Electronic Journal of Differential Equations, Vol. 2009(2009), No. 83, pp. 1–22.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
OPTIMAL BOUNDARY CONTROL FOR A TIME DEPENDENT
THERMISTOR PROBLEM
VOLODYMYR HRYNKIV
Abstract. An optimal control of a two dimensional time dependent thermistor problem is considered. The problem consists of two nonlinear partial differential equations coupled with appropriate boundary conditions which model
the coupling of the thermistor to its surroundings. Based on physical considerations, an objective functional to be minimized is introduced and the
convective boundary coefficient is taken to be the control. Existence of solutions to the state system and existence of the optimal control are proven. To
characterize this optimal control, the optimality system consisting of the state
and adjoint equations is derived.
1. Introduction
Thermistor is a thermally sensitive resistor whose electrical conductivity changes
drastically by orders of magnitude as the temperature reaches a certain threshold.
Thermistors are used as temperature control elements in a wide variety of military
and industrial equipment ranging from space vehicles to air conditioning controllers.
They are also used in the medical field for localized and general body temperature
measurement, in meteorology for weather forecasting as well as in chemical industries as process temperature sensors [11, 13].
We consider the two-dimensional time-dependent thermistor problem
ut − ∆u − σ(u)|∇ϕ|2 = 0
∇ · (σ(u)∇ϕ) = 0
in Ω × (0, T ),
in Ω × (0, T ),
∂u
+ βu = 0 on ΓN × (0, T ),
∂n
u = 0 on ΓD × (0, T ),
ϕ = ϕ0
u = u0
(1.1)
on ∂Ω × (0, T ),
on Ω × {0},
where ϕ(x, t) is the electric potential, u(x, t) is the temperature, and σ(u) is the
electrical conductivity. Here n denotes the outward unit normal and ∂/∂n = n · ∇
is the normal derivative on ∂Ω. The elliptic-parabolic system (1.1) describes the
2000 Mathematics Subject Classification. 49J20, 49K20.
Key words and phrases. Optimal control; thermistor problem; elliptic systems.
c 2009 Texas State University - San Marcos.
Submitted December 10, 2008. Published July 10, 2009.
1
2
V. HRYNKIV
EJDE-2009/83
heating of a conductor produced by an electric current. It is the consequence of
Ohm’s law and Fourier’s law
I = −σ(u)∇ϕ,
Q = −k(u)∇u,
(1.2)
coupled with the conservation laws of energy and current
ρc
∂u
+ ∇ · Q = I · E,
∂t
∇ · I = 0,
(1.3)
(1.4)
where E denotes the electric field, ρ is the density of the conductor, and c is its heat
capacity. Since the nonlinearity k(u) varies only slightly with u, it is of secondary
importance, and therefore we can assume k(u) = 1. As the density ρ of the material
and its specific heat c typically are constants, we assume that ρc = 1 (see [1]). A
more detailed discussion of equations (1.1) can be found in [1, 2, 7, 9, 17]. Boundary
conditions show how the thermistor is connected thermally and electrically to its
surroundings. To describe boundary conditions for the temperature we decompose
the boundary ∂Ω into two disjoint sets Γ̄D ∪ Γ̄N = ∂Ω. The Robin boundary condition for temperature represents the physically significant case of convection and/or
radiation in situations where u is not too different from the ambient temperature.
It is precisely this boundary condition that will be used as a control.
It has been observed that large temperature gradients may cause the thermistor
to crack. Numerical experiments indicate (see [7, 23]) that low values of the heat
transfer coefficient β lead to small temperature variations. On the other hand,
low values of the heat transfer coefficient result in high operating temperatures
of a thermistor which are also undesirable. This motivated us to take the heat
transfer coefficient as a control and to consider the optimal control problem of
minimizing the heat transfer coefficient while keeping the operating temperature
of the thermistor reasonably low. These physical considerations lead us to the
following objective functional
Z TZ
Z
J(β) =
u dx dt +
β 2 ds dt.
0
Ω
ΓN ×(0,T )
Denoting the set of admissible controls by
UM = {β ∈ L∞ (∂Ω × (0, T )) : 0 < λ ≤ β ≤ M },
the optimal control problem is
Find β ∗ ∈ UM such that J(β ∗ ) = min J(β).
β∈UM
(1.5)
Henceforth we use the standard notation for Sobolev spaces, we denote k · kp =
k · kLp (Ω) for each p ∈ [1, ∞]; other norms will be clearly marked. We also denote
Q := Ω × (0, T ) and ΓTN := ΓN × (0, T ).
Analysis of both the steady-state and time-dependent thermistor equations with
different types of boundary and initial conditions has received a great deal of attention. See [1, 2, 3, 4, 7, 9, 17, 18, 19, 22] for existence of weak solutions, existence/nonexistence of classical solutions, uniqueness and related regularity results
in different settings with various assumptions on the coefficients.
