Cγ>0 such that
|γ(x, ϕ)|+|∂ϕγ(x, ϕ)| ≤ Cγ(2.25)
for a.e. x∈Ω and for all ϕ∈[0, N ].
Even if γalso depends on the point x∈Ω, in the following, we will employ the shorter
notation γ(ϕ) instead of γ(x, ϕ). For the same reason, the partial derivative of γwith
respect to the variable ϕwill be denoted by γ′(ϕ).
2.3 Previous results
Definition 2.5 (State system).We say that the quadruplet (ϕ, σ, u, z)is a weak solu-
tion to the state problem (1.1)–(1.3)if
ϕ∈H1(0, T ;H)∩L∞(0, T ;V)∩L2(0, T ;W),0≤ϕ≤N,
σ∈H1(0, T ;V′)∩L∞(0, T ;H)∩L2(0, T ;V),0≤σ≤M,
u∈W1,∞(0, T ;V0),
z∈H1(0, T ;H)∩L∞(0, T ;V)∩L2(0, T ;W),0< z < 1
where M=M(M0, S∗), with
ϕ(0) = ϕ0, σ(0) = σ0,u(0) = u0, z(0) = z0
a.e. in Ω, such that
ZΩ
∂tϕη +∇ϕ· ∇ηdx=ZΩ(p(σ, z)−χ1)ϕ1−ϕ
N−ϕg(σ, z)ηdx , (2.26a)
h∂tσ, ηiV+ZΩ
∇σ· ∇ηdx+ZΩ
k1(ϕ, z)σ
k2(ϕ, z) + σηdx+ZΓ
(σ−σΓ)ηdHd−1
=ZΩ
χ2S(ϕ, z)ηdx ,
(2.26b)
ZΩAε(∂tu) + B(ϕ, z)ε(u):ε(θ) dx=ZΩ
f·θdx , (2.26c)
ZΩ∂tzη +∇z· ∇η+β(z)η+π(z)ηdx=ZΩι−Ψ(ϕ, ε(u))ηdx , (2.26d)
a.e. in (0, T ), for every η∈Vand θ∈V0.
Since χis fixed and regular enough, well-posedness and continuous dependence can be
proved as in [Cav+25]. Note that zhas values in (0,1) because zbelongs to the domain
of β. The precise statement is contained in the theorem below.
8
Theorem 2.6. The following statements hold.
(i) For every χ∈ U, the state system admits a unique solution in the sense of Defini-
tion 2.5, with the following additional regularity
ϕ∈H1(0, T ;V)∩L∞(0, T ;W)∩L2(0, T ;H3),
u∈W1,∞(0, T ;W0),
z∈H1(0, T ;V)∩L∞(0, T ;W).
(ii) For every χ∈ URand its associated solution to the state system (ϕ, σ, u, z), the
following estimate is satisfied
kϕkH1(V)∩L∞(W)∩L2(H3)+kσkH1(V′)∩L∞(H)∩L2(V)
+kukW1,∞(W0)+kzkH1(V)∩L∞(W)≤CR
(2.27)
for a positive constant CRthat depends on R, the initial data, and the assigned
functions, but not on χ.
(iii) For every couple of controls χ,χ∈ URwith associated solutions to the state system
given respectively by (ϕ, σ, u, z)and (ϕ, σ, u, z), the following continuous depen-
dence inequality is satisfied
kϕ−ϕkL∞(H)∩L2(V)+kσ−σkL∞(H)∩L2(V)+ku−ukH1(V0)
+kz−zkL∞(H)∩L2(V)≤CRkχ−χkL2(Q)
for a certain positive constant CRthat depends on R, the initial data, and the
assigned functions, but not on χ,χ.
Remark 2.7. From (ii) and (iii) in Theorem 2.6, employing standard interpolation
results, we are able to prove the following continuity estimates which we will need later
on. Precisely, for every χ,χin UR, we have:
kϕ−ϕkL∞(L4)+kε(u)−ε(u)kL∞(L4)+kz−zkL∞(L4)≤CRkχ−χk
1
4
L2(Q).(2.28)
Applying Gagliardo–Nirenberg interpolation inequality from Lemma 2.1, estimates (ii)
and (iii) from the Theorem above, we have:
kε(u)−ε(u)kL∞(L4)≤Ckε(u)−ε(u)k
1
4
L∞(H)kε(u)−ε(u)k
3
4
L∞(V)
≤Cku−uk
1
4
L∞(V0)ku−uk
3
4
L∞(W0)≤CRku−uk
1
4
H1(V0)≤CRkχ−χk
1
4
L2(Q).
The same can be performed for ϕand z. Similarly, we have
kϕ−ϕkL4(L∞)+kz−zkL4(L∞)≤CRkχ−χk
1
4
L2(Q).(2.29)
9
In fact, applying Gagliardo–Nirenberg inequality and the embedding W1,4֒→L∞(Ω),
we obtain
kϕ−ϕkL∞(Ω) ≤ kϕ−ϕkW1,4≤Ckϕ−ϕk
1
4
Vkϕ−ϕk
3
4
W≤CRkϕ−ϕk
1
4
V,
where the last inequality follows from the energy estimate in (ii). Elevating to the power
4th and integrating in time over (0, T ) leads to
kϕ−ϕk4
L4(L∞)=ZT
0
kϕ−ϕk4
L∞(Ω) dt≤CRZT
0
kϕ−ϕkVdt
=CRkϕ−ϕkL1(V)≤CRkϕ−ϕkL2(V)≤CRkχ−χkL2(Q)
thanks to the continuous estimate (iii). The same holds for the damage. Finally, the
following inequality is satisfied
kϕ−ϕkL4(L3)+kz−zkL4(L3)≤CRkχ−χkL2(Q).(2.30)
Proceeding as before,
kϕ−ϕk4
L4(L3)=ZT
0
kϕ−ϕk4
L3dt≤CZT
0
kϕ−ϕk2
Hkϕ−ϕk2
Vdt
≤Ckϕ−ϕk2
L∞(H)ZT
0
kϕ−ϕk2
Vdt
=Ckϕ−ϕk2
L∞(H)kϕ−ϕk2
L2(V)≤CRkχ−χk4
L2(Q),
and the same for z.
2.4 A strict separation property for the damage
Differently from [Cav+25], we impose more restrictive assumptions on the potential ˆ
β,
as well as boundedness for ιand Ψ, which enable us to establish a separation property
for the damage z.
