This section considers POCP (1.4)−(1.6) with the functionals G(y) := 1

2ky−y_{ds}k^{2}_{L}2(0,T;L^{2}(Ω)) and H(u) := α
2kuk^{2}_{X},

where α is a regularization parameter. The model control problem is stated as follows:

u∈Uminad

J(y, u) := 1 2

ky−y_{ds}k^{2}_{L}2(0,T;L^{2}(Ω))+αkuk^{2}_{X} (4.1)

87

CHAPTER 4. Controls acting on lower dimensional manifolds 88 subject to the state equation

∂y

∂t −∆y=u(x, t)δ_{γ(t)}(x) in Ω_{T},
y(·,0) =y_{0} in Ω,

y= 0 on Γ_{T},

(4.2)

where Ω is an open bounded domain inR^{d}(d = 2 or 3) with Lipschitz boundary Γ :=∂Ω
andI = [0, T] (a fixed realT < ∞). Moreover, we denote Ω_{T} = Ω×I and Γ_{T} =∂Ω×I.

In the above, the initial state y_{0} ∈ H_{0}^{1}(Ω) and the desired state y_{ds} ∈ L^{2}(0, T;L^{2}(Ω)).

Later on, we will specify the control space X. Further, we denote the state variable
by y = y(x, t), control variable u = u(x, t) and denote ^{∂y}_{∂t} = yt. The set of admissible
controls U_{ad} is defined by

U_{ad} :=

u∈X : u_{a} ≤u(x, t)≤u_{b} a.e. on γ(t) a.a. t ∈I , (4.3)
whereu_{a}, u_{b} are real constants withu_{a}< u_{b} or constant vectors according to manifold’s
dimension, andδ_{γ(t)}is a Dirac measure onγ(t). Assume that, for allt∈[0, T], the lower
dimensional manifold γ(t) is precisely contained in Ω. Note that the lower dimensional
manifold represents a point, or a curve if d ≥ 2, or a surface if d = 3 which can be
time independent or evolves in the time horizon. For the well-posedness of the problem
(4.1)−(4.3), we refer to [13, 35].

We now assume that the control variable u(x, t) ∈ L^{2}(0, T;R^{m}) or u(x, t) ∈
L^{2}(0, T;L^{2}(γ(t))) with norm

Z T 0

ku(t)k^{2}_{R}mdt < ∞ if r= 0,
or

Z T 0

Z

γ(t)

|u(x, t)|^{2}dx dt < ∞ if r 6= 0,

respectively, where the norm k · k_{R}^{m} denotes the standard Euclidean norm on m-
dimensional Euclidean space. Henceforth, we denote the control space as X :=

L^{2}(0, T;U) with U = R^{m} if r = 0 ( or U := L^{2}(γ(t)) if r ≥ 1), and the inner product
h·,·i_{X} is a duality pairing between X and (X)^{0}. Here (X)^{0} denotes the dual space of X.

We now define the inner products on L^{2}(Ω) and L^{2}(0, T;L^{2}(Ω)) as
(ψ, φ) =

Z

Ω

ψ φ dx, ∀ψ, φ∈L^{2}(Ω),

CHAPTER 4. Controls acting on lower dimensional manifolds 89 and

(ψ, φ)_{Ω}_{T} =
Z T

0

Z

Ω

ψ φ dxdt= Z

ΩT

ψ φ dxdt, ∀ψ, φ∈L^{2}(0, T;L^{2}(Ω)),
respectively. The bilinear forms a(·,·) and a(·,·)_{Ω}_{T} are defined as

a(v, w) = Z

Ω

∇v · ∇w dx, ∀v, w∈H_{0}^{1}(Ω),
and

a(v, w)_{Ω}_{T} =
Z

ΩT

∇v · ∇w dx dt, ∀v, w∈L^{2}(0, T;H_{0}^{1}(Ω)),

respectively. Then there exists α_{0}, α_{1} >0 such that a(·,·) the bilinear form satisfies

|a(v, w)| ≤ α0kvk1kwk1, ∀v, w∈H_{0}^{1}(Ω),
and

a(v, v) ≥ α_{1}kvk^{2}_{1}, ∀v ∈H_{0}^{1}(Ω),
respectively, i.e., a(·,·) is bounded and coercive.

Next, we discuss the well-posed property for the state equation (4.2) (cf., [35]).

In order to state the existence and uniqueness result for the problem (4.2), we define following notation:

H0 =

H^{−1}(Ω) ifd−r= 1,
L^{2}(Ω) ifd−r >1;

and H2 =

H_{0}^{1}(Ω) ifd−r= 1,
H^{2}(Ω)∩H_{0}^{1}(Ω) ifd−r >1.

For Φ∈L^{2}(0, T;H_{2}), we define

hu(x, t)δ_{γ(t)},Φi_{I} =

m

X

j=1

Z T 0

u_{j}(t)Φ(γ_{j}(t), t)dt if r= 0,
Z T

0

Z

γ(t)

u(x, t)Φ(x, t)dxdt if r≥1.

