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ESAIM: COCV 30 (2024) 13 ESAIM: Control, Optimisation and Calculus of Variations
https://doi.org/10.1051/cocv/2023084 www.esaim-cocv.org
SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE
OPTIMAL CONTROL OF A PHASE FIELD TUMOR GROWTH
MODEL WITH LOGARITHMIC POTENTIAL
J¨
urgen Sprekels1and Fredi Tr¨
oltzsch2,*
Abstract. This paper treats a distributed optimal control problem for a tumor growth model of
viscous Cahn–Hilliard type. The evolution of the tumor fraction is governed by a thermodynamic force
induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable
term like the L1–norm in order to enhance the occurrence of sparsity effects in the optimal controls, i.e.,
of subdomains of the space-time cylinder where the controls vanish. In the context of cancer therapies,
sparsity is very important in order that the patient is not exposed to unnecessary intensive medical
treatment. In this work, we focus on the derivation of second-order sufficient optimality conditions for
the optimal control problem. While in previous works on the system under investigation such conditions
have been established for the case without sparsity, the case with sparsity has not been treated before.
Mathematics Subject Classification. 35K57, 37N25, 49J50, 49J52, 49K20, 49K40.
Received June 22, 2023. Accepted November 15, 2023.
1. Introduction
Let α > 0, β > 0, χ > 0, and let Ω ⊂R3denote some open and bounded domain having a smooth boundary
Γ = ∂Ω and the unit outward normal nwith associated outward normal derivative ∂n. Moreover, we fix some
final time T > 0 and introduce for every t∈(0, T ) the sets Qt:= Ω ×(0, t) and Qt:= Ω ×(t, T ). We also set,
for convenience, Q:= QTand Σ := Γ ×(0, T ). We then consider the following optimal control problem:
(CP) Minimize the cost functional
J((µ, φ, σ),u) := b1
2ZZQ
|φ−bφQ|2+b2
2ZΩ
|φ(T)−bφΩ|2+b3
2ZZQ
(|u1|2+|u2|2) + κg(u)
=: J1((µ, φ, σ),u) + κg(u) (1.1)
Keywords and phrases: Optimal control, tumor growth models, logarithmic potentials, second-order sufficient optimality
conditions, sparsity.
1Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, D-10117 Berlin, Germany.
2Institut f¨ur Mathematik, Technische Universit¨at Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany.
*Corresponding author: troeltzsch@math.tu-berlin.de
©
The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
2J. SPREKELS AND F. TR ¨
OLTZSCH
subject to the state system
α∂tµ+∂tφ−∆µ=P(φ)(σ+χ(1 −φ)−µ)−Hu1in Q, (1.2)
β∂tφ−∆φ+F′
1(φ) + F′
2(φ) = µ+χ σ in Q, (1.3)
∂tσ−∆σ=−χ∆φ−P(φ)(σ+χ(1 −φ)−µ) + u2in Q, (1.4)
∂nµ=∂nφ=∂nσ= 0 on Σ,(1.5)
µ(0) = µ0, φ(0) = φ0, σ(0) = σ0in Ω,(1.6)
and to the control constraint
u= (u1, u2)∈ Uad.(1.7)
Here, g:L2(Q)2→Ris a nonnegative, continuous and convex functional, which is typically of the sparsity-
enhancing form
g((u1, u2)) = ZZQ
(|u1|+|u2|).(1.8)
The constants b1and b2are nonnegative, while b3and κare positive; bφQand bφΩare given target functions.
The term g(u) accounts for possible sparsity effects. Moreover, the set of admissible controls Uad is a nonempty,
closed and convex subset of the control space
U:= L∞(Q)2.(1.9)
We remark at this place that L∞(Q)2is the space in which the Fr´echet derivative of the control-to-state mapping
will turn out to exist. In contrast to this, the space L2(Q)2seems to be more natural in the discussion of second-
order sufficient optimality conditions. This phenomenon is part of the well-known two-norm discrepancy. To
overcome this difficulty, we need to work with both these spaces simultaneously.
The state system (1.2)–(1.6) constitutes a simplified and relaxed version of the four-species thermodynami-
cally consistent model for tumor growth originally proposed by Hawkins-Daruud et al. in [38] that additionally
includes the chemotaxis-like terms χσ in (1.3) and −χ∆φin (1.4). Let us briefly review the role of the occur-
ring symbols. The primary (state) variables φ, µ, σ denote the tumor fraction, the associated chemical potential,
and the nutrient concentration, respectively. Furthermore, the additional term α∂tµcorresponds to a parabolic
regularization of equation (1.2), while β∂tφis the viscosity contribution to the Cahn–Hilliard equation. The
nonlinearity Pdenotes a proliferation function, whereas the positive constant χrepresents the chemotactic
sensitivity and provides the system with a cross-diffusion coupling.
The evolution of the tumor fraction is mainly governed by the nonlinearities F1and F2whose derivatives
occur in (1.3). Here, F2is smooth, typically a concave function. As far as F1is concerned, we admit in this
paper functions of logarithmic type such as
F1,log(r) =
(1 + r) ln(1 + r) + (1 −r) ln(1 −r) for r∈(−1,1)
2 ln(2) for r∈ {−1,1}.
+∞for r∈ [−1,1]
(1.10)
We assume that F=F1+F2is a double-well potential. This is actually the case if F2(r) = k(1 −r2) with a
sufficiently large k > 0. Note also that F′
1,log(r) becomes unbounded as r↘ −1 and r↗1.
The control variable u2occurring in (1.4) can model either an external medication or some nutrient supply,
while u1, which occurs in the phase equation (1.2), models the application of a cytotoxic drug to the system.
SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE OPTIMAL CONTROL 3
Usually, u1is multiplied by a truncation function H(φ) in order to have the action only in the spatial region
where the tumor cells are located. Typically, one assumes that H(−1) = 0,H(1) = 1, and H(φ) is in between
if −1<φ<1; see [29,35,41,42] for some insights on possible choices of H. Also in [15,17,54], this kind
of nonlinear coupling between u1and φhas been admitted. For our analysis, we have decided to make a
simplification because the inclusion of the nonlinearity H(φ) leads in the Lipschitz estimates below for the first
and second Fr´echet derivatives of the control-to-state operator to numerous additional terms causing tedious
estimations that go beyond the scope of this paper without bringing new insights. We have thus chosen to
simplify the original system somewhat by assuming that H=H(x, t) is a bounded nonnegative function that
does not depend on φ. We stress the fact that this simplification does not have any impact on the validity of
the results from [17] to be used below.
As far as well-posedness is concerned, the above model was already investigated in the case χ= 0 in [6–9],
and in [25] with α=β=χ= 0. There the authors also pointed out how the relaxation parameters αand βcan
be set to zero, by providing the proper framework in which a limit system can be identified and uniquely solved.
We also note that in [13] a version has been studied in which the Laplacian in the equations (1.2)–(1.4) has
been replaced by fractional powers of a more general class of selfadjoint operators having compact resolvents.
A model which is similar to the one studied in this note was the subject of [15,54].
For some nonlocal variations of the above model we refer to [27,28,47]. Moreover, in order to better
emulate in-vivo tumor growth, it is possible to include in similar models the effects generated by the fluid flow
development by postulating a Darcy law or a Stokes–Brinkman law. In this direction, we refer to [11,24,27,29–
33,35], and we also mention [36], where elastic effects are included. For further models, discussing the case
of multispecies, we refer the reader to [21,27]. The investigation of associated optimal control problems also
presents a wide number of results of which we mention[10,13,15,22,23,28,34,37,42,48–52,54,56].
Sparsity in the optimal control theory of partial differential equations is a very active field of research. The
use of sparsity-enhancing functionals goes back to inverse problems and image processing. Soon after the seminal
paper [57], many results were published. We mention only very few of them with closer relation to our paper, in
particular [1,39,40], on directional sparsity, and [5] on a general theorem for second-order conditions; moreover,
we refer to some new trends in the investigation of sparsity, namely, infinite horizon sparse optimal control (see,
e.g., [43,44]), and fractional order optimal control (cf. [46], [45]). These papers concentrated on first-order
optimality conditions for sparse optimal controls of single elliptic and parabolic equations. In [3,4], first- and
second-order optimality conditions have been discussed in the context of sparsity for the (semilinear) system of
FitzHugh–Nagumo equations. Moreover, we refer to the measure control of the Navier–Stokes system studied
in [2].
The optimal control problem (CP) has recently been investigated in [17] for the case of logarithmic potentials
F1and without sparsity terms, where second-order sufficient optimality conditions have been derived using the
τ–critical cone and the splitting technique as described in the textbook [58]. In [54] and [18], sparsity terms
have been incorporated, where in the latter paper not only logarithmic nonlinearities but also nondifferentiable
double obstacle potentials have been admitted. However, second-order sufficient optimality conditions have not
been derived.
The derivation of meaningful second-order conditions for locally optimal controls of (CP) in the logarithmic
case with sparsity term is the main object of this paper. In particular, we aim at constructing suitable critical
cones which are as small as possible. In our approach, we follow the recent work [55] on the sparse optimal
control of Allen–Cahn systems, which was based on ideas developed in [4].
The paper is organized as follows. In the next section, we list and discuss our assumptions, and we collect
known results from [18] concerning the properties of the state system (1.2)–(1.6) and of the control-to-state oper-
ator. In Section 3, we study the optimal control problem. We derive first-order necessary optimality conditions
and results concerning the full sparsity of local minimizers, and we establish second-order sufficient optimality
conditions for the optimal control problem (CP). In an appendix, we prove auxiliary results that are needed
for the main theorem on second-order sufficient conditions.