We mention in passing that few authors consider a thermistor problem which
consists of two parabolic equations coupled with some boundary conditions (see [12,
EJDE-2009/83
OPTIMAL CONTROL FOR A THERMISTOR PROBLEM
3
15]). This fully parabolic system comes from (1.2), (1.3), and the time dependent
version of charge conservation (1.4), i.e.,
∂q
+ ∇ · I = 0.
∂t
(1.6)
However, the vast majority of the existing literature on time dependent thermistor
deals with the elliptic-parabolic system (1.1). This reflects a feasible thermistor
scenario where the time derivative in (1.6) is dropped considering that the relaxation
time for the potential is very small, hence giving the equation (1.4) (see [6, 17]).
The first optimal control paper on a thermistor problem is the paper by Lee and
Shilkin [12] where a fully parabolic system is considered and the source is taken to
be the control. Optimal control of a nonlocal elliptic-parabolic case is studied in [5],
where the applied potential is taken to be the control. Optimal control of a steady
state thermistor problem is considered in [10], where the heat transfer coefficient is
the control.
The motivation for our work is threefold. First, the results from [10] are extended
to the time dependent elliptic-parabolic thermistor system. As it was mentioned
before, such an elliptic-parabolic thermistor model represents a reasonably realistic
situation where we neglect the time derivative in (1.6) by taking into consideration
the difference in time scale with the thermal processes. One of the technical difficulties in dealing with such an elliptic-parabolic system lies in the fact that there
is no information on time derivative of the potential ϕ, and as a result one cannot
use directly the standard compactness result from [16] to obtain strong convergence
of sequences of potentials in appropriate spaces. To circumvent this difficulty one
needs to incorporate the structure of the system. This is different from [12] in that
a fully parabolic system and a different type of control were considered there. Secondly, we present a complete and self-contained derivation of the optimality system,
whereas in [5] the optimality system was given only for the special case of constant
σ(u). Thirdly, the choice of the convective boundary condition as a control for
this time dependent problem seems to be quite appropriate from point of view of
practical applications.
In section 2 we derive a priori estimates under the assumption of small boundary
data and prove existence of solutions to the state system. In section 3 we prove
existence of an optimal control. The optimality system is derived and an optimal
control is characterized in section 4.
2. A priori estimates and existence of solutions to the state system
We make the following assumptions:
•
•
•
•
•
Ω ⊂ R2 is a bounded domain with smooth boundary;
σ(s) ∈ C 1 (R), 0 < C1 ≤ σ(s) ≤ C2 for all s ∈ R;
σ(s) is Lipschitz: |σ(s1 ) − σ(s2 )| ≤ K |s1 − s2 | for all s1 , s2 ∈ R;
1,∞
ϕ0 ∈ W (∂Ω × (0, T ));
Extending ϕ0 to the whole domain Ω × (0, T ) and using the same notation,
1,∞
i.e., ϕ0 ∈ W (Q), we assume that kϕ0 kW 1,∞ (Q) is sufficiently small.
The following lemma guarantees a better regularity for solutions of equations in
divergence form. It is taken from [20]. More information about this lemma can be
found in [14].
4
V. HRYNKIV
EJDE-2009/83
Lemma 2.1. Let Ω ⊂ Rn be a bounded domain with a smooth boundary. Assume
that g ∈ (L2 (Ω))n and a ∈ C(Ω̄) with minΩ̄ a > 0. Let u be the weak solution of the
following problem
−∇ · (a∇u) = ∇ · g
u=0
in Ω
on ∂Ω.
Then for each p > 2 there exists a positive constant c∗ depending only on n, Ω, a,
and p such that if g ∈ (L2 (Ω))n then we have
k∇ukp ≤ c∗ (kgkp + k∇uk2 ).
Let VD := {v ∈ H 1 (Ω) : v = 0 on ΓD }. By a weak solution to (1.1) we mean a
pair (u, ϕ) such that u ∈ L2 (0, T ; VD (Ω)), ϕ − ϕ0 ∈ L2 (0, T ; H01 (Ω)) and
Z T
Z
Z
Z
hut , vi +
∇u · ∇v dx dt +
βuv ds dt =
σ(u)|∇ϕ|2 v dx dt,
0
Q
ΓT
Q
N
(2.1)
Z
σ(u)∇ϕ · ∇w dx dt = 0,
Q
for all v ∈ L2 (0, T ; H 1 (Ω)) ∩ L∞ (Q) and w ∈ L2 (0, T ; H01 (Ω)), where h·, ·i denotes
duality bracket between VD0 and VD , with VD0 being the dual of VD .