Note that

m

X

j=1

Z T 0

u_{j}(t)Φ(γ_{j}(t), t)dt

≤ kuk_{L}^{2}_{(0,T;R}^{m}_{)}kΦk_{L}^{2}_{(0,T;L}^{∞}_{(Ω))} ifr= 0,
and

Z T 0

Z

γ(t)

u(x, t)Φ(x, t)dxdt

≤ kuk_{L}^{2}_{(0,T}_{;L}^{2}_{(γ(t)))}kΦk_{L}^{2}_{(0,T}_{;L}^{2}_{(γ(t)))} if r ≥1.

CHAPTER 4. Controls acting on lower dimensional manifolds 90
From the trace theorem and the standard embedding resultH^{2}(Ω)∩H_{0}^{1}(Ω) ,→ C(Ω) we
know that

kΦ|_{γ(t)}(·, t)k_{L}^{2}_{(γ(t))} ≤ C_{4,1}kΦ(·, t)k_{H}^{1}

0(Ω) ford= 2, 3 and d−r = 1, and

kΦk_{L}^{2}_{(0,T}_{;L}^{∞}_{(Ω))} ≤ C_{4,2}kΦk_{L}2(0,T;H^{2}(Ω)∩H_{0}^{1}(Ω)),

respectively, where C_{4,1} and C_{4,2} depend on the manifold γ(t). Hence,

kΦk_{L}^{2}_{(0,T;L}^{2}_{(γ(t)))}≤

C_{4,1}kΦk_{L}2(0,T;H_{0}^{1}(Ω)) ifd= 2,3; d−r = 1,
C_{4,3}kΦk_{L}^{2}_{(0,T}_{;L}^{∞}_{(Ω))} ≤C_{4,4}kΦk_{L}2(0,T;H^{2}(Ω)∩H_{0}^{1}(Ω) ifd= 3; r= 1,
with C_{4,4} =C_{4,2}C_{4,3}. Thus, we find that

hu(x, t)δ_{γ(t)},Φi_{I} ≤

C_{4,5}kuk_{L}^{2}_{(0,T}_{;}_{R}^{m}_{)}kΦk_{L}^{2}_{(0,T;H}_{2}_{)} if r = 0,
C4,6kuk_{L}^{2}_{(0,T}_{;L}^{2}_{(γ(t)))}kΦk_{L}^{2}_{(0,T}_{;H}_{2}_{)} if r≥1,

(4.4)

whereC_{4,5} andC_{4,6} depend on the manifoldγ(t). Sinceh·,·i_{I}denotes the duality pairing
betweenL^{2}(0, T;H_{2}) andL^{2}(0, T; (H_{2})^{0}),u(x, t)δ_{γ(t)} ∈L^{2}(0, T; (H_{2})^{0}), where (H_{2})^{0} is the
dual space of H_{2}.

The following lemma concerns the existence of a solution for the problem (4.2). The weak solution of (4.2) can be defined by employing the transposition technique (cf. [53]

and [55]).

Lemma 4.1.1. Let f ∈ L^{2}(0, T;H_{0}), and let Φ be the solution of (1.18). Then the
control problem (4.2) has a weak solution y∈L^{2}(0, T; (H_{0})^{0}) with u∈U_{ad}, if

(y, f)_{Ω}_{T} = (y_{0},Φ(·,0)) +hu(x, t)δ_{γ(t)},Φi_{I}, (4.5)
for Φ∈L^{2}(0, T;H_{2}), where (H_{0})^{0} denotes the dual of H_{0}.

Invoking the well-known Lax-Milgram theorem, we can show that the control prob-
lem (4.2) admits a unique solution y ∈ L^{2}(0, T; (H_{0})^{0}) in the sense of (4.5). Following
theorem presents the regularity of the solution for the problem (4.2) in various dimension
of the manifold γ(t). For a proof, we refer to [35].

Theorem 4.1.1. Assume that y_{0} ∈ H_{0}^{1}(Ω) and f ∈ L^{2}(0, T;H_{0}). Let y ∈
L^{2}(0, T;L^{2}(Ω)) be the solution of the state equation (4.2) in the sense of (4.5), i.e.,

(y, f)_{Ω}_{T} = huδ_{γ(t)},Φi_{I}+ (y_{0},Φ(·,0)).

CHAPTER 4. Controls acting on lower dimensional manifolds 91 Then, we have the following regularity results:

y ∈L^{2}(0, T;W_{0}^{1,s}(Ω))∩H^{1}(0, T;W^{−1,s}(Ω)), s ∈
1, d

d−1

when r = 0, d= 2,3;

y ∈L^{2}(0, T;W_{0}^{1,σ}(Ω))∩H^{1}(0, T;W^{−1,σ}(Ω), σ∈(1,2) when r= 1, d= 3;

y ∈L^{2}(0, T;H^{3−}^{2} (Ω)∩H_{0}^{1}(Ω))∩H^{1}(0, T;H^{−1+}^{2} (Ω)), ∀ >0 when r ≥1, d−r = 1.