Prior to this, let us fix some notation. For any Banach space X, we denote by ∥·∥Xthe norm in the space X,
by X∗its dual space, and by ⟨ ·,· ⟩Xthe duality pairing between X∗and X. For any 1 ≤p≤ ∞ and k≥0, we
4J. SPREKELS AND F. TR ¨
OLTZSCH
denote the standard Lebesgue and Sobolev spaces on Ω by Lp(Ω) and Wk,p(Ω), and the corresponding norms
by ∥·∥Lp(Ω) =∥·∥pand ∥·∥Wk,p(Ω) , respectively. For p= 2, they become Hilbert spaces, and we employ
the standard notation Hk(Ω) := Wk,2(Ω). As usual, for Banach spaces Xand Ythat are both continuously
embedded in some topological vector space Z, we introduce the linear space X∩Ywhich becomes a Banach
space when equipped with its natural norm ∥v∥X∩Y:= ∥v∥X+∥v∥Y, for v∈X∩Y. Moreover, we recall the
definition (1.9) of the control space Uand introduce the spaces
H:= L2(Ω), V := H1(Ω), W0:= {v∈H2(Ω) : ∂nv= 0 on Γ}.(1.11)
Furthermore, by ( ·,·) and ∥·∥we denote the standard inner product and related norm in H, and, for
simplicity, we also set ⟨ ·,· ⟩ := ⟨ ·,· ⟩V.
Throughout the paper, we make repeated use of H¨older’s inequality, of the elementary Young inequality
ab ≤δ|a|2+1
4δ|b|2∀a, b ∈R,∀δ > 0,(1.12)
as well as of the continuity of the embeddings H1(Ω) ⊂Lp(Ω) for 1 ≤p≤6 and H2(Ω) ⊂C0(Ω).
We close this section by introducing a convention concerning the constants used in estimates within this
paper: we denote by Cany positive constant that depends only on the given data occurring in the state system
and in the cost functional, as well as on a constant that bounds the (L∞(Q)×L∞(Q))–norms of the elements of
Uad. The actual value of such generic constants Cmay change from formula to formula or even within formulas.
Finally, the notation Cδindicates a positive constant that additionally depends on the quantity δ.
2. General setting and properties of the control-to-state
operator
In this section, we introduce the general setting of our control problem and state some results on the state
system (1.2)–(1.6) and the control-to-state operator that in its present form have been established in [17,18].
We make the following assumptions on the data of the system.
(A1) α, β, χ are positive constants.
(A2) F=F1+F2, where F2∈C5(R) has a Lipschitz continuous derivative F′
2, and where F1:R→[0,+∞]
is convex and lower semicontinuous and satisfies F1(0) = 0, F1|(−1,1) ∈C5(−1,1), as well as
lim
r↘−1F′
1(r) = −∞ and lim
r↗1F′
1(r)=+∞.(2.1)
(A3) P∈C3(R)∩W3,∞(R) and H∈L∞(Q) are nonnegative and bounded.
(A4) The initial data satisfy µ0, σ0∈H1(Ω) ∩L∞(Ω), φ0∈W0, as well as
−1<min
x∈Ω
φ0(x)≤max
x∈Ω
φ0(x)<1.(2.2)
(A5) With fixed given constants ui, uisatisfying ui< ui,i= 1,2, we have
Uad ={u= (u1, u2)∈ U :ui≤ui≤uia.e. in Qfor i= 1,2}.(2.3)
(A6) R > 0 is a constant such that Uad ⊂ UR:= {u∈ U :∥u∥U< R}.
SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE OPTIMAL CONTROL 5
Remark 2.1. Observe that (A3) implies that the functions P, P ′, P ′′ are Lipschitz continuous on R. Let us
also note that F1=F1,log satisfies (A2). Moreover, (2.2) implies that initially there are no pure phases. Finally,
(A6) just fixes an open and bounded subset of Uthat contains Uad.
The following result is a consequence of [18], Theorem 2.3.
Theorem 2.2. Suppose that the conditions (A1)–(A6) are fulfilled. Then the state system (1.2)–(1.6)has for
every u= (u1, u2)∈ URa unique strong solution (µ, φ, σ)with the regularity
µ∈H1(0, T ;H)∩C0([0, T ]; V)∩L2(0, T ;W0)∩L∞(Q),(2.4)
φ∈W1,∞(0, T ;H)∩H1(0, T ;V)∩L∞(0, T ;W0)∩C0(Q),(2.5)
σ∈H1(0, T ;H)∩C0([0, T ]; V)∩L2(0, T ;W0)∩L∞(Q).(2.6)
Moreover, there is a constant K1>0, which depends on Ω, T, R, α, β and the data of the system, but not on the
choice of u∈ UR, such that
∥µ∥H1(0,T ;H)∩C0([0,T ];V)∩L2(0,T ;W0)∩L∞(Q)
+∥φ∥W1,∞(0,T ;H)∩H1(0,T ;V)∩L∞(0,T ;W0)∩C0(Q)
+∥σ∥H1(0,T ;H)∩C0([0,T ];V)∩L2(0,T ;W0)∩L∞(Q)≤K1.(2.7)
Furthermore, there are constants r∗, r∗, which depend on Ω, T, R, α, β and the data of the system, but not on
the choice of u∈ UR, such that
−1< r∗≤φ(x, t)≤r∗<1for all (x, t)∈Q. (2.8)
Also, there is some constant K2>0having the same dependencies as K1such that
max
i=0,1,2,3
P(i)(φ)
L∞(Q)+ max
i=0,1,2,3,4,5max
j=1,2
F(i)
j(φ)
L∞(Q)≤K2.(2.9)
Finally, if ui∈ URare given controls and (µi, φi, σi)the corresponding solutions to (1.2)–(1.6), for i= 1,2,
then, with a constant K3>0having the same dependencies as K1,
∥µ1−µ2∥H1(0,T ;H)∩C0([0,T ];V)∩L2(0,T ;W0)+∥φ1−φ2∥H1(0,T ;H)∩C0([0,T ];V)∩L2(0,T ;W0)
+∥σ1−σ2∥H1(0,T ;H)∩C0([0,T ];V)∩L2(0,T ;W0)≤K3∥u1−u2∥L2(Q)2.(2.10)
Remark 2.3. Condition (2.8), known as the separation property, is especially important for the case of singular
potentials such as F1=F1,log, since it guarantees that the phase variable φalways stays away from the critical
values −1,1. The singularity of F′
1is therefore no longer an obstacle for the analysis, as the values of φrange
in some interval in which F′
1is smooth.
Owing to Theorem 2.2, the control-to-state operator
S:u= (u1, u2)7→ (µ, φ, σ)
is well defined as a mapping between U=L∞(Q)2and the Banach space specified by the regularity results (2.4)–
(2.6). We now discuss its differentiability properties. For this purpose, some functional analytic preparations
6J. SPREKELS AND F. TR ¨
OLTZSCH
are in order. We first define the linear spaces
X:= X×e
X×X, where
X:= H1(0, T ;H)∩C0([0, T ]; V)∩L2(0, T ;W0)∩L∞(Q),
e
X:= W1,∞(0, T ;H)∩H1(0, T ;V)∩L∞(0, T ;W0)∩C0(Q),(2.11)
which are Banach spaces when endowed with their natural norms. Next, we introduce the linear space
Y:= (µ, φ, σ)∈ X :α∂tµ+∂tφ−∆µ∈L∞(Q), β∂tφ−∆φ−µ∈L∞(Q),
∂tσ−∆σ+χ∆φ∈L∞(Q),(2.12)
which becomes a Banach space when endowed with the norm
∥(µ, φ, σ)∥Y:= ∥(µ, φ, σ)∥X+∥α∂tµ+∂tφ−∆µ∥L∞(Q)+∥β∂tφ−∆φ−µ∥L∞(Q)
+∥∂tσ−∆σ+χ∆φ∥L∞(Q).(2.13)
Finally, we put
Z:= H1(0, T ;H)∩C0([0, T ]; V)∩L2(0, T ;W0),(2.14)
Z:= Z×Z×Z. (2.15)
For fixed (µ∗, φ∗, σ∗), we first discuss an auxiliary result for the linear initial-boundary value problem
α∂tµ+∂tφ−∆µ=λ1[P(φ∗)(σ−χφ −µ) + P′(φ∗)(σ∗+χ(1 −φ∗)−µ∗)φ]
−λ2Hh1+λ3f1in Q, (2.16)
β∂tφ−∆φ−µ=λ1[χ σ −F′′(φ∗)φ] + λ3f2in Q, (2.17)
∂tσ−∆σ+χ∆φ=λ1[−P(φ∗)(σ−χφ −µ)−P′(φ∗)(σ∗+χ(1 −φ∗)−µ∗)φ]
+λ2h2+λ3f3in Q, (2.18)
∂nµ=∂nφ=∂nσ= 0 on Σ,(2.19)
µ(0) = λ4µ0, φ(0) = λ4φ0, σ(0) = λ4σ0in Ω,(2.20)
which for λ1=λ2= 1 and λ3=λ4= 0 coincides with the linearization of the state equation at ((µ∗, φ∗, σ∗),
(u∗
1, u∗
2)). We remark at this place that the functions h1, h2play the role of control increments, while the role
of (f1, f2, f3) will be become clear during the proof of a number of Lipschitz properties (see, e.g., (2.37)–(2.39),
(2.42)–(2.43)). We have the following result.