Proposition 2.8. There exist 0< r∗≤r∗<1which may depend on the data of the
problem and on Rsuch that, for every χ∈ URwith the associated solution to the state
problem (ϕ, σ, u, z),
r∗≤z≤r∗(2.31)
a.e. in Q.
Proof. We claim that there exist 0 < r∗≤r∗<1 such that
1. r∗≤ess inf(z0),
2. ess sup(z0)≤r∗,
3. β(r) + π(r) + kιkL∞+kΨkL∞≤0 for all r∈(0, r∗),
10
4. β(r) + π(r)− kιkL∞− kΨkL∞≥0 for all r∈(r∗,1).
We can find such r∗, r∗satisfying conditions 1and 2because of hypothesis (H9) ensuring
that z0remains bounded away from 0 and 1. Moreover, regarding points 3and 4, we
recall that, from the specific choice we made for the potential ˆ
β, it holds
β(r)→ −∞ if r→0+, β(r)→+∞if r→1−
where, from hypothesis (H5),π∈C0(R) so is bounded over [0,1] and, from hypotheses
(H6) and (H7),ιand Ψ are bounded. Let χbe an arbitrary control in URand S(χ) =
(ϕ, σ, u, z). We test the equation (2.26d) with (z−r∗)+, obtaining:
0 = 1
2
d
dtZΩ
|(z−r∗)+|2dx+ZΩ
|∇(z−r∗)+|2dx
+ZΩβ(z) + π(z)−ι+ Ψ(ϕ, ε(u))(z−r∗)+dx
≥1
2
d
dtZΩ
|(z−r∗)+|2dx ,
where we used the property 4in order to obtain the inequality. Integrating in time over
the interval (0, t), it follows that
ZΩ
|(z−r∗)+|2dx≤ZΩ
|(z0−r∗)+|2dx= 0
because, accordingly to property 2,z0is smaller or equal to r∗almost everywhere, thus
(z0−r∗)+= 0. This proves that (z−r∗)+= 0 and, equivalently, that zis smaller
or equal to r∗almost everywhere in Q. Proceeding in the same way, we test equation
(2.26d) with −(z−r∗)−. Employing inequality 3and then integrating in time and using
1, we obtain that (z−r∗)−is equal to 0. Thus, we have that zis bigger or equal to r∗
almost everywhere in Q.
Remark 2.9. Let us highlight the fact that thanks to the strict separation property
we have just proved, from now on we will treat βas a regular potential. Moreover, we
trivially deduce that
kβ(z)kL∞+kβ′(z)kL∞+kβ′′(z)kL∞≤CR,
since β∈C2(0,1) by hypothesis (H4).
3 The optimal control problem
In view of the previous Theorem 2.6, we introduce the so-called control-to-state operator
or solution operator, which maps every control χto the unique solution to the associated
11
state problem. More precisely, we introduce the state-space
VS:=hH1(0, T ;H)∩L∞(0, T ;V)∩L2(0, T ;W)i
×hH1(0, T ;V′)∩L∞(0, T ;H)∩L2(0, T ;V)i×W1,∞(0, T ;W)
×hH1(0, T ;H)∩L∞(0, T ;V)∩L2(0, T ;W)i
which is continuously embedded into the larger
V:=hC0([0, T ]; H)∩L2(0, T ;V)i×hC0([0, T ]; H)∩L2(0, T ;V)i
×H1(0, T ;V0)×hC0([0, T ]; H)∩L2(0, T ;V)i.
We define
S:U → V,χ7→ (ϕ, σ, u, z).(3.1)
Notice that Sis well-defined over Uand Lipschitz continuous over UR. We introduce
the reduced cost functional as
J(χ):=J(S(χ),χ).(3.2)
Then, the optimal control problem can be stated as
min
χ∈Uad
J(χ),(3.3)
which means that we search a minimizer for the functional Jsubject to the PDE system
(1.1)–(1.3) and constrained to Uad .
Theorem 3.1. There exists at least one minimizer χ∗∈ Uad to the optimal control
problem (3.3).
Proof. The reduced cost functional is proper and non-negative, so infχ∈Uad J(χ) is finite
and non-negative. Let {χn}n∈N⊆ Uad be a minimizing sequence for J, meaning that
inf
χ∈Uad
J(χ) = lim
n→+∞J(χn).
We denote the corresponding solution to the state system as (ϕn, σn,un, zn) = S(χn).
Since the sequence {χn}n∈N∈ Uad, it is uniformly bounded in Uand, consequentially,
there exists a χ∗∈ U such that, along a subsequence that we do not relabel,
χn,1→χ∗
1weakly-∗in L2(0, T ;V)∩L∞(Q),
χn,2→χ∗
2weakly-∗in L∞(Q).
Notice that Uad is convex and closed in the space L2(0, T ;V)×L2(Q) thus, it is sequen-
tially weakly closed. This justifies the fact that the limit χ∗belongs to Uad. Moreover,
12
by the uniform boundedness of the solution sequence from (ii) in Theorem 2.6, applying
Banach–Alaouglu (see, e.g., [Br´e11]) and Aubin–Lions Theorems (see [Sim86, Section 8,
Corollary 4]), there exists a (ϕ∗, σ∗,u∗, z∗)∈ VSsuch that, along a further subsequence,
ϕn→ϕ∗weakly-∗in H1(0, T ;H)∩L∞(0, T ;V)∩L2(0, T ;W),
strong in L2(0, T ;V),
a.e. in Q,
σn→σ∗weakly-∗in H1(0, T ;V′)∩L∞(0, T ;H)∩L2(0, T ;V),
strong in L2(0, T ;H),
a.e. in Q,
un→u∗weakly-∗in W1,∞(0, T ;W0),
strong in C0([0, T ]; V0),
zn→z∗weakly-∗in H1(0, T ;H)∩L∞(0, T ;V)∩L2(0, T ;W),
strong in L2(0, T ;V),
a.e. in Q.
The first step of the proof consists of proving that S(χ∗) = (ϕ∗, σ∗,u∗, z∗). To do so,
thanks to the convergences above, we can pass to the limit in the PDE system satisfied
by (ϕn, σn,un, zn) and χn, from which (ϕ∗, σ∗,u∗, z∗) is the unique solution to the state
system associated with χ∗. The second step is showing that
inf
χ∈Uad
J(χ) = J(χ∗).