The quadratic term on the right hand side of the first equation in (2.1) can be
dealt with as in [21, 17], and we obtain
Z T
Z
Z
hut , vi dt +
∇u · ∇v dx dt +
βuv ds dt
0
ΓT
Q
ZN
=
(ϕ0 − ϕ)σ(u)∇ϕ · ∇v dx dt + (σ(u)∇ϕ · ∇ϕ0 ) v dx dt,
Q
Q
Z
σ(u)∇ϕ · ∇w dx dt = 0,
Z
(2.2)
Q
for all v ∈ L (0, T ; H (Ω)) ∩ L∞ (Q) and w ∈ L2 (0, T ; H01 (Ω)). By taking p = r in
Lemma 2.1, we can say that for each r > 2, ϕ(·, t) ∈ W 1,r (Ω) for a.e. t ∈ (0, T ).
Therefore, since ∇ϕ(·, t) ∈ Lr (Ω) and ∇ϕ0 (·, t) ∈ L2 (Ω) a.e. t ∈ (0, T ), it follows
0
that there exists s0 such that ∇ϕ(·, t) · ∇ϕ0 (·, t) ∈ Ls (Ω) and
1
1 1
= + .
(2.3)
s0
2 r
Next, since Ω ⊂ R2 it follows v(·, t) ∈ H 1 (Ω) ⊂⊂ Ls (Ω) for s ∈ [1, ∞), conjugate
of s0 in (2.3):
1
1
+ = 1.
(2.4)
0
s
s
Also, by the weak maximum principle (see [17, formula (2.14)])
2
1
kϕkL∞ (Q) ≤ sup |ϕ0 |.
(2.5)
ΓD
Note that r > 2 is chosen from Lemma 2.1, then s0 and s are determined and satisfy
s0 < 2 < s. First, we show existence of solutions to the state system. Define
W (0, T ) := {v ∈ L2 (0, T ; VD ) : vt ∈ L2 (0, T ; VD0 )}.
Theorem 2.2. There exists a weak solution to the state system (1.1).
EJDE-2009/83
OPTIMAL CONTROL FOR A THERMISTOR PROBLEM
5
We show existence of solutions to the state system by using the Schauder’s fixed
point theorem (see, for example [8]). Let R : W (0, T ) → W (0, T ) be the operator
defined as follows R := S ◦ T : ϑ 7→ u, with T : L2 (0, T ; H 1 (Ω)) ∩ L∞ (Q) 3 ϑ 7→
ϕ ∈ L2 (0, T ; H01 (Ω)) where for a given ϑ, the function ϕ solves uniquely
Z
σ(ϑ)∇ϕ · ∇w dx dt = 0,
(2.6)
Q
2
(0, T ; H01 (Ω)),
∞
for all w ∈ L
and where S is defined as follows S : L2 (0, T ; H01 (Ω)) ×
2
1
L (0, T ; H (Ω)) ∩ L (Q) 3 (ϕ, ϑ) 7→ u ∈ L2 (0, T ; H 1 (Ω)) ∩ L∞ (Q), with u being
the unique solution to
Z
Z
Z T
hut , vi dt +
∇u · ∇v dx dt +
βuv ds dt
0
Q
ΓT
N
(2.7)
Z
Z
=
(ϕ0 − ϕ)σ(ϑ)∇ϕ · ∇v dx dt + (σ(ϑ)∇ϕ · ∇ϕ0 ) v dx dt,
Q
Q
for all v ∈ L2 (0, T ; H 1 (Ω)) ∩ L∞ (Q). Standard theory implies that there exists a
unique solution to (2.6), (2.7) which means that the map R is well-defined. Before
we proceed with the proof of the theorem, we need to derive a priori estimates.
Lemma 2.3. Given ϑ ∈ L2 (0, T ; H 1 (Ω)) ∩ L∞ (Q), let ϕ = T (ϑ) and u = S(ϕ, ϑ)
be solutions satisfying (2.6) and (2.7). Then there exist constants c1 , c2 , c3 , and c4
independent of ϑ such that
kϕkL2 (0,T ;H 1 (Ω)) ≤ c1 ,
(2.8)
kukL2 (0,T ;VD (Ω)) ≤ c2 ,
(2.9)
kukL2 (0,T ;L2 (ΓN )) ≤ c3 ,
(2.10)
kut kL2 (0,T ;VD0 (Ω)) ≤ c4 .
(2.11)
Proof of Lemma 2.3. Taking w = ϕ − ϕ0 in (2.6) and using properties of σ, we
obtain (2.8). Taking v = u in (2.7), we have
Z T
Z
Z
βu2 ds dt
hut , ui dt +
|∇u|2 dx dt +
0
ΓT
N
Q
Z
Z
(ϕ0 − ϕ)σ(ϑ)∇ϕ · ∇u dx dt +
=
(σ(ϑ)∇ϕ · ∇ϕ0 ) u dx dt.