Lemma 2.4. Suppose that λ1, λ2, λ3, λ4∈ {0,1}are given and that the assumptions (A1)–(A6) are fulfilled.
Moreover, let u∗= (u∗
1, u∗
2)∈ URbe given and (µ∗, φ∗, σ∗) = S(u∗). Then (2.16)–(2.20)has for every h=
(h1, h2)∈L2(Q)2and (f1, f2, f3)∈L2(Q)3a unique solution (µ, φ, σ )∈Z×e
X×Z. Moreover, the linear
mapping
((h1, h2),(f1, f2, f3)) 7→ (µ, φ, σ) (2.21)
is continuous from L2(Q)2×L2(Q)3into Z×e
X×Z. Moreover, if h∈L∞(Q)2and (f1, f2, f3)∈L∞(Q)3, in
addition, then it holds (µ, φ, σ)∈ Y, and the mapping (2.21)is continuous from L∞(Q)2×L∞(Q)3into Y.
SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE OPTIMAL CONTROL 7
Proof. The existence result and the continuity of the mapping (2.21) between the spaces L∞(Q)2×L∞(Q)3
and Ydirectly follow from the statement of [17], Lemma 4.1 and Remark 4.2. Moreover, from the estimates
(4.36)–(4.38) and (4.43) in [17] we can conclude that the mapping (2.21) is also continuous between the spaces
L2(Q)2×L2(Q)3and Z×e
X×Z.
Now let u∗= (u∗
1, u∗
2)∈ URbe arbitrary and (µ∗, φ∗, σ∗) = S(u∗). Then, according to [17], Theorem 4.4, the
control-to-state operator Sis twice continuously Fr´echet differentiable at u∗as a mapping from Uinto Y.
Moreover, for every h= (h1, h2)∈ U , the first Fr´echet derivative S′(u∗)∈ L(U,Y) of Sat u∗is given by the
identity S′(u∗)[h]=(ηh, ξh, θh), where (ηh, ξh, θh)∈ Y is the unique solution to the linearization of the state
system given by the initial-boundary value problem (2.16)–(2.20) with λ1=λ2= 1 and λ3=λ4= 0.
Remark 2.5. Observe that, in view of the continuity of the embedding Y ⊂ Z×e
X×Z, the operator S′(u∗)∈
L(U,Y) also belongs to the space L(U, Z ×e
X×Z) and, owing to the density of Uin L2(Q)2, can be extended
continuously to an element of L(L2(Q)2, Z ×e
X×Z) without changing its operator norm. Denoting the extended
operator still by S′(u∗), we see that the identity S′(u∗)[h]=(ηh, ξh, θh) is also valid for every h∈L2(Q)2,
only that (ηh, ξh, θh)∈Z×e
X×Z, in general. In addition, it also follows from the proof of [17], Lemma 4.1
that there is a constant K4>0, which depends only on Rand the data, such that
∥S′(u)[h]∥Z×
e
X×Z≤K4∥h∥L2(Q)2for all u∈ URand every h∈L2(Q)2.(2.22)
Next, we show a Lipschitz property for the extended operator S′.
Lemma 2.6. The mapping S′:U → L(L2(Q)2, Z ×e
X×Z),u7→ S′(u), is Lipschitz continuous in the following
sense: there is a constant K5>0, which depends only on Rand the data, such that, for all controls u1,u2∈ UR
and all increments h∈L2(Q)2,
∥(S′(u1)− S′(u2)) [h]∥Z≤K5∥u1−u2∥L2(Q)2∥h∥L2(Q)2.(2.23)
Proof. We put (µi, φi, σi) := S(ui), (ηi, ξi, θi) := S′(ui)[h], i= 1,2, as well as
u:= u1−u2, µ := µ1−µ2, φ := φ1−φ2, σ := σ1−σ2,
η:= η1−η2, ξ := ξ1−ξ2, θ := θ1−θ2.
Then it follows from (2.10) in Theorem 2.2 that
∥(µ, φ, σ)∥Z≤K3∥u∥L2(Q)2.(2.24)
Moreover, (η, ξ, θ) solves the problem
α∂tη+∂tξ−∆η=P(φ1)(θ−χξ −η) + P′(φ1)(σ1+χ(1 −φ1)−µ1)ξ+f1in Q, (2.25)
β∂tξ−∆ξ=χθ −F′′ (φ1)ξ+f2in Q, (2.26)
∂tθ−∆θ+χ∆ξ=−P(φ1)(θ−χξ −η)−P′(φ1)(σ1+χ(1 −φ1)−µ1)ξ+f3in Q, (2.27)
∂nη=∂nξ=∂nθ= 0 on Σ,(2.28)
η(0) = ξ(0) = θ(0) = 0 in Ω,(2.29)
which is of the form (2.16)–(2.20) with λ1=λ3= 1 and λ2=λ4= 0, and where
f1:= −f3:= ((P(φ1)−P(φ2))(θ2−χξ2−η2) + P′(φ1)(σ−χφ −µ)ξ2
8J. SPREKELS AND F. TR ¨
OLTZSCH
+ (P′(φ1)−P′(φ2))(σ2+χ(1 −φ2)−µ2)ξ2,(2.30)
f2:= −(F′′(φ1)−F′′ (φ2))ξ2.(2.31)
We therefore conclude from Lemma 2.4 that
∥(η, ξ, θ)∥Z≤C∥f1∥L2(Q)+∥f2∥L2(Q).
Hence, the proof will be finished once we can show that
∥f1∥L2(Q)+∥f2∥L2(Q)≤C∥u∥L2(Q)2∥h∥L2(Q)2.(2.32)
To this end, we first use the mean value theorem, (2.9), H¨older’s inequality, the continuity of the embedding
V⊂L4(Ω), as well as (2.10) and (2.24), to find that
∥f2∥2
L2(Q)≤CZZQ
|φ|2|ξ2|2≤CZT
0
∥φ∥2
4∥ξ2∥2
4ds ≤C∥φ∥2
C0([0,T ];V)∥ξ2∥2
C0([0,T ];V)
≤C∥φ∥2
Z∥S′(u2)[h]∥2
Z≤C∥u∥2
L2(Q)2∥h∥2
L2(Q)2.(2.33)
Here, we have for convenience omitted the argument sin the third integral. We will do this repeatedly in the
following. For the three summands on the right-hand side of (2.30), which we denote by A1, A2, A3, in this
order, we obtain by similar reasoning the estimates
ZZQ
|A1|2≤CZZQ
|φ|2|θ2−χξ2−η2|2≤CZT
0
∥φ∥2
4∥η2∥2
4+∥ξ2∥2
4+∥θ2∥2
4ds
≤C∥u∥2
L2(Q)2∥h∥2
L2(Q)2,(2.34)
ZZQ
|A2|2≤CZT
0
|ξ2|2|µ|2+|φ|2+|σ|2ds ≤CZT
0
∥ξ2∥2
4∥µ∥2
4+∥φ∥2
4+∥σ∥2
4ds
≤C∥u∥2
L2(Q)2∥h∥2
L2(Q)2,(2.35)
ZZQ
|A3|2≤C∥σ2∥2
L∞(Q)+∥φ2∥2
L∞(Q)+∥µ2∥2
L∞(Q)+ 1ZZQ
|φ|2|ξ2|2
≤C∥u∥2
L2(Q)2∥h∥2
L2(Q)2,(2.36)
where in the last estimate we also used (2.7) and (2.33). With this, the assertion is proved.