To this end, we write the reduced cost functional as the sum of the following terms
J1(χ) = α1
2kϕ−ϕQk2
L2(Q)+α4
2kσ−σQk2
L2(Q)+α7
2kz−zQk2
L2(Q)+α9
2kχk2
L2(Q),
J2(χ) = α2
2kϕ(T)−ϕΩk2
H+α3kϕ(T)kL1(Ω)
+α5
2kσ(T)−σΩk2
H+α6
2kpγ(ϕ)ε(u)k2
L2(Q)+α8kz(T)kL1(Ω).
By the weak convergences above and weak lower semicontinuity of the L2-norm, we
obtain
lim inf
n→+∞J1(χn)≥J1(χ∗).(3.4)
Notice that, by uniform boundedness of the sequence from (ii) in Theorem 2.6, and
Aubin–Lions Theorem (see [Sim86, Section 8, Corollary 4]), we have the strong conver-
gences
ϕn→ϕ∗strongly in C0([0, T ]; H),(3.5)
σn→σ∗strongly in C0([0, T ]; V′),(3.6)
zn→z∗strongly in C0([0, T ]; H).(3.7)
13
Again from (ii) in Theorem 2.6,{σ∗(T)}n∈Nis uniformly bounded in H. Thus, extracting
a further subsequence,
σn(T)→σ∗(T) weakly in H, (3.8)
where we are able to identify the limit with σ∗(T) because of convergence (3.6). Since
ϕn→ϕ∗a.e. and γis bounded by hypothesis (H12),
pγ(ϕn)η→pγ(ϕ∗)ηstrongly in L2(0, T ;H),(3.9)
for every η∈L2(0, T ;H) thanks to Dominated Convergence Theorem. Moreover, from
the convergences enlisted above, we know that
ε(un)→ε(u∗) weakly in L2(0, T ;H).(3.10)
Putting together (3.9) and (3.10), we deduce that
pγ(ϕn)ε(un)→pγ(ϕ∗)ε(u∗) weakly in L2(0, T ;H).(3.11)
By the weak lower semicontinuity of the L2-norm and the weak convergences (3.8),
(3.11), and by the strong continuity of the L2-norm and the strong convergences (3.5),
(3.7), we have
lim inf
n→+∞J2(χn)≥J2(χ∗)
from which the thesis follows.
We aim to establish first-order optimality conditions for the optimal control problem
(3.3). To do so, the standard procedure is to prove the Fr´echet differentiability of the
control-to-state operator. With this purpose, we linearize the state system.
4 The linearized state system
We consider a fixed control χ∈ Uad with the associated state given by (ϕ, σ, u, z) = S(χ).
For every small h= (h1, h2)∈ U, we consider the perturbed variables
ϕ+ξ, σ +ρ, u+ω, z +ζ, χ+h,(4.1)
and the corresponding state system. Linearizing it near (ϕ, σ, u, z), χand approximating
the nonlinearities with their first order Taylors’expantions, we have that (ξ, ρ, ω, ζ ), h
satisfy the linear PDE system
∂tξ−∆ξ=a1ξ+a2ρ+a3ζ+a4h1,(4.2a)
∂tρ−∆ρ=b1ξ+b2ρ+b3ζ+b4h2,(4.2b)
−div Aε(∂tω) + B(ϕ, z)ε(ω)=−div [c1ξ+c2ζ],(4.2c)
∂tζ−∆ζ=d1ξ+d2:ε(ω) + d3ζ, (4.2d)
14
where, for the sake of better readability, we introduced the following notation:
a1=U,ϕ = (p(σ, z)−χ1)1−2ϕ
N−g(σ, z),
a2=U,σ =p,σ(σ, z)ϕ1−ϕ
N−ϕg,σ(σ, z),
a3=U,z =p,z(σ, z)ϕ1−ϕ
N−ϕg,z(σ, z),
a4=U,χ1=−ϕ1−ϕ
N,
b1=−K,ϕ(ϕ, σ, z) + χ2S,ϕ(ϕ, z )
=−k1,ϕ(ϕ, z)σ
k2(ϕ, z) + σ+k1(ϕ, z)σk2,ϕ (ϕ, z)
(k2(ϕ, z) + σ)2+χ2S,ϕ (ϕ, z),
b2=−K,σ(ϕ, σ, z) = −k1(ϕ, z)
k2(ϕ, z) + σ+k1(ϕ, z)σ
(k2(ϕ, z) + σ)2,
b3=−K,z(ϕ, σ, z) + χ2S,z (ϕ, z)
=−k1,z(ϕ, z)σ
k2(ϕ, z) + σ+k1(ϕ, z)σk2,z (ϕ, z)
(k2(ϕ, z) + σ)2+χ2S,z (ϕ, z),
b4=S(ϕ, z),
c1=−B,ϕ(ϕ, z)ε(u),
c2=−B,z(ϕ, z)ε(u),
d1=−Ψ,ϕ(ϕ, ε(u)),
d2=−Ψ,ε(ϕ, ε(u)),
d3=−β′(z)−π′(z).
Remark 4.1. Notice that, since all the assigned functions are Lipschitz continuous
and bounded, and because of the regularity we have already proved for (ϕ, σ, u, z) in
Theorem 2.6,
a1,...,a4, b1,...,b4, d1,d2∈L∞(Q),
and they are uniformly bounded by a constant that depends on R. Moreover,
c1,c2∈L∞(0, T ;Lp)
for any p∈[1,6] and their norm is uniformly bounded by a certain CR, because
|c1|+|c2| ≤ C|ε(u)|,
and u∈W1,∞(0, T ;W)֒→W1,∞(0, T ;W1,p ). Regarding the last term, thanks to the
regularity of βin its domain and to the separation property we proved in Proposition 2.8,
d3∈L∞(Q),
and its norm is bounded by a constant that depends on R.
15
We couple the system (4.2) with the following boundary and initial conditions
∂νξ=∂νζ= 0 ∂νρ=−ρ, ω= 0 on Σ,(4.4)
ξ(0) = ρ(0) = ζ(0) = 0,ω(0) = 0in Ω.(4.5)
Notice that, even if for the formal derivation of the linearized system we started from
a small perturbation h∈ Uad , the obtained system (4.2)–(4.5) makes sense for every
h∈L2(Q)×L2(Q) .