Q
Q
Using properties of β we obtain
Z
Z
Z TZ
Z TZ
u2
u2
2
−
+
|∇u| dx dt + λ
u2 ds dt
Ω×{T } 2
Ω×{0} 2
0
Ω
0
ΓN
Z TZ
Z TZ
≤
(ϕ0 − ϕ)σ(ϑ)∇ϕ · ∇u dx dt +
(σ(ϑ)∇ϕ · ∇ϕ0 ) u dx dt.
0
Ω
We can write
Z
Ω×{T }
Z
≤
Ω
0
u2
+
2
Z
u20
+ C3
2
T
Z
0
|∇u|2 dx dt + λ
Ω
Z
T
Z
0
T
Ω
Z
u2 ds dt
ΓN
Z
Z
T
Z
|∇ϕ| · |∇u| dx dt + C4
0
Ω
|∇ϕ| · |u| dx dt
0
Ω
6
V. HRYNKIV
≤ C5 + C6
Z
Z
≤ C5 + C8
EJDE-2009/83
T
{ k∇ϕk2 · k∇uk2 + k∇ϕk2 · kuk2 } dt
0
T
k∇ϕk22
0
1
dt +
2
T
Z
0
Z
|∇u|2 dx dt,
Ω
where we used Poincare inequality. We obtained
Z
Z Z
Z TZ
u2
1 T
|∇u|2 dx dt + λ
+
u2 ds dt
2 0 Ω
Ω×{T } 2
0
ΓN
Z T
≤ C5 + C8
k∇ϕk22 dt ≤ C9 ,
(2.12)
0
where the last inequality in (2.12) follows from (2.8) (also, see the estimate (2.15)
on p. 245 of [17]). This implies estimates (2.9) and (2.10). Using the PDE and the
previous estimates one can show (2.11). This ends the proof of Lemma 2.3.
We continue the proof of Theorem 2.2. Let W0 := {v ∈ W (0, T ) : kvkL2 (0,T ;VD ) ≤
c2 , kvkL2 (0,T ;VD0 ) ≤ c4 , v(x, 0) = u0 (x, 0)}. We show that the operator R : W0 → W0
is continuous with respect to weak convergence in W (0, T ). Then, since W0 is
weakly compact, it will follow that R has a fixed point in W0 . Let {ϑj } ⊂ W0
w
be a sequence such that ϑj * ϑ in W (0, T ), and let uj = u(ϑj ) and ϕj = ϕ(ϑj )
be the corresponding sequences of solutions. A priori estimates imply that on a
subsequence
w
in L2 (0, T ; VD (Ω)),
ϑj * ϑ
∂ϑj w ∂ϑ
*
in L2 (0, T ; VD0 (Ω)),
∂t
∂t
w
uj * u in L2 (0, T ; VD (Ω)),
∂uj w ∂u
*
in L2 (0, T ; VD0 (Ω)),
∂t
∂t
w
ϕj * ϕ in L2 (0, T ; H 1 (Ω)),
w
uj * u
2
2
in L (0, T ; L (ΓN )).
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
Note that by the standard compactness result (see [16]), convergence in (2.13),
(2.14), (2.15), and (2.16), imply that on a subsequence, not relabeled, we have
s
in L2 (Q),
(2.19)
s
2
(2.20)
ϑj → ϑ
uj → u
s
in L (Q),
σ(ϑj ) → σ(ϑ)
in L2 (Q).
(2.21)
Moreover, it can be shown that
s
ϕj → ϕ
s
∇ϕj → ∇ϕ
in L2 (Q),
in L2 (Q).
(2.22)
(2.23)
Note that (2.22) and (2.23) can be obtained by incorporating the structure of the
PDE system as we do not have information on ϕt . For the proof of (2.22) and
(2.23) the reader is referred to [1], p. 1132-1133.
EJDE-2009/83
OPTIMAL CONTROL FOR A THERMISTOR PROBLEM
We have
Z
Z
Z T
h(uj )t , vi dt +
∇uj · ∇v dx dt +
0
Q
βuj v ds dt
ΓT
N
Q
Z
=
7
Z
(2.24)
(ϕ0 − ϕj )σ(ϑj )∇ϕj · ∇v dx dt + (σ(ϑj )∇ϕj · ∇ϕ0 ) v dx dt,
Q
Z
σ(ϑj )∇ϕj · ∇w dx dt = 0,
Q
for all v ∈ L (0, T ; H (Ω)) ∩ L∞ (Q) and w ∈ L2 (0, T ; H01 (Ω)). It is immediate that
Z T
Z T
hujt , vi dt →
hut , vi dt,
0
0
Z
Z
∇uj · ∇v dx dt →
∇u · ∇v dx dt,
Q
Q
Z
Z
βuj v ds dt →
βuv ds dt as j → ∞.