Next, we turn our interest to the second Fr´echet derivative S′′(u∗) of Sat u∗. Let h= (h1, h2)∈ U and
k= (k1, k2)∈ U. Then, (ηh, ξh, θh) := S′(u∗)[h] and (ηk, ξ k, θk) := S′(u∗)[k] both belong to Yand, by virtue
of [17], Theorem 4.6, (ν, ψ, ρ) = S′′ (u∗)[h,k]∈ Y is the unique solution to the bilinearization of the state
system at ((µ∗, φ∗, σ∗),(u∗
1, u∗
2)), which is given by the linear initial-boundary value problem
α∂tν+∂tψ−∆ν=P(φ∗)(ρ−χψ −ν) + P′(φ∗)(σ∗+χ(1 −φ∗)−µ∗)ψ+f1in Q, (2.37)
β∂tψ−∆ψ−ν=χρ −F′′ (φ∗)ψ+f2in Q, (2.38)
∂tρ−∆ρ+χ∆ψ=−P(φ∗)(ρ−χψ −ν)−P′(φ∗)(σ∗+χ(1 −φ∗)−µ∗)ψ+f3in Q, (2.39)
∂nν=∂nψ=∂nρ= 0 on Σ,(2.40)
SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE OPTIMAL CONTROL 9
ν(0) = ψ(0) = ρ(0) = 0 in Ω,(2.41)
and which is again of the form (2.16)–(2.20) with λ1=λ3= 1 and λ2=λ4= 0, where
f1:= −f3:= P′(φ∗)ξk(θh−χξh−ηh) + ξh(θk−χξk−ηk)
+P′′(φ∗)ξkξh(σ∗+χ(1 −φ∗)−µ∗),(2.42)
f2:= −F(3)(φ∗)ξhξk.(2.43)
Now assume that h,k∈L2(Q)2are given. Then the expressions (ηh, ξh, θh) := S′(u∗)[h] and (ηk, ξk, θk) :=
S′(u∗)[k] are well-defined elements of the space Z×e
X×Z, where S′(u∗) now denotes the extension of the
Fr´echet derivative introduced in Remark 2.5. We now claim that there is a constant b
C > 0 that depends only
on Rand the data, such that
∥f1∥L2(Q)+∥f2∥L2(Q)≤b
C∥h∥L2(Q)2∥k∥L2(Q)2.(2.44)
Indeed, arguing as in the derivation of the estimates (2.33)–(2.36), we obtain
∥f1∥2
L2(Q)≤CZT
0
∥ξk∥2
4∥θh∥2
4+∥ξh∥2
4+∥ηh∥2
4ds
+CZT
0
∥ξh∥2
4∥θk∥2
4+∥ξk∥2
4+∥ηk∥2
4ds
+C∥σ∗∥2
L∞(Q)+∥φ∗∥2
L∞(Q)+∥µ∗∥2
L∞(Q)+ 1ZT
0
∥ξk∥2
4∥ξh∥2
4ds
≤C∥S′(u∗)[h]∥2
C0([0,T ];V)∥S′(u∗)[k]∥2
C0([0,T ];V)≤C∥h∥2
L2(Q)2∥k∥2
L2(Q)2,
∥f2∥2
L2(Q)≤CZT
0
∥ξh∥2
4∥ξk∥2
4ds ≤C∥h∥2
L2(Q)2∥k∥2
L2(Q)2,
which proves the claim. At this point, we can conclude from Lemma 2.4 that the system (2.37)–(2.41) has for
every h,k∈L2(Q)2a unique solution (ν, ψ, ρ)∈Z×e
X×Z. Moreover, we have, with a constant K6>0 that
depends only on Rand the data,
∥(ν, ψ, ρ)∥Z×
e
X×Z≤K6∥h∥L2(Q)2∥k∥L2(Q)2∀h,k∈L2(Q)2.(2.45)
Remark 2.7. Similarly as in Remark 2.5, the operator S′′ (u∗)∈ L(U,L(U,Y)) can be extended continuously
to an element of L(L2(Q)2,L(L2(Q)2, Z ×e
X×Z)) without changing its operator norm. Denoting the extended
operator still by S′′(u∗), we see that the identity S′′(u∗)[h,k] = (ν, ψ , ρ) is also valid for every h,k∈L2(Q)2,
only that (ν, ψ, ρ)∈Z×e
X×Z, in general. In addition, we have
∥S′′ (u∗)[h,k]∥Z×
e
X×Z≤K6∥h∥L2(Q)2∥k∥L2(Q)2for all u∗∈ URand h,k∈L2(Q)2.(2.46)
We conclude our preparatory work by showing a Lipschitz property for the extended operator S′′ that
resembles (2.23).
10 J. SPREKELS AND F. TR ¨
OLTZSCH
Lemma 2.8. The mapping S′′ :U → L(L2(Q)2,L(L2(Q)2, Z ×e
X×Z)),u7→ S′′ (u), is Lipschitz continuous
in the following sense: there is a constant K7>0, which depends only on Rand the data, such that, for all
controls u1,u2∈ URand all increments h,k∈L2(Q)2,
∥(S′′(u1)− S ′′(u2)) [h,k]∥Z≤K7∥u1−u2∥L2(Q)2∥h∥L2(Q)2∥k∥L2(Q)2.(2.47)
Proof. We put (µi, φi, σi) := S(ui), (ηh
i, ξh
i, θh
i) := S′(ui)[h], (ηk
i, ξk
i, θk
i) := S′(ui)[k], (νi, ψi, ρi) :=
S′′(ui)[h,k], for i= 1,2, as well as
u:= u1−u2, µ := µ1−µ2, φ := φ1−φ2, σ := σ1−σ2,
ηh:= ηh
1−ηh
2, ξh:= ξh
1−ξh
2, θh:= θh
1−θh
2,
ηk:= ηk
1−ηk
2, ξk:= ξk
1−ξk
2, θk:= θk
1−θk
2,
ν:= ν1−ν2, ψ := ψ1−ψ2, ρ := ρ1−ρ2.
Then it follows from (2.10) and (2.23) that
∥(µ, φ, σ)∥Z≤C∥u∥L2(Q)2,∥(ηh, ξh, θh)∥Z≤C∥u∥L2(Q)2∥h∥L2(Q)2,
∥(ηk, ξk, θk)∥Z≤C∥u∥L2(Q)2∥k∥L2(Q)2.(2.48)
We also recall the estimates (2.22) and (2.46). Moreover, (ν, ψ, ρ) solves the problem
α∂tν+∂tψ−∆ν=P(φ1)(ρ−χψ −ν) + P′(φ1)(σ1+χ(1 −φ1)−µ1)ψ+g1in Q, (2.49)
β∂tψ−∆ψ=χρ −F′′ (φ1)ψ+g2in Q, (2.50)
∂tρ−∆ρ+χ∆ψ=−P(φ1)(ρ−χψ −ν)−P′(φ1)(σ1+χ(1 −φ1)−µ1)ψ+g3in Q, (2.51)
∂nν=∂nψ=∂nρ= 0 on Σ,(2.52)
ν(0) = ψ(0) = ρ(0) = 0 in Ω,(2.53)
which is again of the form (2.16)–(2.20) with λ1=λ3= 1 and λ2=λ4= 0, where
g1:= −g3:= ((P(φ1)−P(φ2))(ρ2−χψ2−ν2) + P′(φ1)(σ−χφ −µ)ψ2
+ (P′(φ1)−P′(φ2))(σ2+χ(1 −φ2)−µ2)ψ2+ (P′(φ1)−P′(φ2)) ξk
1(θh
1−χξh
1−ηh
1)
+P′(φ2)ξk(θh
1−χξh
1−ηh
1) + P′(φ2)ξk
2(θh−χξh−ηh)
+ (P′(φ1)−P′(φ2)) ξh
1(θk
1−χξk
1−ηk
1) + P′(φ2)ξh(θk
1−χξk
1−ηk
1)
+P′(φ2)ξh
2(θk−χξk−ηk)+(P′′(φ1)−P′′(φ2)) ξh
1ξk
1(σ1+χ(1 −φ1)−µ1)
+P′′(φ2)ξkξh
1(σ1+χ(1 −φ1)−µ1) + P′′(φ2)ξk
2ξh(σ1+χ(1 −φ1)−µ1)
+P′′(φ2)ξk
2ξh
2(σ−χφ −µ) =:
13
X
i=1
Bi,(2.54)
g2:= −(F′′(φ1)−F′′ (φ2))ψ2−F(3)(φ1)−F(3)(φ2)ξh
1ξk
1−F(3)(φ2)ξhξk
1+ξh
2ξk,(2.55)
where Bidenotes the ith summand on the right-hand side of (2.54).
SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE OPTIMAL CONTROL 11
At this point, we infer from the proof of [17], Lemma 4.1 that the assertion follows once we can show that
13
X
i=1
∥Bi∥L2(Q)+∥g2∥L2(Q)≤C∥u∥L2(Q)2∥h∥L2(Q)2∥k∥L2(Q)2.
We only show the corresponding estimate for the terms B1, B4, B11 and leave the others to the interested
reader. In the following, we make use of the mean value theorem, H¨older’s inequality, the continuity of the
embeddings V⊂L6(Ω) ⊂L4(Ω), and the global estimates (2.7), (2.8), (2.22), and (2.46). We have
∥B1∥2
L2(Q)≤CZT
0
∥φ∥2
4∥ρ2∥2
4+∥ψ2∥2
4+∥ν2∥2
4ds
≤C∥φ∥2
C0([0,T ];V)∥S′′ (u2)[h,k]∥2
C0([0,T ];V)3≤C∥u∥2
L2(Q)2∥h∥2
L2(Q)2∥k∥2
L2(Q)2,
∥B4∥2
L2(Q)≤CZT
0
∥φ∥2
6∥ξk
1∥2
6∥θh
1∥2
6+∥ξh
1∥2
6+∥ηh
1∥2
6ds
≤C∥φ∥2
C0([0,T ];V)∥ξk
1∥2
Z∥S′(u1)[h]∥2
Z≤C∥u∥2
L2(Q)2∥h∥2
L2(Q)2∥k∥2
L2(Q)2,
as well as
∥B11∥2
L2(Q)≤C∥σ1∥2
L∞(Q)+∥φ1∥2
L∞(Q)+∥µ1∥2
L∞(Q)+ 1ZT
0
∥φ∥2
6∥ξh
1∥2
6∥ξk
1∥2
6ds
≤C∥φ∥2
C0([0,T ];V)∥ξh
1∥2
C0([0,T ];V)∥ξk
1∥2
C0([0,T ];V)≤C∥u∥2
L2(Q)2∥h∥2
L2(Q)2∥k∥2
L2(Q)2.
The assertion of the lemma is thus proved.
3. The optimal control problem
We now begin to investigate the control problem (CP). In addition to (A1)–(A6), we make the following
assumptions:
(C1) The constants b1, b2are nonnegative, while b3, κ are positive.
(C2) It holds bφΩ∈H1(Ω) and bφQ∈L2(Q).
(C3) g:L2(Q)2→Ris nonnegative, continuous and convex.
Observe that (C3) implies that gis weakly sequentially lower semicontinuous on L2(Q)2. Moreover, denoting
in the following by ∂the subdifferential mapping in L2(Q)2, it follows from standard convex analysis that
∂g is defined on the entire space L2(Q)2and is a maximal monotone operator. In addition, the mapping
((µ, φ, σ),u)7→ J((µ, φ, σ),u) defined by the cost functional (1.1) is obviously continuous and convex (and thus
weakly sequentially lower semicontinuous) on the space L2(Q)×C0([0, T ]; L2(Ω)) ×L2(Q)×L2(Q)2. From a
standard argument (which needs no repetition here) it then follows that the problem (CP) has a solution.