Proposition 4.2. For every χ∈ URwith associated S(χ) = (ϕ, σ, u, z)∈ V and for
every h∈L2(Q)×L2(Q)there exists a unique solution (ξ, ρ, ω, ζ )to the linearized state
system (4.2)–(4.5)in the sense that
ξ∈H1(0, T ;H)∩L∞(0, T ;V)∩L2(0, T ;W),
ρ∈H1(0, T ;V′)∩L∞(0, T ;H)∩L2(0, T ;V),
ω∈W1,∞(0, T ;V0),
ζ∈H1(0, T ;H)∩L∞(0, T ;V)∩L2(0, T ;W)
with
ξ(0) = ρ(0) = ζ(0) = 0,ω(0) = 0
such that
ZΩ
∂tξη dx+ZΩ
∇ξ· ∇ηdx=ZΩ
[a1ξ+a2ρ+a3ζ+a4h1]ηdx , (4.6a)
h∂tρ, ηiV+ZΩ
∇ρ· ∇ηdx+ZΓ
ρη dHd−1=ZΩ
[b1ξ+b2ρ+b3ζ+b4h2]ηdx , (4.6b)
ZΩAε(∂tω) + B(ϕ, z)ε(ω):ε(θ) dx=ZΩ
[c1ξ+c2ζ]:ε(θ) dx , (4.6c)
ZΩ
∂tζη dx+ZΩ
∇ζ· ∇ηdx=ZΩd1ξ+d2:ε(ω) + d3ζηdx , (4.6d)
a.e. in (0, T ), for every η∈Vand θ∈V0. Moreover, the solution satisfies the following
estimate:
kξkH1(H)∩L∞(V)∩L2(W)+kρkH1(H)∩L∞(V)∩L2(W)+kωkW1,∞(V0)
+kξkH1(H)∩L∞(V)∩L2(W)≤CRkhkL2(Q)2.(4.7)
Proof. Existence can be proved using a Galerkin scheme. Since it is a standard pro-
cedure, we will show only the formal a priori estimates that are necessary to pass to
the limit from the discrete to the continuous system. We test (4.2a) with ξ, (4.2b)
with ρ, (4.2d) with ζ, and sum the three equations. Applying the Young inequality and
Remark 4.1, we obtain
d
dtkξk2
H+kρk2
H+kζk2
H+k∇ξk2
H+k∇ρk2
H+k∇ζk2
H
≤CRkξk2
H+kρk2
H+kε(ω)k2
H+kζk2
H+kh1k2
H+kh2k2
H.
(4.8)
16
Testing (4.2c) with ε(∂tω) and employing the fact that Ais positive definite, we have
CAkε(∂tω)k2
H≤ZΩ
Aε(∂tω):ε(∂tω) dx
=ZΩ−B(ϕ, z)ε(ω):ε(∂tω) + ξc1:ε(∂tω) + ζc2:ε(∂tω)dx .
Recalling that Bis bounded and c1,c2are uniformly bounded in L∞(0, T ;L6) thanks to
Remark 4.1, we estimate the right-hand side with the H¨older and the Young inequalities.
We get
CAkε(∂tω)k2
H≤Ckε(ω)kH+kξkL3kc1kL6+kζkL3kc2kL6kε(∂tω)kH
≤δkε(∂tω)k2
H+CR,δ kε(ω)k2
H+kξk2
L3+kζk2
L3,
where δis a small positive constant yet to be determined. By means of the interpolation
inequality in Lemma 2.1 and again the Young inequality, we have
CAkε(∂tω)k2
H≤δkε(∂tω)k2
H+k∇ξk2
H+k∇ζk2
H
+CR,δ kε(ω)k2
H+kξk2
H+kζk2
H.
(4.9)
Recalling that
kε(ω)k2
H≤Zt
0
kε(∂tω)k2
Hds
because ω(0) = 0, summing equations (4.8) and (4.9), and fixing δsmall enough leads
to
d
dtkξk2
H+kρk2
H+kζk2
H+k∇ξk2
H+k∇ρk2
H+kε(∂tω)k2
H+k∇ζk2
H
≤CR kξk2
H+kρk2
H+kζk2
H+Zt
0
kε(∂tω)k2
Hds+khk2
L2(Q)!.
Applying Gronwall inequality and then standard parabolic regularity estimates, we re-
cover the a priori estimates we need to pass in the Galerkin discretization. This way,
existence is proved as well as the estimate (4.7) in the statement. Uniqueness follows
from the estimate we have just proved and the fact that the system is linear.
Remark 4.3. For χ∈ URfixed, it is convenient to denote the solution to the linearized
state system associated with a perturbation h∈ U as (ξh, ρh,ωh, ζ h). From this result,
it follows that the map
U ⊆ L2(Q)×L2(Q)→ V,h7→ (ξh, ρh,ωh, ζh),
is linear and continuous.
17
5 Differentiability of the control-to-state operator
In this section, we will prove the Fr´echet differentiability of the control-to-state operator.
Theorem 5.1. The control-to-state operator S:U → V is Fr´echet differentiable in UR
and the Fr´echet derivative of Sin χ∈ URis given by
DS(χ)h= (ξh, ρh,ωh, ζh)
for every h∈ U.
Proof. We consider a fixed and arbitrary χ∈ Uad with S(χ) = (ϕ, σ, u, z). Our goal is
to prove that
lim
khkU→0
kS(χ+h)− S(χ)−(ξh, ρh,ωh, ζh)kV
khkU
= 0,(5.1)
from which the thesis follows. We introduce the notation S(χ+h) = (ϕh, σh,uh, zh)
and
Φh:=ϕh−ϕ−ξh, λh:=σh−σ−ρh,wh:=uh−u−ωh, µh:=zh−z−ζh.
Since χbelongs to Uad which is in turn contained in the open set UR, there exists a
constant Cχsuch that, for every h∈ URwith khkU≤Cχ, the control χ+hstill belongs
to UR. Without loss of generality, since our aim is to pass to the limit as khkUgoes to
0, we will consider only hwith a small norm in this sense. We are going to prove that
k(Φh, λh,wh, µh)kV≤CRkhk
5
4
U,(5.2)
which yields to the limit (5.1). To do so, we consider the PDE system satisfied by
(Φh, λh,wh, µh) which can be trivially obtained by Theorem 2.6 and Proposition 4.2.