2
1
ΓT
N
ΓT
N
We show
Z
Z
(ϕ0 − ϕj )σ(ϑj )∇ϕj · ∇v dx dt →
(ϕ0 − ϕ)σ(ϑ)∇ϕ · ∇v dx dt.
Q
(2.25)
Q
We estimate the difference between the sequence terms and the limit
Z
Z
(ϕ0 − ϕj )σ(ϑj )∇ϕj · ∇v dx dt − (ϕ0 − ϕ)σ(ϑ)∇ϕ · ∇v dx dt
Q
Q
Z
≤
σ(ϑj )(ϕ0 − ϕj )∇ϕj · ∇v − σ(ϑ)(ϕ0 − ϕj )∇ϕ · ∇v dx dt
Q
Z
σ(ϑ)(ϕ0 − ϕj )∇ϕ · ∇v − σ(ϑ)(ϕ0 − ϕ)∇ϕ · ∇v dx dt := A + B,
+
Q
where B :=
w
R
Q
σ(ϑ)(ϕ−ϕj )∇ϕ·∇v dx dt → 0 by the weak convergence ϕj * ϕ in
0
L2 (0, T ; Ls (Ω)) (by Lemma 2.1 and the fact that σ(ϑ)∇ϕ · ∇v ∈ L2 (0, T ; Ls (Ω))).
For term A, we have
Z
A:=
(ϕ0 − ϕj )(σ(ϑj ) − σ(ϑ))∇ϕj · ∇v dx dt
Q
Z
+
(ϕ0 − ϕj )σ(ϑ)(∇ϕj − ∇ϕ) · ∇v dx dt = A1 + A2 .
Q
Note that
Z
(ϕ0 − ϕj )(σ(ϑj ) − σ(ϑ))∇ϕj · ∇v dx dt
A1 : =
Q
Z
T
Z
≤ 2M̃ K
|ϑj − ϑ| · |∇ϕj | · |∇v| dx dt
0
Ω
T
Z
≤C
kϑj − ϑks k∇ϕj kr k∇vk2 dt
0
Z
≤ C sup k∇ϕj kr
t
T
kϑj − ϑks k∇vk2 dt
0
8
V. HRYNKIV
EJDE-2009/83
≤ Ckϑj − ϑkL2 (0,T ;Ls (Ω)) k∇vkL2 (Q) → 0
by the Simon compactness result [16], i.e., W (0, T ) ⊂⊂ L2 (0, T ; Ls (Ω)). Also, note
sup k∇ϕj kr ≤ Cϕ0 ,
(2.26)
t
by [10, formula (2.9)], and Cϕ0 here denotes a constant which depends on the
s
boundary data ϕ0 . Hence on a subsequence ϑj → ϑ in L2 (0, T ; Ls (Ω)). In the
above, C is a generic constant and M̃ := supQ |ϕ0 |. For A2 , we write
Z
A2 ≤
(ϕ0 − ϕ)σ(ϑ)(∇ϕj − ∇ϕ) · ∇v dx dt
Q
Z
+
(ϕ − ϕj )σ(ϑ)(∇ϕj − ∇ϕ) · ∇v dx dt → 0
Q
by (2.23). Similarly, it can be shown that
Z
Z
σ(ϑj )(∇ϕj · ∇ϕ0 )v dx dt →
σ(ϑ)(∇ϕ · ∇ϕ0 )v dx dt,
Q
Q
Z
Z
σ(ϑj )∇ϕj · ∇w dx dt →
σ(ϑ)∇ϕ · ∇w dx dt.
Q
Q
Letting j → ∞ in (2.24), we conclude by the Schauder theorem that R has a fixed
point in W0 and hence (u, ϕ) with ϕ = T (u) solves (1.1).
3. Existence of an optimal control
Now we proceed to the proof of existence of an optimal control.
Theorem 3.1. There exists a solution to the optimal control problem (1.5).
Proof. Let {βn }∞
n=1 ⊂ UM be a minimizing sequence,
lim J(βn ) = inf J(β).
n→∞
β∈UM
Let un = u(βn ) and ϕn = ϕ(βn ) be the corresponding solutions to
Z T
Z
Z
hunt , vi dt +
∇un ∇v dx dt +
βn un v ds dt
0
ΓT
N
Q
Z
=
Q
Z
(ϕ0 − ϕn )σ(un )∇ϕn ∇v dx dt + (σ(un )∇ϕn · ∇ϕ0 ) v dx dt,
Q
Z
σ(un )∇ϕn · ∇w dx dt = 0,
(3.1)
Q
for all v ∈ L (0, T ; H (Ω)) ∩ L∞ (Q) and w ∈ L2 (0, T ; H01 (Ω)). Note that the
existence of an optimal control can be derived from the proof of Theorem 2.2 and
the estimates (2.8), (2.9), (2.10), and (2.11). In passing to the limit as n → ∞ in
(3.1), we only need to check the term with βn . We have (in addition to convergences
from the previous section):
2
1
βn * β ∗
w∗
in L∞ (∂Ω × (0, T )),
s
in L2 (0, T ; L2 (∂Ω)).
un → u∗
EJDE-2009/83
OPTIMAL CONTROL FOR A THERMISTOR PROBLEM
9
This implies
Z
Z
βn un v ds →
ΓT
N
β ∗ u∗ v ds as n → ∞.