In the following, we often denote by u∗= (u∗
1, u∗
2)∈ Uad a local minimizer in the sense of Uand by
(µ∗, φ∗, σ∗) = S(u∗) the associated state. The corresponding adjoint state variables solve the adjoint system,
which is given by the backward-in-time parabolic system
−∂tp−β∂tq−∆q+χ∆r+F′′ (φ∗)q−P′(φ∗)(σ∗+χ(1 −φ∗)−µ∗)(p−r)
+χP (φ∗)(p−r) = b1(φ∗−bφQ) in Q, (3.1)
−α∂tp−∆p−q+P(φ∗)(p−r) = 0 in Q, (3.2)
12 J. SPREKELS AND F. TR ¨
OLTZSCH
−∂tr−∆r−χq −P(φ∗)(p−r) = 0 in Q, (3.3)
∂np=∂nq=∂nr= 0 on Σ,(3.4)
(p+βq)(T) = b2(φ∗(T)−bφΩ), αp(T) = 0, r(T) = 0 in Ω.(3.5)
According to [17], Theorem 5.2, the adjoint system has a unique weak solution (p, q, r ) satisfying
p+βq ∈H1(0, T ;V∗),(3.6)
p∈H1(0, T ;H)∩C0([0, T ]; V)∩L2(0, T ;W0)∩L∞(Q),(3.7)
q∈L∞(0, T ;H)∩L2(0, T ;V),(3.8)
r∈H1(0, T ;H)∩C0([0, T ]; V)∩L2(0, T ;W0)∩L∞(Q),(3.9)
as well as
− ⟨∂t(p+βq), v⟩+ZΩ
∇q· ∇v−χZΩ
∇r· ∇v+ZΩ
F′′(φ∗)q v
−ZΩ
P′(φ∗)(σ∗+χ(1 −φ∗)−µ∗)(p−r)v+χZΩ
P(φ∗)(p−r)v=b1ZΩ
(φ∗−bφQ)v, (3.10)
−αZΩ
∂tp v +ZΩ
∇p· ∇v−ZΩ
q v +ZΩ
P(φ∗) (p−r)v= 0,(3.11)
−ZΩ
∂tr v +ZΩ
∇r· ∇v−χZΩ
q v −ZΩ
P(φ∗) (p−r)v= 0,(3.12)
for every v∈Vand almost every t∈(0, T ), and
(p+βq)(T) = b2(φ∗(T)−bφΩ), p(T) = 0, r(T) = 0,a.e. in Ω.(3.13)
Moreover, it follows from the proof of [17], Theorem 5.2 that there exists a constant K8>0, which depends
only on Rand the data (but not on the special choice of u∗∈ Uad ), such that
∥p∥H1(0,T ;H)∩C0([0,T ];V)∩L2(0,T ;W0)∩L∞(Q)+∥q∥H1(0,T ;V∗)∩L∞(0,T ;H)∩L2(0,T ;V)
+∥r∥H1(0,T ;H)∩C0([0,T ];V)∩L2(0,T ;W0)∩L∞(Q)
≤K8∥φ∗−bφQ∥L2(Q)+∥φ∗(T)−bφΩ∥V.(3.14)
3.1. First-order necessary optimality conditions
In this section, we aim at deriving associated first-order necessary optimality conditions for local minima of
the optimal control problem (CP). We assume that (A1)–(A6) and (C1)–(C3) are fulfilled and define the
reduced cost functionals associated with the functionals Jand J1introduced in (1.1) by
b
J(u) = J(S(u),u),b
J1(u) = J1(S(u),u).(3.15)
Since Sis twice continuously Fr´echet differentiable from Uinto Yand Yis continuously embedded in
C0([0, T ]; L2(Q)3), Sis also twice continuously Fr´echet differentiable from Uinto C0([0, T ]; L2(Q)3). It thus
follows from the chain rule that the smooth part b
J1of b
Jis a twice continuously Fr´echet differentiable mapping
from Uinto R, where, for every u∗= (u∗
1, u∗
2)∈ U and every h= (h1, h2)∈ U, it holds with (µ∗, φ∗, σ∗) = S(u∗)
SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE OPTIMAL CONTROL 13
that
b
J′
1(u∗)[h] = b1ZZQ
ξh(φ∗−bφQ) + b2ZΩ
ξh(T)(φ∗(T)−bφΩ)
+b3ZZQ
(u∗
1h1+u∗
2h2),(3.16)
where (ηh, ξh, θh) = S′(u∗)[h] is the unique solution to the linearized system (2.16)–(2.20), with λ1=λ2= 1
and λ3=λ4= 0, associated with h.
Remark 3.1. Observe that the right-hand side of (3.16) is meaningful also for arguments h= (h1, h2)∈L2(Q)2,
where in this case (ηh, ξh, θh) = S′(u∗)[h] with the extension of the operator S′(u∗) to L2(Q)2introduced in
Remark 2.5. Hence, by means of the identity (3.16) we can extend the operator b
J′
1(u∗)∈ U∗to L2(Q)2. The
extended operator, which we again denote by b
J′
1(u∗), then becomes an element of (L2(Q)2)∗. In this way,
expressions of the form b
J′
1(u∗)[h] have a proper meaning also for h∈L2(Q)2.
In the following, we assume that u∗= (u∗
1, u∗
2)∈ Uad is a given locally optimal control for (CP) in the sense
of U, that is, there is some ε > 0 such that
b
J(u)≥b
J(u∗) for all u∈ Uad satisfying ∥u−u∗∥U≤ε. (3.17)
Notice that any locally optimal control in the sense of Lp(Q)2with 1 ≤p < ∞is also locally optimal in the
sense of U. Therefore, a result proved for locally optimal controls in the sense of Uis also valid for locally
optimal controls in the sense of Lp(Q)2. It is of course also valid for (globally) optimal controls.
Now, in the same way as in [55], we infer that then the variational inequality
b
J′
1(u∗)[u−u∗] + κ(g(u)−g(u∗)) ≥0∀u∈ Uad (3.18)
is satisfied. Moreover, denoting by the symbol ∂the subdifferential mapping in L2(Q)2(recall that gis a
convex continuous functional on L2(Q)2), we conclude from [55], Theorem 4.5 that there is some λ∗= (λ∗
1, λ∗
2)∈
∂g(u∗)⊂L2(Q)2such that
b
J′
1(u∗)[u−u∗] + ZZQ
κ(λ∗
1(u1−u∗
1) + λ∗
2(u2−u∗
2)) ≥0∀u= (u1, u2)∈ Uad.(3.19)
As usual, we simplify the expression b
J′
1(u∗)[u−u∗] in (3.19) by means of the adjoint state variables defined in
(3.1)–(3.5). A standard calculation (see the proof of [17], Thm. 5.4) then leads to the following result.
Theorem 3.2. (Necessary optimality condition) Suppose that (A1)–(A6) and (C1)–(C3) are fulfilled. More-
over, let u∗= (u∗
1, u∗
2)∈ Uad be a locally optimal control of (CP) in the sense of Uwith associated state
(µ∗, φ∗, σ∗) = S(u∗)and adjoint state (p∗, q ∗, r∗). Then there exists some λ∗= (λ∗
1, λ∗
2)∈∂g(u∗)such that, for
all u= (u1, u2)∈ Uad ,
ZZQ
(−Hp∗+κλ∗
1+b3u∗
1)(u1−u∗
1) + ZZQ
(r∗+κλ∗
2+b3u∗
2)(u2−u∗
2)≥0.(3.20)
Remark 3.3. We underline again that (3.20) is also necessary for all globally optimal controls and all controls
which are even locally optimal in the sense of Lp(Q)×Lp(Σ) with p≥1. Observe also that the variational
inequality (3.20) is equivalent to two independent variational inequalities for u∗
1and u∗
2that have to hold
14 J. SPREKELS AND F. TR ¨
OLTZSCH
simultaneously, namely,
ZZQ
(−Hp∗+κλ∗
1+b3u∗
1) (u1−u∗
1)≥0∀u1∈U1
ad,(3.21)
ZZQ
(r∗+κλ∗
2+b3u∗
2) (u2−u∗
2)≥0∀u2∈U2
ad,(3.22)
where
Ui
ad := {ui∈L∞(Q) : ui≤ui≤uia.e. in Q}, i = 1,2.(3.23)
3.2. Sparsity of controls
The convex function gin the ob jective functional accounts for the sparsity of optimal controls, i.e., any
locally optimal control can vanish in some region of the space-time cylinder Q. The form of this region depends
on the particular choice of the functional gwhich can differ in different situations. The sparsity properties can
be deduced from the variational inequalities (3.21) and (3.22) and the form of the subdifferential ∂g. In this
paper, we restrict our analysis to the case of full sparsity which is characterized by the functional (recall (1.1))
g(u) = g(u1, u2) := ZZQ
(|u1|+|u2|).(3.24)
Other important choices leading to the directional sparsity with respect to time and the directional sparsity with
respect to space are not considered here. It is well known (see, e.g., [54]) that the subdifferential of gis given
by
∂g(u) = ∂g(u1, u2)
:=
(λ1, λ2)∈L2(Q)2:λi∈
{1}if ui>0
[−1,1] if ui= 0
{−1}if ui<0
a.e. in Q, i = 1,2
.(3.25)
The following sparsity result can be proved in exactly the same way as [55], Theorem 4.9.