Explicitly, the following equations are satisfied
ZΩ
∂tΦhηdx+ZΩ
∇Φh· ∇ηdx=ZΩ
[A1+A2]ηdx , (5.3a)
h∂tλh, ηiV+ZΩ
∇λh· ∇ηdx+ZΓ
λhηdHd−1=ZΩ
[B1+B2+B3]ηdx , (5.3b)
ZΩhAε(∂twh) + C1i:ε(θ) dx= 0,(5.3c)
ZΩ
∂tµhηdx+∇µh· ∇ηdx=ZΩ
[D1+D2]ηdx , (5.3d)
for every η∈Vand θ∈V0as well as the initial conditions
Φh(0) = 0, λh(0) = 0,wh(0) = 0, µh(0) = 0.(5.4)
Here we have introduced the notation:
A1=U(ϕh, σh, zh, χ1)−U(ϕ, σ, z, χ1)
18
−hU,ϕ(ϕ, σ, z, χ1)ξh+U,σ (ϕ, σ, z, χ1)ρh+U,z (ϕ, σ, z, χ1)ζhi,
A2=−ϕh1−ϕh
N−ϕ1−ϕ
Nh1,(5.5)
B1=K(ϕh, σh, zh)−K(ϕ, σ, z)−hK,ϕ(ϕ, σ, z)ξh+K,σ (ϕ, σ, z)ρh+K,z(ϕ, σ, z)ζhi,
B2=S(ϕh, zh)−S(ϕ, z)−S,ϕ(ϕ, z)ξh+S,z (ϕ, z)ζhχ2,
B3=[S(ϕh, zh)−S(ϕ, z)]h2,(5.6)
C1=B(ϕh, zh)ε(uh)− B(ϕ, z)ε(u)
−hB,ϕ(ϕ, z)ε(u)ξh+B(ϕ, z)ε(ωh) + B,z (ϕ, z)ε(u)ζhi,
D1=−hβ(zh) + π(zh)−(β(z) + π(z)) −(β′(z) + π′(z))ζhi,
D2=−hΨ(ϕh, ε(uh)) −Ψ(ϕ, ε(u)) −(Ψ,ϕ(ϕ, ε(u))ξh+ Ψ,ε(ϕ, ε(u)) :ε(ωh))i.
The next step is testing each equation in (5.3) with a proper term and doing some
estimates. For this reason, it is convenient to rewrite some of the known coefficient
functions we have just introduced. To this end, we recall that, according to Taylor’s
theorem with an integral reminder, for a function l∈W2,2([0,1]) it holds
l(1) = l(0) + l′(0) + Z1
0
l′′(s)(1 −s) ds .
Let’s take A1into account. We introduce yh= (ϕh, σh, zh, χ1), y= (ϕ, σ, z, χ1), and
the function
l(s) = U(syh+ (1 −s)y),
which is W2,∞because Uhas this regularity and syh+ (1 −s)yis L∞. If we apply the
formula above, we get
U(yh) = U(y) + ∇U(y)·(yh−y)
+Z1
0hD2U(syh+ (1 −s)y)(yh−y)·(yh−y)i(1 −s) ds
=:U(y) + ∇U(y)·(yh−y) + A1(yh−y)·(yh−y).
Notice that the matrix
A1=Z1
0
D2U(syh+ (1 −s)y)(1 −s) ds
as well as ∇Uare bounded and their L∞-norm are uniformly controlled by a constant
that depends on R. Comparing the equality we have just obtained with A1leads to
A1=U,ϕ(ϕ, σ, z, χ1)Φh+U,σ (ϕ, σ, z, χ1)λh+U,z(ϕ, σ, z, χ1)µh
+A1(ϕh−ϕ, σh−σ, zh−z, 0) ·(ϕh−ϕ, σh−σ, zh−z, 0).(5.7)
19
Proceeding in the same way, we have:
B1=K,ϕ(ϕ, σ, z)Φh+K,σ (ϕ, σ, z)λh+K,z (ϕ, σ, z)µh
+B1(ϕh−ϕ, σh−σ, zh−z)·(ϕh−ϕ, σh−σ, zh−z),(5.8)
B2=hS,ϕ(ϕ, z)Φh+S,z(ϕ, z)µh+B2(ϕh−ϕ, zh−z)·(ϕh−ϕ, zh−z)iχ2,(5.9)
C1=B,ϕ(ϕ, z)ε(u)Φh+B(ϕ, z)ε(wh) + B,z (ϕ, z)ε(u)µh
+C1(ϕh−ϕ, ε(uh)−ε(u), zh−z)·(ϕh−ϕ, ε(uh)−ε(u), zh−z),(5.10)
D1=−h(β′(z) + π′(z))µh+D1(zh−z)2i,(5.11)
D2=−hΨ,ϕ(ϕ, ε(u))Φh+ Ψ,ε(ϕ, ε(u)) :ε(wh)
+D2(ϕh−ϕ, ε(uh)−ε(u)) ·(ϕh−ϕ, ε(uh)−ε(u))i.
(5.12)
Notice that the terms written in Fraktur font are the ones related to the integral of the
hessian of the auxiliary function l, and, therefore, their dimensions change from case
to case: for example, B1is a matrix in R3×3,B2a matrix in R2×2, and D1is a scalar.
Moreover, it is easy to check that all these terms are uniformly bounded in L∞by a
constant that depends on Rwith the only exception of C1, which is uniformly bounded
in L∞(L6) because, even if B ∈ W2,∞, the terms ε(uh), ε(u) are uniformly bounded
only in this weaker norm. We will examine C1more closely later, addressing the estimate
of C1. We test equation (5.3a) with Φh, obtaining
1
2
d
dtkΦhk2
H+k∇Φhk2
H=ZΩ
(A1+A2)Φhdx .
From equation (5.7),
|A1| ≤ CR|Φh|+|λh|+|µh|+|ϕh−ϕ|2+|σh−σ|2+|zh−z|2,
and from equation (5.5),
|A2| ≤ CR|ϕh−ϕ| |h1|.
Putting these elements together and using the H¨older inequality, we obtain
1
2
d
dtkΦhk2
H+k∇Φhk2
H
≤CRkΦhkH+kλhkH+kµhkHkΦhkH
+CRkϕh−ϕkHkϕh−ϕkL6+kσh−σkHkσh−σkL6
+kzh−zkHkzh−zkL6+kϕh−ϕkL6kh1kHkΦhkL3.