ΓT
N
Letting n → ∞ in (3.1), we obtain that (u∗ , ϕ∗ ) is a weak solution associated with
β ∗ , i.e., u∗ = u(β ∗ ) and ϕ∗ = ϕ(β ∗ ). Using weak lower semicontinuity of J(β) with
respect to L2 norm one can show that the infimum is achieved at β ∗ .
4. Derivation of the optimality system
To characterize an optimal control, we need to derive an optimality system which
consists of the original state system coupled with an adjoint system. To obtain
the necessary conditions for the optimality system, we differentiate the objective
functional with respect to the control. Since the objective functional also depends
on u which depends on the control β and is coupled to ϕ through the PDE (1.1),
we will need to differentiate u and ϕ with respect to control β.
Theorem 4.1 (Sensitivities). If the boundary data ϕ0 are sufficiently small; i.e.,
if kϕ0 kW 1,∞ (Q) is small enough, then the mapping β 7→ (u, ϕ) is differentiable in
the following sense:
u(β + ε`) − u(β) w
* ψ1 in L2 (0, T ; H01 (Ω)),
ε
ϕ(β + ε`) − ϕ(β) w
* ψ2 in L2 (0, T ; H01 (Ω)) as ε → 0
ε
(4.1)
for any β ∈ UM and ` ∈ L∞ (∂Ω × (0, T )) such that (β + ε`) ∈ UM for small ε.
Moreover, the sensitivities, ψ1 ∈ L2 (0, T ; H01 (Ω)) and ψ2 ∈ L2 (0, T ; H01 (Ω)), satisfy
−(ψ1 )t + ∆ψ1 + σ 0 (u)|∇ϕ|2 ψ1 + 2σ(u)∇ϕ · ∇ψ2 = 0
0
∇ · [σ (u)ψ1 ∇ϕ + σ(u)∇ψ2 ] = 0
in Ω × (0, T ),
in Ω × (0, T ),
∂ψ1
+ βψ1 + `u = 0 on ΓN × (0, T ),
∂n
ψ1 = 0 on ΓD × (0, T ),
ψ2 = 0
ψ1 = 0
(4.2)
on ∂Ω × (0, T ),
on Ω × {0}.
Proof. Recall that we denoted: u = u(β) and ϕ = ϕ(β). Now we also denote
uε = u(β ε ), ϕε = ϕ(β ε ), where β ε := β + ε`. The weak formulation for (uε , ϕε ) is
Z
T
huεt , vi dt +
0
Z
∇uε ∇v dx dt +
β ε uε v ds dt
ΓT
N
Q
Z
=
Z
Z
(ϕ0 − ϕε )σ(uε )∇ϕε ∇v dx dt + (σ(uε )∇ϕε · ∇ϕ0 ) v dx dt,
Q
Q
Z
ε
ε
σ(u )∇ϕ · ∇w dx dt = 0,
Q
(4.3)
10
V. HRYNKIV
EJDE-2009/83
for all v ∈ L2 (0, T ; H 1 (Ω)) ∩ L∞ (Q) and w ∈ L2 (0, T ; H01 (Ω)). Similarly, for (u, ϕ)
Z T
Z
Z
hut , ṽi dt +
∇u∇ṽ dx dt +
βuṽ ds dt
0
Q
ΓT
ZN
=
(ϕ0 − ϕ)σ(u)∇ϕ∇ṽ dx dt + (σ(u)∇ϕ · ∇ϕ0 ) ṽ dx dt,
Q
Q
Z
σ(u)∇ϕ · ∇w̃ dx dt = 0,
Z
(4.4)
Q
for all ṽ ∈ L (0, T ; H (Ω)) ∩ L∞ (Q) and w̃ ∈ L2 (0, T ; H01 (Ω)). Take test functions
(uε − u)/ε and (ϕε − ϕ)/ε, subtract the corresponding equations, and divide by ε:
Z T ε
Z
ut − ut uε − u
uε − u uε − u ,
dt +
∇
dx dt
∇
ε
ε
ε
ε
0
Q
Z
uε − u uε − u ds dt
+
β
ε
ε
ΓT
N
Z
uε − u =−
`uε
ds dt
ε
ΓT
N
(4.5)
Z h
i
1
uε − u ε
ε
ε
+
(ϕ0 − ϕ )σ(u )∇ϕ − (ϕ0 − ϕ)σ(u)∇ϕ · ∇
dx dt
ε Q
ε
Z h
i
uε − u 1
σ(uε )∇ϕε − σ(u)∇ϕ · ∇ϕ0
dx dt,
+
ε Q
ε
Z h
ϕε − ϕ i
ϕε − ϕ 1
σ(uε )∇ϕε · ∇
− σ(u)∇ϕ · ∇
dx dt = 0.
