This section considers POCP (1.4)−(1.6) with the functionals G(y) := 1
2ky−ydsk2L2(0,T;L2(Ω)) and H(u) := α 2kuk2X,
where α is a regularization parameter. The model control problem is stated as follows:
u∈Uminad
J(y, u) := 1 2
ky−ydsk2L2(0,T;L2(Ω))+αkuk2X (4.1)
87
CHAPTER 4. Controls acting on lower dimensional manifolds 88 subject to the state equation
∂y
∂t −∆y=u(x, t)δγ(t)(x) in ΩT, y(·,0) =y0 in Ω,
y= 0 on ΓT,
(4.2)
where Ω is an open bounded domain inRd(d = 2 or 3) with Lipschitz boundary Γ :=∂Ω andI = [0, T] (a fixed realT < ∞). Moreover, we denote ΩT = Ω×I and ΓT =∂Ω×I.
In the above, the initial state y0 ∈ H01(Ω) and the desired state yds ∈ L2(0, T;L2(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 Uad is defined by
Uad :=
u∈X : ua ≤u(x, t)≤ub a.e. on γ(t) a.a. t ∈I , (4.3) whereua, ub are real constants withua< ub or constant vectors according to manifold’s dimension, andδγ(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) ∈ L2(0, T;Rm) or u(x, t) ∈ L2(0, T;L2(γ(t))) with norm
Z T 0
ku(t)k2Rmdt < ∞ if r= 0, or
Z T 0
Z
γ(t)
|u(x, t)|2dx dt < ∞ if r 6= 0,
respectively, where the norm k · kRm denotes the standard Euclidean norm on m- dimensional Euclidean space. Henceforth, we denote the control space as X :=
L2(0, T;U) with U = Rm if r = 0 ( or U := L2(γ(t)) if r ≥ 1), and the inner product h·,·iX 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 L2(Ω) and L2(0, T;L2(Ω)) as (ψ, φ) =
Z
Ω
ψ φ dx, ∀ψ, φ∈L2(Ω),
CHAPTER 4. Controls acting on lower dimensional manifolds 89 and
(ψ, φ)ΩT = Z T
0
Z
Ω
ψ φ dxdt= Z
ΩT
ψ φ dxdt, ∀ψ, φ∈L2(0, T;L2(Ω)), respectively. The bilinear forms a(·,·) and a(·,·)ΩT are defined as
a(v, w) = Z
Ω
∇v · ∇w dx, ∀v, w∈H01(Ω), and
a(v, w)ΩT = Z
ΩT
∇v · ∇w dx dt, ∀v, w∈L2(0, T;H01(Ω)),
respectively. Then there exists α0, α1 >0 such that a(·,·) the bilinear form satisfies
|a(v, w)| ≤ α0kvk1kwk1, ∀v, w∈H01(Ω), and
a(v, v) ≥ α1kvk21, ∀v ∈H01(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, L2(Ω) ifd−r >1;
and H2 =
H01(Ω) ifd−r= 1, H2(Ω)∩H01(Ω) ifd−r >1.
For Φ∈L2(0, T;H2), we define
hu(x, t)δγ(t),ΦiI =
m
X
j=1
Z T 0
uj(t)Φ(γj(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
uj(t)Φ(γj(t), t)dt
≤ kukL2(0,T;Rm)kΦkL2(0,T;L∞(Ω)) ifr= 0, and
Z T 0
Z
γ(t)
u(x, t)Φ(x, t)dxdt
≤ kukL2(0,T;L2(γ(t)))kΦkL2(0,T;L2(γ(t))) if r ≥1.