We estimate the term kΦkL3by means of the Gagliardo–Nirenberg inequality (see
20
Lemma 2.1), and then we apply the H¨older inequality. We have
1
2
d
dtkΦhk2
H+k∇Φhk2
H≤CRkλhk2
H+kµhk2
H
+kϕh−ϕk2
Hkϕh−ϕk2
L6+kσh−σk2
Hkσh−σk2
L6+kzh−zk2
Hkzh−zk2
L6
+CRkϕh−ϕk2
Hkh1k2
L6+CR,δkΦhk2
H+δk∇Φhk2
H
(5.13)
for a small parameter δ > 0. Testing (5.3b) with λhleads to
1
2
d
dtkλhk2
H+k∇λhk2
H≤1
2
d
dtkλhkH+k∇λhk2
H+kλhk2
L2
Γ=ZΩ
(B1+B2+B3)λhdx .
Proceeding as before, from equations (5.8) and (5.9) we get
|B1|+|B2| ≤ CR|Φh|+|λh|+|µh|+|ϕh−ϕ|2+|σh−σ|2+|zh−z|2,
where we have also employed the fact that kχ2kL∞≤R. From equation (5.6) we derive
|B3| ≤ C|ϕh−ϕ|+|zh−z||h2|.
Consequentially, through the H¨older and the Young inequalities, then the Gagliardo–
Nirenberg inequality, and again the Young inequality with a small positive parameter δ,
we have
1
2
d
dtkλhk2
H+k∇λhk2
H≤CRkΦhk2
H+kµhk2
H
+kϕh−ϕk2
Hkϕh−ϕk2
L6+kσh−σk2
Hkσh−σk2
L6+kzh−zk2
Hkzh−zk2
L6
+CRkϕh−ϕk2
L6+kzh−zk2
L6kh2k2
H+CR,δkλhk2
H+δk∇λhk2
H.
(5.14)
We test equation (5.3c) with ∂twh, obtaining:
CAkε(∂twh)k2
H≤ZΩ
Aε(∂twh):ε(∂twh) dx=−ZΩ
C1:ε(∂twh) dx .
Our goal is to perform suitable estimates of the right-hand side of this inequality. By
equation (5.10),
|C1| ≤C|ε(u)||Φh|+|µh|+C|ε(wh)|
+|C1(ϕh−ϕ, ε(uh)−ε(u), zh−z)·(ϕh−ϕ, ε(uh)−ε(u), zh−z)|.
Let us analyze the last term on the right-hand side. We define yh= (ϕh, ε(uh), zh),
y= (ϕ, ε(u), z), the function
L(y) = B(ϕ, z)ε(u),
21
and the associated
l(s) = Lsyh+ (1 −s)y=Bsϕh+ (1 −s)ϕ, s zh+ (1 −s)hs ε(uh) + (1 −s)ε(u)i.
As done before,
C1=Z1
0
D2Lsyh+ (1 −s)y(1 −s) ds .
Notice that Lis linear in ε(u), so the related second derivative vanishes. Thus, the
quadratic term in ε(uh)−ε(u) will not appear in C1. Explicitly,
|C1(yh−y)·(yh−y)| ≤C|ε(uh)|+|ε(u)|h|ϕh−ϕ|2+|zh−z|2i
+C|ε(uh)−ε(u)|h|ϕh−ϕ|+|zh−z|i,
because Bbelongs to W2,∞. Employing this inequality, we have
CAkε(∂twh)k2
H≤CZΩ|ε(u)||Φh|+|µh||ε(∂twh)|+|ε(wh)||ε(∂twh)|dx
+CZΩ|ε(uh)−ε(u)||ϕh−ϕ|+|zh−z||ε(∂twh)|dx
+CZΩ|ε(uh)|+|ε(u)||ϕh−ϕ|2+|zh−z|2|ε(∂twh)|dx
≤Chkε(u)kL6kΦhkL3+kµhkL3+kε(wh)kH
+kε(uh)−ε(u)kL4kϕh−ϕkL4+kzh−zkL4
+kε(uh)kL6+kε(u)kL6kϕh−ϕk2
L6+kzh−zk2
L6ikε(∂twh)kH.
We recall that by (ii) the terms kε(u)kL6,kε(uh)kL6are bounded by a constant that
depends on R. By the Young inequality and the Gagliardo–Nirenberg interpolation
inequality from Lemma 2.1, we deduce
CAkε(∂twh)k2
H
≤δk∇Φhk2
H+k∇µhk2
H+kε(∂twh)k2
H+CR,δhkΦhk2
H+kµhk2
H
+kε(wh)k2
H+kε(uh)−ε(u)k2
L4(kϕh−ϕk2
L4+kzh−zk2
L4)
+kϕh−ϕk4
L6+kzh−zk4
L6i
(5.15)
for a small δ > 0. Testing equation (5.3d) with µhleads to
1
2
d
dtkµhk2
H+k∇µhk2
H=ZΩ
(D1+D2)µhdx .
Regarding D1, we recall that thanks to the separation property we proved for z, the
term β′(z) + π′(z) is bounded. We have
|D1| ≤ C|µh|+|zh−z|2.
22
Turning our attention to D2, we get
|D2| ≤ C|Φh|+|ε(wh)|+|ϕh−ϕ|2+|ε(uh)−ε(u)|2.
Thus, with standard argumentation, we deduce
1
2
d
dtkµhk2
H+k∇µhk2
H
≤δk∇µhk2
H+Cδkµhk2
H+kΦhk2
H+kε(wh)k2
H+kzh−zk2
Hkzh−zk2
L6
+kϕh−ϕk2
Hkϕh−ϕk2
L6+kε(uh)−ε(u)k2
Hkε(uh)−ε(u)k2
L4.
(5.16)
We sum inequalities (5.13)–(5.16), integrate in time over (0, t) remembering that the
initial values of the unknowns are zero, and move the terms multiplied by δto the
left-hand side, fixing a parameter small enough. This way we get the following:
kΦhk2
H+kλhk2
H+kµhk2
H+Zt
0k∇Φhk2
H+k∇λhk2
H+kε(∂twh)k2
H+k∇µhk2
Hds
≤CRZt
0hkΦhk2
H+kλhk2
H+kε(wh)k2
H+kµhk2
H
+kϕh−ϕk2
Hkϕh−ϕk2
L6+kσh−σk2
Hkσh−σk2
L6+kzh−zk2
Hkzh−zk2
L6
+kϕh−ϕk2
Hkh1k2
L6+kϕh−ϕk2
L6+kzh−zk2
L6kh2k2
H
+kε(uh)−ε(u)k2
L4(kϕh−ϕk2
L4+kε(uh)−ε(u)k2
H+kzh−zk2
L4)
+kϕh−ϕk4
L6+kzh−zk4
L6ids .