ε Ω
ε
ε
2
1
We present the detailed derivation of the L2 (0, T ; H 1 (Ω)) estimate for (ϕε − ϕ)/ε.
Since (ϕε − ϕ)/ε ∈ L2 (0, T ; H01 (Ω)) it follows from Poincaré’s inequality that it is
enough to derive a bound on k∇(ϕε − ϕ)/εkL2 (Q) .
From the second equation in (4.5) we obtain
Z
Z
ϕε ϕε − ϕ ϕ
ϕε − ϕ ε
σ(u )∇
·∇
dx dt =
σ(u)∇
·∇
dx dt.
(4.6)
ε
ε
ε
ε
Q
Q
Taking into account (4.6) we can write
Z
ϕε − ϕ 2
σ(u) ∇
dx dt
ε
Q
Z
ϕε − ϕ ϕε − ϕ =
σ(u)∇
·∇
dx dt
ε
ε
Q
Z
Z
ϕε ϕε − ϕ ϕ
ϕε − ϕ =
σ(u)∇
·∇
dx dt −
σ(u)∇
·∇
dx dt
(4.7)
ε
ε
ε
ε
Q
Q
Z
Z
ϕε ϕε − ϕ ϕε ϕε − ϕ =
σ(u)∇
·∇
dx dt −
σ(uε )∇
·∇
dx dt
ε
ε
ε
ε
Q
Q
Z
σ(u) − σ(uε ) ε
ϕε − ϕ ∇ϕ · ∇
dx dt.
=
ε
ε
Q
Using (4.6) and (4.7) we obtain
Z TZ
ϕε − ϕ 2
C1
∇
dx dt
ε
0
Ω
EJDE-2009/83
OPTIMAL CONTROL FOR A THERMISTOR PROBLEM
Z
T
11
ϕε − ϕ 2
dx dt
ε
Ω
σ(u) − σ(uε ) ε
ϕε − ϕ ∇ϕ · ∇
dx dt
ε
ε
Z
σ(u) ∇
≤
Z0
=
Q
T
ϕε − ϕ uε − u
ε
·
k∇ϕ
k
·
∇
dt
r
s
2
ε
ε
0
Z T
uε − u
ϕε − ϕ ≤ K sup k∇ϕε kr
· ∇
dt
s
2
ε
ε
t
0
uε − u
ϕε − ϕ ≤ Cϕ0
∇
,
2
s
L (0,T ;L (Ω))
L2 (0,T ;L2 (Ω))
ε
ε
where we used (2.26). We can write
ϕε − ϕ uε − u
uε − u
∇
≤
C
≤
C̃
.
2
2
s
L (Q)
L (0,T ;L (Ω))
L2 (0,T ;H01 (Ω))
ε
ε
ε
Z
≤K
(4.8)
Since L2 (0, T ; H 1 (Ω)) ⊂ L2 (0, T ; Ls (Ω)), it follows by Poincaré’s inequality
ϕε − ϕ
ε
L2 (0,T ;H 1 (Ω))
≤ Cϕ0
uε − u
ε
L2 (0,T ;H 1 (Ω))
.
(4.9)
Now we proceed to the derivation of H 1 -norm estimate of (uε − u)/ε using (4.5):
Z
Z
Z
1
uε − u 2
u ε − u 2
uε − u 2
∇
dx dt + λ
dx +
ds dt
2 Ω×{T }
ε
ε
ε
ΓT
Q
N
Z
uε − u ≤ −
`uε
ds dt
ε
ΓT
Z Nh
i uε − u 1
(ϕ0 − ϕε )σ(uε )∇ϕε − (ϕ0 − ϕ)σ(u)∇ϕ ∇
dx dt
+
ε Q
ε
Z h
i
1
uε − u +
σ(uε )∇ϕε − σ(u)∇ϕ ∇ϕ0
dx dt
ε Q
ε
Z
uε − u = −
`uε
ds dt + C + D .
ε
ΓT
N
We have
Z
Z
Z
1
uε − u 2
u ε − u 2
u ε − u 2
dx +
∇
dx dt + λ
ds dt
2 Ω×{T }
ε
ε
ε
Q
ΓT
N
Z
Z
uε − u
uε − u
ε
≤
|`| · |u | ·
ds dt + |C| + |D| ≤ M
|uε | ·
ds dt + |C| + |D|
ε
ε
ΓT
ΓT
N
N
uε − u
+ |C| + |D|
L2 (0,T ;L2 (ΓN ))
ε
ε
u −u
≤ c8 kuε kL2 (0,T ;H 1 (Ω)) ·
+ |C| + |D|.