CHAPTER 4. Controls acting on lower dimensional manifolds 90 From the trace theorem and the standard embedding resultH2(Ω)∩H01(Ω) ,→ C(Ω) we know that
kΦ|γ(t)(·, t)kL2(γ(t)) ≤ C4,1kΦ(·, t)kH1
0(Ω) ford= 2, 3 and d−r = 1, and
kΦkL2(0,T;L∞(Ω)) ≤ C4,2kΦkL2(0,T;H2(Ω)∩H01(Ω)),
respectively, where C4,1 and C4,2 depend on the manifold γ(t). Hence,
kΦkL2(0,T;L2(γ(t)))≤
C4,1kΦkL2(0,T;H01(Ω)) ifd= 2,3; d−r = 1, C4,3kΦkL2(0,T;L∞(Ω)) ≤C4,4kΦkL2(0,T;H2(Ω)∩H01(Ω) ifd= 3; r= 1, with C4,4 =C4,2C4,3. Thus, we find that
hu(x, t)δγ(t),ΦiI ≤
C4,5kukL2(0,T;Rm)kΦkL2(0,T;H2) if r = 0, C4,6kukL2(0,T;L2(γ(t)))kΦkL2(0,T;H2) if r≥1,
(4.4)
whereC4,5 andC4,6 depend on the manifoldγ(t). Sinceh·,·iIdenotes the duality pairing betweenL2(0, T;H2) andL2(0, T; (H2)0),u(x, t)δγ(t) ∈L2(0, T; (H2)0), where (H2)0 is the dual space of H2.
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 ∈ L2(0, T;H0), and let Φ be the solution of (1.18). Then the control problem (4.2) has a weak solution y∈L2(0, T; (H0)0) with u∈Uad, if
(y, f)ΩT = (y0,Φ(·,0)) +hu(x, t)δγ(t),ΦiI, (4.5) for Φ∈L2(0, T;H2), where (H0)0 denotes the dual of H0.
Invoking the well-known Lax-Milgram theorem, we can show that the control prob- lem (4.2) admits a unique solution y ∈ L2(0, T; (H0)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 y0 ∈ H01(Ω) and f ∈ L2(0, T;H0). Let y ∈ L2(0, T;L2(Ω)) be the solution of the state equation (4.2) in the sense of (4.5), i.e.,
(y, f)ΩT = huδγ(t),ΦiI+ (y0,Φ(·,0)).
CHAPTER 4. Controls acting on lower dimensional manifolds 91 Then, we have the following regularity results:
y ∈L2(0, T;W01,s(Ω))∩H1(0, T;W−1,s(Ω)), s ∈ 1, d
d−1
when r = 0, d= 2,3;
y ∈L2(0, T;W01,σ(Ω))∩H1(0, T;W−1,σ(Ω), σ∈(1,2) when r= 1, d= 3;
y ∈L2(0, T;H3−2 (Ω)∩H01(Ω))∩H1(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)∈L2(0, T;L2(Ω))×Uad such that
u∈Uminad 1 2
ky−ydsk2L2(0,T;L2(Ω))+αkuk2X (4.6) subject to
−(y, vt)ΩT −(y,∆v)ΩT = huδγ(t), viI+ (y0, v(·,0)) ∀v ∈ W1(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∈L2(0, T;H01(Ω)) such that the following optimality conditions are satisfied:
−(y, vt)ΩT −(y,∆v)ΩT = huδγ(t), viI+ (y0, v(·,0)) ∀v ∈ W1(0, T), (4.8)
−(pt, v)ΩT +a(p, v)ΩT = (y−yds, v)ΩT ∀v ∈ W2(0, T), (4.9)
p(·, T) = 0, (4.10)
(αu+p|γ, w−u)X ≥ 0 ∀w∈Uad, (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
−α1p(γj(t))(t)m j=1
for r = 0, PUad
−α1p(x, t)|γ
for r ≥1, where PUad is the orthogonal projection onto Uad.
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 y0 ∈H01(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∈L2(I;W01,s(Ω))∩H1(I;W−1,s(Ω)) s∈(1,d−1d ), p∈L2(I;H2(Ω)∩H01(Ω))∩H1(I;L2(Ω)) u∈L2(I;Rm).
Case II (if r = 1, d= 3):
y∈L2(I;W01,σ(Ω))∩H1(I;W−1,σ(Ω)) σ∈(1,2),
p∈L2(I;H2(Ω)∩H01(Ω))∩H1(I;L2(Ω)) u∈L2(I;H1(γ(t)).
Case III (if r≥1, d−r= 1):
y∈L2(I;H3−2 (Ω)∩H01(Ω))∩H1(I;H−1+2 (Ω)) for any >0, p∈L2(I;H2(Ω)∩H01(Ω))∩H1(I;L2(Ω)) u∈L2(I;H1(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 L2(Ω)-norm while the error for the control vari- able in the U-norm with U = L2(γ(t)) or U = Rm 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 L2(0, T;L2(Ω))-norm while er- ror for the control variable in the L2(0, T;Rm) or L2(0, T;L2(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.