We recall that
kε(wh)k2
H≤CZs
0
kε(∂twh)k2
Hdτ ,
and that, thanks to (iii) in Theorem 2.6, it holds
kϕh−ϕkL∞(H)+kσh−σkL∞(H)+kε(uh)−ε(u)kL∞(H)
+kzh−zkL∞(H)≤CRkhkL2(Q),
and that, by (2.28) in Remark 2.7, we have
kϕh−ϕkL∞(L4)+kε(uh)−ε(u)kL∞(L4)+kzh−zkL∞(L4)≤CRkhk
1
4
L2(Q).
Finally, we observe that
Zt
0
kϕh−ϕk4
L6ds≤Zt
0
kϕh−ϕk2
L∞(Ω)kϕh−ϕk2
L3ds
≤ kϕh−ϕk2
L4(L∞)kϕh−ϕk2
L4(L3)≤CRkhk
1
2
L2(Q)khk2
L2(Q)=CRkhk
5
2
L2(Q)
23
where we have applied the H¨older inequality and, in the last passage, we have combined
(2.29) and (2.30) from Remark 2.7. The same inequality holds for zh−z. Thus, we
obtain
kΦhk2
H+kλhk2
H+kµhk2
H+Zt
0k∇Φhk2
H+k∇λhk2
H+kε(∂twh)k2
H+k∇µhk2
Hds
≤CRZt
0hkΦhk2
H+kλhk2
H+kµhk2
Hids+Zt
0Zs
0
kε(∂twh)k2
Hdτds
+khk2
L2(Q)Zt
0hkϕh−ϕk2
L6+kσh−σk2
L6+kzh−zk2
L6ids
+khk2
L2(Q)kh1k2
L2(L6)+kh2k2
L∞(H)Zt
0hkϕh−ϕk2
L6+kzh−zk2
L6ids
+khk
1
2
L2(Q)Zt
0hkϕh−ϕk2
L4+kzh−zk2
L4ids+khk4
L2(Q)+khk
5
2
L2(Q).
Again, we recall that from (iii) in Theorem 2.6 we know
kϕh−ϕkL2(V)+kσh−σkL2(V)+kzh−zkL2(V)≤CRkhkL2(Q),
and that V ֒→L4, L6. Finally, we obtain:
kΦhk2
H+kλhk2
H+kµhk2
H+Zt
0k∇Φhk2
H+k∇λhk2
H+kε(∂twh)k2
H+k∇µhk2
Hds
≤CRZt
0hkΦhk2
H+kλhk2
H+kµhk2
Hids+Zt
0Zs
0
kε(∂twh)k2
Hdτds
+khk4
L2(Q)+khk2
L2(Q)kh1k2
L2(V)+kh2k2
L∞(H)+khk
5
2
L2(Q).
By means of the Gronwall inequality, we have:
kΦhk2
H+kλhk2
H+kµhk2
H+Zt
0k∇Φhk2
H+k∇λhk2
H+kε(∂twh)k2
H+k∇µhk2
Hds
≤CRkhk4
L2(Q)+khk2
L2(Q)kh1k2
L2(V)+kh2k2
L∞(H)+khk
5
2
L2(Q),
from which (5.2) follows. Therefore, the proof of Theorem 5.1 is complete.
From this result, it follows that the reduced cost functional J is Fr´echet differentiable
over the set UR. Since Uad is a closed and convex subset of U, we can prove the following
result.
24
Corollary 5.2. Let χ∗∈ Uad be an optimal control for the control problem with the
associated state S(χ∗) = (ϕ∗, σ∗,u∗, z∗). Then, the following inequality is satisfied
α1ZT
0ZΩ
(ϕ∗−ϕQ)ξdxdt+α2ZΩ
(ϕ∗(T)−ϕΩ)ξ(T) dx+α3ZΩ
ξ(T) dx
+α4ZT
0ZΩ
(σ∗−σQ)ρdxdt+α5ZΩ
(σ∗(T)−σΩ)ρ(T) dx
+α6ZT
0ZΩ1
2γ′(ϕ∗)ε(u∗):ε(u∗)ξ+γ(ϕ∗)ε(u∗):ε(ω)dxdt
+α7ZT
0ZΩ
(z∗−zQ)ζdxdt+α8ZΩ
z∗(T) dx
+α9ZT
0ZΩ
χ·(χ−χ∗) dxdt≥0
(5.17)
for every χ∈ Uad, where (ξ, ρ, ω, ζ )is the unique solution to the linearized system in χ∗
for h=χ−χ∗.
Proof. First of all, we recall that the cost functional Jis well-defined over the space
hC0([0, T ]; H)×C0([0, T ]; H)×L2(0, T ;V)×C0([0, T ]; H)i
×hL2(0, T ;H)×L2(0, T ;H)i,
where it is also Fr´echet differentiable. Moreover, from Theorem 5.1, the solution operator
S:U → V is Fr´echet differentiable in UR. Since, from standard embedding results, Vis
continuously embedded in
C0([0, T ]; H)×C0([0, T ]; H)×L2(0, T ;V)×C0([0, T ]; H),
Sis as well Fr´echet differentiable in URif it is seen as a functional between Uand this
larger space. Thus, the reduced cost functional J is Fr´echet differentiable in URand, by
the chain rule,
DJ(χ∗)[χ−χ∗] = α1ZT
0ZΩ
(ϕ∗−ϕQ)ξdxdt+α2ZΩ
(ϕ∗(T)−ϕΩ)ξ(T) dx
+α3ZΩ
ξ(T) dx+α4ZT
0ZΩ
(σ∗−σQ)ρdxdt+α5ZΩ
(σ∗(T)−σΩ)ρ(T) dx
+α6ZT
0ZΩ1
2γ′(ϕ∗)ε(u∗):ε(u∗)ξ+γ(ϕ∗)ε(u∗):ε(ω)dxdt
+α7ZT
0ZΩ
(z∗−zQ)ζdxdt+α8ZΩ
ζ(T) dx+α9ZT
0ZΩ
χ·(χ−χ∗) dxdt,
where χ∗a the optimal control, χis any admissible control, and (ξ, ρ, ω, ζ) is the solution
to the system linearized in χ∗with h=χ−χ∗. From the optimality of χ∗and the
25
convexity of Uad, we obtain the trivial inequality
J(χ∗+t(χ−χ∗)) −J(χ∗)≥0
for every t∈(0,1). Dividing by tand passing to the limit as t→0+leads to
DJ(χ∗)[χ−χ∗]≥0.