The weak formulation of the optimal control problem (4.1)−(4.3) is to seek a pair
(y, u)∈L^{2}(0, T;L^{2}(Ω))×U_{ad} such that

u∈Umin_{ad}
1
2

ky−y_{ds}k^{2}_{L}2(0,T;L^{2}(Ω))+αkuk^{2}_{X} (4.6)
subject to

−(y, v_{t})_{Ω}_{T} −(y,∆v)_{Ω}_{T} = huδ_{γ(t)}, vi_{I}+ (y_{0}, v(·,0)) ∀v ∈ W_{1}(0, T). (4.7)
It is well known (cf. [53]) that the control problem (4.6)−(4.7) has a unique optimal
pair (y, u) iff there exists a co-state variablep∈L^{2}(0, T;H_{0}^{1}(Ω)) such that the following
optimality conditions are satisfied:

−(y, v_{t})_{Ω}_{T} −(y,∆v)_{Ω}_{T} = huδ_{γ(t)}, vi_{I}+ (y_{0}, v(·,0)) ∀v ∈ W_{1}(0, T), (4.8)

−(p_{t}, v)_{Ω}_{T} +a(p, v)_{Ω}_{T} = (y−y_{ds}, v)_{Ω}_{T} ∀v ∈ W_{2}(0, T), (4.9)

p(·, T) = 0, (4.10)

(αu+p|_{γ}, w−u)_{X} ≥ 0 ∀w∈U_{ad}, (4.11)

where p|_{γ} stands for the restriction of p on the manifold γ(t). From (4.11), the control
variable ucan be expressed in terms of the co-state variablep as follows: For t∈[0, T],

u(t) =

PUad

−_{α}^{1}p(γ_{j}(t))(t)m
j=1

for r = 0, PUad

−_{α}^{1}p(x, t)|_{γ}

for r ≥1,
where PUad is the orthogonal projection onto U_{ad}.

The following theorem collects the regularity results of the solutions (y, p, u) to the control problem (4.8)−(4.11).

Theorem 4.1.2. Let (y, p, u)be the solution of the optimization problem (4.8)−(4.11).

Assume that y_{0} ∈H_{0}^{1}(Ω), then we have the following regularity results:

CHAPTER 4. Controls acting on lower dimensional manifolds 92 Case I (if r= 0, d= 2, 3):

y∈L^{2}(I;W_{0}^{1,s}(Ω))∩H^{1}(I;W^{−1,s}(Ω)) s∈(1,_{d−1}^{d} ),
p∈L^{2}(I;H^{2}(Ω)∩H_{0}^{1}(Ω))∩H^{1}(I;L^{2}(Ω)) u∈L^{2}(I;R^{m}).

Case II (if r = 1, d= 3):

y∈L^{2}(I;W_{0}^{1,σ}(Ω))∩H^{1}(I;W^{−1,σ}(Ω)) σ∈(1,2),

p∈L^{2}(I;H^{2}(Ω)∩H_{0}^{1}(Ω))∩H^{1}(I;L^{2}(Ω)) u∈L^{2}(I;H^{1}(γ(t)).

Case III (if r≥1, d−r= 1):

y∈L^{2}(I;H^{3−}^{2} (Ω)∩H_{0}^{1}(Ω))∩H^{1}(I;H^{−1+}^{2} (Ω)) for any >0,
p∈L^{2}(I;H^{2}(Ω)∩H_{0}^{1}(Ω))∩H^{1}(I;L^{2}(Ω)) u∈L^{2}(I;H^{1}(γ(t)).

In this direction, for time-independent control problem, Gong et al. [33] have ex-
tensively discussed the finite element approximations to elliptic control problems with
controls acting on the lower dimensional manifold. They have derived a priori error
estimates for the state variable in the L^{2}(Ω)-norm while the error for the control vari-
able in the U-norm with U = L^{2}(γ(t)) or U = R^{m} according to the dimension of the
manifold. Gong et al. [31] have studied a priori error analysis for the control problem
governed by parabolic equations, where the control acts only on finitely many points
which are time-independent. Thereafter, Gong and Yan [35] have generalized the a pri-
ori error analysis results for the optimal control problem governed by parabolic partial
differential equations where the manifold evolves with time. In all of the above studies,
a priori error estimates are derived and the support of the controls requires to be very
small compared to the actual size of the domain Ω if we are restricted by the cost of
controls. To the best of author’s knowledge, thea posteriorierror analysis are yet to be
explored. This chapter attempts to studya posteriorierror analysis of the fully discrete
finite element approximations to the state, co-state and control variables for the control
problems (4.1)−(4.3) with controls acting on a lower dimensional manifold. We derive
a posteriori error estimates for the state variable in the L^{2}(0, T;L^{2}(Ω))-norm while er-
ror for the control variable in the L^{2}(0, T;R^{m}) or L^{2}(0, T;L^{2}(γ(t)))-norm according to
dimension of the manifold.

CHAPTER 4. Controls acting on lower dimensional manifolds 93 The layout of the chapter is as follows. Section 4.2 is devoted to the fully discrete finite element approximation of the model problem (4.1)− (4.3). A posteriori error estimates for the state, co-state and control variables are derived in Section 4.3. We perform numerical tests to illustrate the performance of the derived estimators in Section 4.4. In the last section, we present some concluding remarks.