Theorem 3.4. (Full sparsity) Suppose that the assumptions (A1)–(A6) and (C1)–(C3) are fulfilled, and
assume that ui<0< ui,i= 1,2. Let u∗= (u∗
1, u∗
2)∈ Uad be a locally optimal control in the sense of Ufor the
problem (CP) with the sparsity functional gdefined in (3.24), and with associated state (µ∗, φ∗, σ∗) = S(u∗)
solving (1.2)–(1.6)and adjoint state (p∗, q∗, r∗)solving (3.1)–(3.5). Then there exists some (λ∗
1, λ∗
2)∈∂g(u∗)
such that (3.21)–(3.22)are satisfied. In addition, we have that
u∗
1(x, t)=0 ⇐⇒ | − H(x, t)p∗(x, t)| ≤ κ, for a.e. (x, t)∈Q, (3.26)
u∗
2(x, t)=0 ⇐⇒ |r∗(x, t)| ≤ κ, for a.e. (x, t)∈Q. (3.27)
Moreover, if (p∗, q∗, r∗)and (λ∗
1, λ∗
2)are given, then (u∗
1, u∗
2)is obtained from the projection formulas
u∗
1(x, t) = max u1,min u1,−b−1
3(−Hp∗+κ λ∗
1) (x, t) for a.e. (x, t)∈Q, (3.28)
u∗
2(x, t) = max u2,min u2,−b−1
3(r∗+κ λ∗
2) (x, t) for a.e. (x, t)∈Q.(3.29)
SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE OPTIMAL CONTROL 15
We remark in this connection that the projection formulas above are standard conclusions from the variational
inequalities (3.21)–(3.22). Moreover, it should be noted that, in order to prove sparsity, the control u= 0 has
to lie in (ui, ui), i= 1,2.
3.3. Second-order sufficient optimality conditions
In this section, we establish the main results of this paper, using auxiliary results collected in the Appendix.
We provide conditions that ensure local optimality of pairs u∗= (u∗
1, u∗
2) obeying the first-order necessary
optimality conditions of Theorem 3.2. Second-order sufficient optimality conditions are based on a condition
of coercivity that is required to hold for the smooth part b
J1of b
Jin a certain critical cone. The nonsmooth
part gcontributes to sufficiency by its convexity. In the following, we generally assume that (A1)–(A6),
(C1)–(C3), and the conditions u1<0< u1and u2<0< u2are fulfilled. Our analysis will follow closely the
lines of [55], which in turn follows [4], where a second-order analysis was performed for sparse control of the
FitzHugh–Nagumo system. In particular, we adapt the proof of [4], Theorem 3.4 to our setting of less regularity.
To this end, we fix a pair of controls u∗= (u∗
1, u∗
2) that satisfies the first-order necessary optimality conditions,
and we set (µ∗, φ∗, σ∗) = S(u∗). Then the cone
C(u∗) = {(v1, v2)∈L2(Q)2satisfying the sign conditions (3.30) a.e. in Q},
where
vi(x, t)(≥0 if u∗
i(x, t) = ui
≤0 if u∗
i(x, t) = ui
, i = 1,2,(3.30)
is called the cone of feasible directions, which is a convex and closed subset of L2(Q)2. We also need the
directional derivative of gat u∈L2(Q)2in the direction v= (v1, v2)∈L2(Q)2, which is given by
g′(u,v) = lim
τ↘0
1
τ(g(u+τv)−g(u)).(3.31)
Following the definition of the critical cone in [4], Section 3.1, we define
Cu∗={v∈C(u∗) : b
J′
1(u∗)[v] + κg′(u∗,v)=0},(3.32)
which is also a closed and convex subset of L2(Q)2. According to [4], Section 3.1, it consists of all v= (v1, v2)∈
C(u∗) satisfying
v1(x, t)
= 0 if | − H(x, t)p∗(x, t) + b3u∗
1(x, t)| =κ
≥0 if u∗
1(x, t) = u1or (−H(x, t)p∗(x, t) = −κand u∗
1(x, t) = 0)
≤0 if u∗
1(x, t) = u1or (−H(x, t)p∗(x, t) = κand u∗
1(x, t) = 0)
,(3.33)
v2(x, t)
= 0 if |r∗(x, t) + b3u∗
2(x, t)| =κ
≥0 if u∗
2(x, t) = u2or (r∗(x, t) = −κand u∗
2(x, t) = 0)
≤0 if u∗
2(x, t) = u2or (r∗(x, t) = κand u∗
2(x, t) = 0)
.(3.34)
Remark 3.5. Let us compare the first condition in (3.33) with the situation in the differentiable control problem
without sparsity terms obtained for κ= 0. Then this condition leads to the requirement that v1(x, t) = 0 if
| − H(x, t)p∗(x, t) + b3u∗
1(x, t)|>0, or, since κ= 0,
v1(x, t) = 0 if | − H(x, t)p∗(x, t) + κλ∗
1(x, t) + b3u∗
1(x, t)|>0.(3.35)
16 J. SPREKELS AND F. TR ¨
OLTZSCH
An analogous condition results for v2.
One might be tempted to define the critical cone using (3.35) and its counterpart for v2also in the case κ > 0.
This, however, is not a good idea, because it leads to a critical cone that is larger than needed, in general. As an
example, we mention the particular case when the control u∗=0satisfies the first-order necessary optimality
conditions and when | − Hp∗|< κ and |r∗|< κ hold a.e. in Q. Then the upper relation of (3.33), and its
counterpart for v2, lead to Cu∗={0}, the smallest possible critical cone.
However, thanks to u∗
1= 0, the variational inequality (3.21) implies that −Hp∗+κλ∗
1+b3u∗
1= 0 a.e. in Q,
i.e., the condition | − H(x, t)p∗(x, t) + κλ∗
1(x, t) + b3u∗
1(x, t)|>0 can only be satisfied on a set of measure zero.
Moreover, also the sign conditions (3.30) do not restrict the critical cone. Hence, the largest possible critical
cone Cu∗=L2(Q)2would be obtained, provided that analogous conditions hold for u∗
2and r∗in Q.
In this example, the quadratic growth condition (3.43) below is valid for the choice (3.32) as critical cone
even without assuming the coercivity condition (3.42) below (here the so-called first-order sufficient conditions
apply), while the use of a cone based on (3.35) leads to postulating (3.42) on the whole space L2(Q)2for the
quadratic growth condition to be valid. This shows that the choice of (3.32) as critical cone is essentially better
than of one based on (3.35).
At this point, we derive an explicit expression for b
J′′
1(u)[v,w] for arbitrary u= (u1, u2),v= (v1, v2),w=
(w1, w2)∈ U. In the following, we argue similarly as in [58], Section 5.7 (see also [17], Sect. 6). At first, we
readily infer that, for every ((µ, φ, σ),u)∈(C0([0, T ]; H))3× U and v= (v1, v2, v3),w= (w1, w2, w3) such that
(v,h),(w,k)∈(C0(0, T ;H))3× U , we have
J′′
1((µ, φ, σ),u)[(v,h),(w,k)] = b1ZZQ
v2w2+b2ZΩ
v2(T)w2(T) + b3ZZQ
h·k,(3.36)
where the dot denotes the Euclidean scalar product in R2. For the second-order derivative of the reduced cost
functional b
J1at a fixed control u∗we then find with (µ∗, φ∗, σ∗) = S(u∗) that
b
J′′
1(u∗)[h,k] = D(µ,φ,σ)J1((µ∗, φ∗, σ∗),u∗)[(ν, ψ , ρ)]
+J′′
1((µ∗, φ∗, σ∗),u∗)[((ηh, ξh, θh),h),((ηk, ξk, θk),k)],(3.37)
where (ηh, ξh, θh), (ηk, ξ k, θk), and (ν, ψ, ρ) stand for the unique corresponding solutions to the linearized system
associated with hand k, and to the bilinearized system, respectively. From the definition of the cost functional
(1.1) we readily infer that
D(µ,φ,σ)J1((µ∗, φ∗, σ∗),u∗)[(ν, ψ , ρ)] = b1ZZQ
(φ∗−bφQ)ψ+b2ZΩ
(φ∗(T)−bφΩ)ψ(T).(3.38)
We now claim that, with the associated adjoint state (p∗, q∗, r∗),
b1ZZQ
(φ∗−bφQ)ψ+b2ZΩ