The next step is simplifying the expression (5.17), removing the linearized variables. In
fact, even if the inequality just derived represents a first-order necessary condition for
optimality, it does not provide a practically efficient characterization. Specifically, for
every element χof the admissible control space Uad , it requires solving the corresponding
linearized system for h=χ−χ∗. This approach is computationally demanding, partic-
ularly in high-dimensional control spaces, highlighting the need for a reformulation. To
this end, we need to introduce the adjoint system.
6 The adjoint system and first-order necessary optimality
conditions
The adjoint system associated with an optimal control χ∗∈ Uad and its corresponding
solution to the state system (ϕ∗, σ∗,u∗, z∗) = S(χ∗) is given by
−∂tq−∆q = a1q + b1r + d1s + c1:ε(v)
+α1(ϕ∗−ϕQ) + α6
2γ′(ϕ∗)ε(u∗):ε(u∗),(6.1a)
−∂tr−∆r = a2q + b2r + α4(σ∗−σQ),(6.1b)
−div −Aε(∂tv) + B(ϕ∗, z∗)ε(v)=−div d2s + α6γ(ϕ∗)ε(u∗),(6.1c)
−∂ts−∆s = a3q + b3r + d3s + c2:ε(v) + α7(z∗−zQ),(6.1d)
coupled with the boundary conditions
∂νq = 0, ∂νr = −r,v= 0, ∂νs = 0,(6.2)
and with the final conditions
q(T) = α2(ϕ∗(T)−ϕΩ) + α3,r(T) = α5(σ∗(T)−σΩ),v(T) = 0,s(T) = α8.(6.3)
Proposition 6.1. Let χ∗be an optimal control with the associated solution to the state
system (ϕ∗, σ∗,u∗, z∗) = S(χ∗). The adjoint system (6.1)–(6.3)has a unique weak
solution (q,r,v,s) in the sense that
q∈H1(0, T ;H)∩L∞(0, T ;V)∩L2(0, T ;W),
r∈H1(0, T ;V′)∩L∞(0, T ;H)∩L2(0, T ;V),
v∈W1,∞(0, T ;V0),
s∈H1(0, T ;H)∩L∞(0, T ;V)∩L2(0, T ;W)
26
with
q(T) = α2(ϕ∗(T)−ϕΩ) + α3,r(T) = α5(σ∗(T)−σΩ),v(T) = 0,s(T) = α8,
such that
ZΩ
(−∂tqη+∇q· ∇η) dx=ZΩa1q + b1r + d1s + c1:ε(v)ηdx
+ZΩα1(ϕ∗−ϕQ) + α6
2γ′(ϕ∗)ε(u∗):ε(u∗)ηdx ,
(6.4a)
− h∂tr, ηiV+ZΩ
∇r· ∇ηdx+ZΓ
rηdHd−1
=ZΩa2q + b2r + α4(σ∗−σQ)ηdx ,
(6.4b)
ZΩ−Aε(∂tv) + B(ϕ∗, z∗)ε(v):ε(θ) dx=ZΩd2s + α6γ(ϕ∗)ε(u∗):ε(θ) dx , (6.4c)
ZΩ
(−∂tsη+∇s· ∇η) dx=ZΩa3q + b3r + d3s + c2:ε(v) + α7(z∗−zQ)ηdx , (6.4d)
a.e. in (0, T ), for every η∈Vand θ∈V0.
The well-posedness of the adjoint system can be proved as in Proposition 4.2, thus we
will omit the proof.
Theorem 6.2 (First-order necessary conditions of optimality).Let χ∗∈ Uad be an
optimal control for the control problem with the associated state S(χ∗) = (ϕ∗, σ∗,u∗, z∗).
Then, the following inequality is satisfied
−ZT
0ZΩ
ϕ∗1−ϕ∗
N(χ1−χ∗
1)q dxdt
+ZT
0ZΩ
S(ϕ∗, z∗)(χ2−χ∗
2)r dxdt+α9ZT
0ZΩ
χ·(χ−χ∗) dxdt≥0
(6.5)
for every χ∈ Uad, where qand rare respectively the solution’s first and third components
to the adjoint system associated with χ∗.
Proof. We test the equations of the linearized system with the solution of the adjoint
system and subtract the equations of the adjoint system tested with the solution of the
linearized system. Then, we sum up all the equations we have obtained. Explicitly, we
test (4.6a) with q, (4.6b) with r, (4.6c) with vand (4.6d) with s. In the same way, we
test (6.4a) with ξ, (6.4b) with ρ, (6.4c) with ωand (6.4d) with ζ. Some terms cancel
out and we have:
d
dtZΩξq + ρr + Aε(ω):ε(v) + ζsdx
=ZΩ
(a4h1q + b4h2r) dx−α1ZΩ
(ϕ∗−ϕQ)ξdx−α4ZΩ
(σ∗−σQ)ρdx
−α6ZΩ1
2γ′(ϕ∗)ε(u∗):ε(u∗)ξ+γ(ϕ∗)ε(u∗):ε(ω)dx−α7ZΩ
(z∗−zQ)ζdx .
27
Integrating in time over the interval (0, T ), exploiting the initial and final conditions,
and moving some terms to the left-hand side, we finally have
α2ZΩ
(ϕ∗(T)−ϕΩ)ξ(T) dx+α3ZΩ
ξ(T) dx+α5ZΩ
(σ∗(T)−σΩ)ρ(T) dx
+α8ZΩ
ζ(T) dx+α1ZT
0ZΩ
(ϕ∗−ϕQ)ξdxdt+α4ZT
0ZΩ
(σ∗−σQ)ρdxdt
+α6ZT
0ZΩ1
2γ′(ϕ∗)ε(u∗):ε(u∗)ξ+γ(ϕ∗)ε(u∗):ε(ω)dxdt
+α7ZT
0ZΩ
(z∗−zQ)ζdxdt=ZT
0ZΩ
(a4h1q + b4h2r) dxdt.
Combining this equality with the inequality stated in Corollary 5.2, we obtain the thesis.
Acknowledgments
G.C. wishes to acknowledge the partial financial support received from International
Research Laboratory LYSM, IRL 2019 CNRS/INdAM. G.C. is a member of GNAMPA
(Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni) of
INdAM (Istituto Nazionale di Alta Matematica).
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