(φ∗(T)−bφΩ)ψ(T)
=ZZQP′(φ∗)ξk(θh−χξh−ηh) + ξh(θk−χξk−ηk)(p∗−r∗)
+P′′(φ∗)ξkξh(σ∗+χ(1 −φ∗)−µ∗)(p∗−r∗)−F(3) (φ∗)ξhξkq∗.(3.39)
SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE OPTIMAL CONTROL 17
To prove this claim, we multiply (2.37) by p∗, (2.38) by q∗, (2.39) by r∗, add the resulting equalities, and
integrate over Q, to obtain that
0 = ZZQ
p∗hα∂tν+∂tψ−∆ν−P(φ∗)(ρ−χψ −ν)
−P′(φ∗)(σ∗+χ(1 −φ∗)−µ∗)ψ−f1i
+ZZQ
q∗hβ∂tψ−∆ψ−ν−χρ +F′′ (φ∗)ψ+F(3)(φ∗)ξhξki
+ZZQ
r∗h∂tρ−∆ρ+χ∆ψ+P(φ∗)(ρ−χψ −ν)
+P′(φ∗)(σ∗+χ(1 −φ∗)−µ∗)ψ+f1i
with the function f1defined in (2.42). Then, we integrate by parts and make use of the initial and terminal
conditions (2.41) and (3.13) to find that
0 = ZZQ
νh−α∂tp∗−∆p∗−q∗+P(φ∗)(p∗−r∗)i
+ZT
0
⟨−∂t(p∗+βq∗)(t), ψ(t)⟩dt +b2ZΩ
(φ∗(T)−bφΩ)ψ(T)
+ZZQ
ψh−∆q∗+χ∆r∗+F′′(φ∗)q∗+χP (φ∗)(p∗−r∗)
−P′(φ∗)(σ∗+χ(1 −φ∗)−µ∗)(p∗−r∗)i
+ZZQ
ρh−∂tr∗−∆r∗−χq∗−P(φ∗)(p∗−r∗)i
+ZZQh−P′(φ∗)ξk(θh−χξh−ηh) + ξh(θk−χξk−ηk)(p∗−r∗)
−P′′(φ∗)ξkξh(σ∗+χ(1 −φ∗)−µ∗)(p∗−r∗) + F(3) (φ∗)ξhξkq∗i,
whence the claim follows, since (p∗, q∗, r∗) solves the adjoint system (3.10)–(3.13). From this characterization,
along with (3.37) and (3.38), we conclude that
b
J′′
1(u∗)[h,k] = b1ZZQ
ξhξk+b2ZΩ
ξh(T)ξk(T) + b3ZZQ
h·k
+ZZQhP′(φ∗)ξk(θh−χξh−ηh) + ξh(θk−χξk−ηk)(p∗−r∗)
+P′′(φ∗)(σ∗+χ(1 −φ∗)−µ∗)(p∗−r∗)ξhξk−F(3) (φ∗)ξhξkq∗i.(3.40)
Observe that the expression on the right-hand side of (3.40) is meaningful also for increments h,k∈
L2(Q)2. Indeed, in this case the expressions (ηh, ξh, θh) = S′(u∗)[h], (ηk, ξk, θk) = S′(u∗)[k], and (ν, ψ , ρ) =
S′′(u∗)[h,k] have an interpretation in the sense of the extended operators S′(u∗) and S′′(u∗) introduced in
Remark 2.5 and Remark 2.7. Therefore, the operator b
J′′
1(u∗) can be extended by the identity (3.40) to the
space L2(Q)2×L2(Q)2. This extension, which will still be denoted by b
J′′
1(u∗), will be frequently used in the
18 J. SPREKELS AND F. TR ¨
OLTZSCH
following. We now show that it is continuous. Indeed, we claim that for all h,k∈L2(Q)2it holds
b
J′′
1(u∗)[h,k]≤b
C∥h∥L2(Q)2∥k∥L2(Q)2,(3.41)
where the constant b
C > 0 is independent of the choice of u∗∈ UR. Obviously, only the last integral on the
right-hand side of (3.40) needs some treatment, and we estimate just its third summand, leaving the others as
an exercise to the reader. We have, by virtue of H¨older’s inequality, the continuity of the embedding V⊂L4(Ω),
and the global bounds (2.9), (2.22), and (3.14),
ZZQ
F(3)(φ∗)ξhξkq∗≤CZT
0
∥ξh∥4∥ξk∥4∥q∗∥2dt
≤C∥ξh∥C0([0,T ];V)∥ξk∥C0([0,T ];V)∥q∗∥L∞(0,T ;H)≤C∥h∥L2(Q)2∥k∥L2(Q)2,
as asserted.
In the following, we will employ the following coercivity condition:
b
J′′
1(u∗)[v,v]>0∀v∈Cu∗\ {0}.(3.42)
Condition (3.42) is a direct extension of associated conditions that are standard in finite-dimensional nonlinear
optimization. In the optimal control of partial differential equation, it was first used in [5]. As in [4], Theorem
3.3 or [5], it can be shown that (3.42) is equivalent to the existence of a constant δ > 0 such that b
J′′
1(u∗)[v,v]≥
δ∥v∥2
L2(Q)2for all v∈Cu∗.
We have the following result.
Theorem 3.6. (Second-order sufficient condition) Suppose that (A1)–(A6) and (C1)–(C3) are fulfilled and
that ui<0< ui,i= 1,2. Moreover, let u∗= (u∗
1, u∗
2)∈ Uad, together with the associated state (µ∗, φ∗, σ∗) =
S(u∗)and the adjoint state (p∗, q∗, r∗), fulfill the first-order necessary optimality conditions of Theorem 3.2. If,
in addition, u∗satisfies the coercivity condition (3.42), then there exist constants ε > 0and ζ > 0such that
the quadratic growth condition
b
J(u)≥b
J(u∗) + ζ∥u−u∗∥2
L2(Q)2(3.43)
holds for all u∈ Uad with ∥u−u∗∥L2(Q)2< ε. Consequently, u∗is a locally optimal control in the sense of
L2(Q)2.
Proof. The proof follows that of [4], Theorem 3.4. We argue by contradiction, assuming that the claim of the
theorem is not true. Then there exists a sequence of controls {uk}⊂Uad such that, for all k∈N,
∥uk−u∗∥L2(Q)2<1
kwhile b
J(uk)<b
J(u∗) + 1
2k∥uk−u∗∥2
L2(Q)2.(3.44)
Noting that uk=u∗for all k∈N, we define
τk=∥uk−u∗∥L2(Q)2and vk=1
τk
(uk−u∗).
Then ∥vk∥L2(Q)2= 1 and, possibly after selecting a subsequence, we can assume that
vk→vweakly in L2(Q)2
SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE OPTIMAL CONTROL 19
for some v∈L2(Q)2. As in [4], the proof is split into three parts.
(i) v∈Cu∗: Obviously, each vkobeys the sign conditions (3.30) and thus belongs to C(u∗). Since C(u∗) is
convex and closed in L2(Q)2, it follows that v∈C(u∗). We now claim that
b
J′
1(u∗)[v] + κg′(u∗,v) = 0.(3.45)
Notice that by Remark 3.1 the expression b
J′
1(u∗)[v] is well defined. For every ϑ∈(0,1) and all v= (v1, v2),u=
(u1, u2)∈L2(Q)2, we infer from the convexity of gthat
g(v)−g(u)≥g(u+ϑ(v−u)) −g(u)
ϑ≥g′(u,v−u)
= max
(λ1,λ2)∈∂g(u)ZZQλ1(v1−u1) + λ2(v2−u2).(3.46)
In particular, with uk= (uk1, uk2),
b
J′
1(u∗)[v] + κg′(u∗,v)≥b
J′
1(u∗)[v] + ZZQ
κλ∗
1v1+λ∗
2v2
=ZZQ(−Hp∗+b3u∗
1+κλ∗
1)v1+ (r∗+b3u∗
2+κλ∗
2)v2
= lim
k→∞
1
τkZZQ(−Hp∗+b3u∗
1+κλ∗
1)(uk1−u∗
1)+(r∗+b3u∗
2+κλ∗
2)(uk2−u∗
2)
≥0,(3.47)
by the variational inequality (3.20). Next, we prove the converse inequality. By (3.44), we have
b
J1(uk)−b
J1(u∗) + κ(g(uk)−g(u∗)) <1
2kτ2
k,
whence, owing to the mean value theorem, and since uk=u∗+τkvk, we get
b
J1(u∗) + τkb
J′
1(u∗+θkτkvk)[vk] + κg(u∗+τkvk)<b
J1(u∗) + κg(u∗) + 1
2kτ2
k
with some 0 < θk<1. From (3.46), we obtain κ(g(u∗+τkvk)−g(u∗)) ≥κg′(u∗, τkvk), and thus
τkb
J′
1(u∗+θkτkvk)[vk] + τkκg′(u∗,vk)<τ2
k
2k.
We divide this inequality by τkand pass to the limit k→ ∞. Here, we invoke Corollary A.2 of the Appendix,
and we use that lim inf k→∞ g′(u∗,vk)≥g′(u∗,v). We then obtain the desired converse inequality
b
J′
1(u∗)[v] + κg′(u∗,v)≤0,
which completes the proof of (i).
(ii) v=0: We again invoke (3.44), now performing a second-order Taylor expansion on the left-hand side,
b
J1(u∗) + τkb
J′
1(u∗)[vk] + τ2
k
2b
J′′
1(u∗+θkτkvk)[vk,vk] + κg(u∗+τkvk)
20 J. SPREKELS AND F. TR ¨
OLTZSCH
<b
J1(u∗) + κg(u∗) + τ2
k
2k.
We subtract b
J1(u∗) + κg(u∗) from both sides and use (3.46) once more to find that
τkb
J′
1(u∗)[vk] + κg′(u∗,vk)+τ2
k
2b
J′′
1(u∗+θkτkvk)[vk,vk]<τ2
k
2k.(3.48)
From the right-hand side of (3.46), and the variational inequality (3.20), it follows that
b
J′
1(u∗)[vk] + κg′(u∗,vk)≥0,
and thus, by (3.48),
b
J′′
1(u∗+θkτkvk)[vk,vk]<1
k.(3.49)
Passing to the limit k→ ∞, we apply Lemma A.3 and deduce that b
J′′
1(u∗)[v,v]≤0.Since we know that
v∈Cu∗, the second-order condition (3.42) implies that v=0.
(iii) Contradiction: From the previous step we know that vk→0weakly in L2(Q)2. Moreover, (3.40) yields
that
b
J′′
1(u∗)[vk, vk] = b3ZZQ
|vk|2+b1ZZQ
|ξk|2+b2ZΩ
|ξk(T)|2
+ZZQh2P′(φ∗)ξk(θk−χξk−ηk)(p∗−r∗)−F(3)(φ∗)q∗|ξk|2i
+ZZQ
P′′(φ∗)(σ∗+χ(1 −φ∗)−µ∗)(p∗−r∗)|ξk|2,(3.50)
where we have set (ηk, ξk, θk) = S′(u∗)[vk], for k∈N. By virtue of Lemma A.3, the sum of the last four integrals
on the right-hand side converges to zero. On the other hand, ∥vk∥L2(Q)2= 1 for all k∈N, by construction.
The weak sequential semicontinuity of norms then implies that
lim inf
k→∞ b
J′′
1(u∗)[vk,vk]≥lim inf
k→∞ b3ZZQ
|vk|2=b3>0.
On the other hand, it is easily deduced from (3.49) and Lemma A.3 that
lim inf
k→∞ b
J′′
1(u∗)[vk,vk]≤0,
a contradiction. The assertion of the theorem is thus proved.
Remark 3.7. We note at this place that the formula (6.5) in [17], which resembles (3.50), contains three sign
errors: indeed, the term in the second line of [17], (6.5) involving P′′ should carry a “plus” sign, while the two
terms in the third line should carry “minus” signs. These sign errors, however, do not have an impact on the
validity of the results established in [17].
SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE OPTIMAL CONTROL 21
Appendix A.
In the following, we assume that (A1)–(A6) and (C1)–(C3) are fulfilled and that u∗∈ Uad is fixed with
associated state (µ∗, φ∗, σ ∗) = S(u∗) and adjoint state (p∗, q∗, r∗). We also recall the definitions of the spaces
used below given in (2.11), (2.14), and (2.15).
Lemma A.1. Let {uk}⊂Uad converge strongly in L2(Q)2to u∗, and let (µk, φk, σk) = S(uk)and (pk, qk, rk),
k∈N, denote the associated states and adjoint states. Then, as k→ ∞,
µk→µ∗strongly in Z, (A.1)
φk→φ∗strongly in Z∩C0(Q),(A.2)
σk→σ∗strongly in Z, (A.3)
pk→p∗weakly-star in Zand strongly in C0([0, T ]; Lp(Ω)) for 1≤p < 6,(A.4)
qk→q∗weakly-star in H1(0, T ;V∗)∩L∞(0, T ;H)∩L2(0, T ;V),(A.5)
rk→r∗weakly-star in Zand strongly in C0([0, T ]; Lp(Ω)) for 1≤p < 6.(A.6)
Proof. The strong convergence ∥S(uk)− S (u∗)∥Z→0 follows directly from (2.10). In addition, the global
bound (2.7) implies that {φk}is bounded in the space e
Xdefined in (2.11), which, thanks to the compactness
of the embedding W0⊂C0(Ω) and [53], Section 8, Corollary 4, is compactly embedded in C0(Q). Therefore it
holds ∥φk−φ∗∥C0(Q)→0 (at first only for a suitable subsequence, but then, owing to the uniqueness of the
limit point, eventually for the entire sequence). The convergence properties (A.1)–(A.3) of the state variables
are thus shown. In addition, it immediately follows from the mean value theorem and (2.9) that, as k→ ∞,
max
i=1,2,3∥F(i)(φk)−F(i)(φ∗)∥C0(Q)→0,
max
i=0,1,2∥P(i)(φk)−P(i)(φ∗)∥C0(Q)→0.(A.7)
Next, we conclude from the bounds (3.14) and (2.7) that there are a subsequence, which is again labeled by
k∈N, and some triple (p, q, r ) such that, as k→ ∞,
pk→pweakly-star in Z∩L∞(Q),(A.8)
qk→qweakly-star in H1(0, T ;V∗)∩L∞(0, T ;H)∩L2(0, T ;V),(A.9)
rk→rweakly-star in Z∩L∞(Q).(A.10)
Moreover, by [53], Section 8, Corollary 4 and the compactness of the embedding V⊂Lp(Ω) for 1 ≤p < 6, we
also have
pk→p, rk→r, both strongly in C0([0, T ]; Lp(Ω)) for 1 ≤p < 6.(A.11)
From these estimates and (A.7) we can easily conclude that, as k→ ∞,
F′′(φk)qk→F′′ (φ∗)q, P (φk)(pk−rk)→P(φ∗)(p−r),
P′(φk)(σk−χ(1 −φk)−µk)(pk−rk)→P′(φ∗)(σ∗−χ(1 −φ∗)−µ∗)(p−r),(A.12)
all weakly in L2(Q).
At this point, we consider the time-integrated version of the adjoint system (3.10)–(3.13) with test functions in
L2(0, T ;V), written for φk, µk, σk, pk, qk, rk,k∈N. Passage to the limit as k→ ∞, using the above convergence
results, immediately leads to the conclusion that (p, q, r) solves the time-integrated version of (3.10)–(3.13)
22 J. SPREKELS AND F. TR ¨
OLTZSCH
with test functions in L2(0, T ;V), which is equivalent to saying that (p, q , r) is a solution to (3.10)–(3.13). By
the uniqueness of this solution, we must have (p, q, r) = (p∗, q∗, r∗). The convergence properties (A.4)–(A.6)
are therefore valid for a suitable subsequence, and since the limit is uniquely determined, also for the entire
sequence.
Corollary A.2. Let {uk}⊂Uad converge strongly in L2(Q)2to u∗, and let {vk}converge weakly to vin
L2(Q)2. Then
lim
k→∞ b
J′
1(uk)[vk] = b
J′
1(u∗)[v].(A.13)
Proof. We have, with uk= (uk1, uk2) and vk= (vk1, vk2),
b
J′
1(uk)[vk] = ZZQ
(−Hpk+b3uk1)vk1+ZZQ
(rk+b3uk2)vk2.
Owing to Lemma A.1, we have, in particular, that {−Hpk+b3uk1}and {rk+b3uk2}converge strongly in
L2(Q) to −Hp∗+b3u∗
1and r∗+b3u∗
2, respectively, whence the assertion immediately follows.
Lemma A.3. Let {uk}and {vk}satisfy the conditions of Corollary A.2, and assume that b3= 0. Then
lim
k→∞ b
J′′
1(uk)[vk,vk] = b
J′′
1(u∗)[v,v].(A.14)
Proof. Let vk= (vk1, vk2), v= (v1, v2), (ηk, ξk, θk) = S′(uk)[vk], and (η, ξ , θ) = S′(u∗)[v]. Since b3= 0, we
infer from (3.50) that we have to show that, as k→ ∞,
b1ZZQ
|ξk|2+b2ZΩ
|ξk(T)|2+ZZQ
2P′(φk)ξk(θk−χξk−ηk)(pk−rk)
+ZZQhP′′(φk)(σk+χ(1 −φk)−µk)|ξk|2−F(3) (φk)qk|ξk|2i
→b1ZZQ
|ξ∗|2+b2ZΩ
|ξ∗(T)|2+ZZQ
2P′(φ∗)ξ∗(θ∗−χξ∗−η∗)(p∗−r∗)
+ZZQhP′′(φ∗)(σ∗+χ(1 −φ∗)−µ∗)(p∗−r∗)|ξ∗|2−F(3) (φ∗)q∗|ξ∗|2i,(A.15)
where (pk, qk, rk) and (p∗, q∗, r∗) are the associated adjoint states. By Lemma 4.1 and its proof, the convergence
properties (A.1)–(A.6) and (A.7) are valid. Moreover, we have
(ηk, ξk, θk)−(η∗, φ∗, θ∗)=(S′(uk)− S′(u∗)) [vk] + S′(u∗)[vk−v].
By virtue of (2.23) and the boundedness of {vk}in L2(Q)2, the first summand on the right-hand side of this
identity converges strongly to zero in Z. The second converges to zero weakly in Z×e
X×Z. Hence, thanks to
the compactness of the embedding Z⊂C0([0, T ]; Lp(Ω)) for 1 ≤p < 6 (see, e.g., [53], Sect. 8, Cor. 4),
(ηk, ξk, θk)→(η∗, ξ∗, θ∗) strongly in C0([0, T ]; Lp(Ω))3for 1 ≤p < 6.(A.16)
In particular, as k→ ∞,
b1ZZQ
|ξk|2+b2ZΩ
|ξk(T)|2→b1ZZQ
|ξ∗|2+b2ZΩ
|ξ∗(T)|2.(A.17)
SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE OPTIMAL CONTROL 23
Moreover, owing to the strong convergences in C0([0, T ]; Lp(Ω)) for 1 ≤p < 6, it is easily checked, using H¨older’s
inequality, that
P′(φk)ξk(θk−χξk−ηk)(pk−rk)→P′(φ∗)ξ∗(θ∗−χξ∗−η∗)(p∗−r∗),
P′′(φk)(σk+χ(1 −φk)−µk)(pk−rk)|ξk|2→P′′ (φ∗)(σ∗+χ(1 −φ∗)−µ∗)(p∗−r∗)|ξ∗|2,(A.18)
both strongly in L1(Q). It remains to show that, as k→ ∞,
ZZQ
F(3)(φk)qk|ξk|2→ZZQ
F(3)(φ∗)q∗|ξ∗|2.
Since qk→q∗weakly in L2(Q) by (A.9), it thus suffices to show that F(3)(φk)|ξk|2→F(3)(φ∗)|ξ∗|2strongly
in L2(Q). However, this is a simple consequence of (A.7) and (A.16). The assertion is thus proved.
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