L2 (0,T ;L2 (ΓN ))
ε
We estimate C term first
Z
σ(uε )∇ϕε − σ(u)∇ϕ
uε − u |C| ≤
|ϕ0 − ϕ| ·
· ∇
dx dt
ε
ε
Q
Z
uε − u ϕ − ϕε
σ(uε )|∇ϕε | · ∇
dx dt
+
ε
ε
Q
≤ c7 kuε kL2 (0,T ;L2 (ΓN )) ·
12
V. HRYNKIV
EJDE-2009/83
σ(uε ) − σ(u)
uε − u · |∇ϕε | · ∇
dx dt
ε
ε
Q
Z
ϕε − ϕ uε − u + 2M̃
σ(u) · ∇
· ∇
dx dt
ε
ε
Q
Z
uε − u ϕ − ϕε
+ C2
· |∇ϕε | · ∇
dx dt
ε
ε
Q
Z
uε − u
uε − u ≤ c2
· |∇ϕε | · ∇
dx dt
ε
ε
Q
Z
uε − u ϕε − ϕ · ∇
dx dt
+ c3
∇
ε
ε
Q
Z
uε − u ϕ − ϕε
+ c4
· |∇ϕε | · ∇
dx dt.
ε
ε
Q
Z
≤ 2M̃
Note that c2 , c3 , c5 , and c6 depend on the data ϕ0 . We continue estimating |C|:
Z
Z
uε − u
uε − u ϕε − ϕ uε − u |C| ≤ c9
· ∇
dx dt + c3
∇
· ∇
dx dt
ε
ε
ε
ε
Q
Q
Z
uε − u ϕε − ϕ
+ c10
· ∇
dx dt
ε
ε
Q
Z T
Z T
uε − u
uε − u ϕε − ϕ ≤ c9
·
∇
dt
+
c
∇
3
2
2
2
ε
ε
ε
0
0
Z T
ε
ε
ε
u − u
u − u
ϕ −ϕ
dt + c10
· ∇
dt
× ∇
2
2
2
ε
ε
ε
0
Z T
Z T
uε − u uε − u ϕε − ϕ ≤ c11
∇
·
∇
dt
+
c
∇
3
2
2
2
ε
ε
ε
0
0
Z
T
uε − u ϕε − ϕ
uε − u dt + c10
· ∇
dt
× ∇
2
2
2
ε
ε
ε
0
uε − u uε − u ϕε − ϕ ∇
≤ c11 ∇
·
+
c
∇
3
L2 (Q)
L2 (Q)
L2 (Q)
ε
ε
ε
uε − u ϕε − ϕ
uε − u × ∇
+ c10
· ∇
L2 (Q)
L2 (Q)
L2 (Q)
ε
ε
ε
uε − u 2
uε − u 2
uε − u 2
≤ c11 ∇
+ c12 ∇
+ c13 ∇
L2 (Q)
L2 (Q)
L2 (Q)
ε
ε
ε
uε − u 2
,
≤ c14 ∇
L2 (Q)
ε
where we used Poincaré’s inequality on (uε − u)/ε and the Cauchy inequality. Similarly, for D we obtain
Z
Z
uε − u
uε − u
ϕε − ϕ uε − u
|D| ≤ c5
· |∇ϕε | ·
dx dt + c6
·
dx dt
∇
ε
ε
ε
ε
Q
Q
uε − u 2
≤ c15 ∇
.
L2 (Q)
ε
Notice that c14 and c15 depend on ϕ0 . Hence, we obtain
Z
Z
uε − u 2
uε − u 2
uε − u 2
1
dx + ∇
+
λ
ds dt
2 (Q)
L
2 Ω×{T }
ε
ε
ε
ΓT
N
EJDE-2009/83
OPTIMAL CONTROL FOR A THERMISTOR PROBLEM
13
uε − u
uε − u 2
+
c
∇
16
2
2
L (0,T ;L (ΓN ))
L2 (Q)
ε
ε
ε
ε
λ u −u 2
u − u 2
≤ c17 kuε k2L2 (0,T ;H 1 (Ω)) +
.
2 (0,T ;L2 (Γ )) + c16 ∇
L
L2 (Q)
N
2
ε
ε
Observe that kuε k2L2 (0,T ;H 1 (Ω)) is bounded by (2.9). Take ϕ0 small enough to get
Z
1
λ uε − u 2
u ε − u 2
uε − u 2
≤ c18 ,
dx + k ∇
2 (Q) +
L
L2 (0,T ;L2 (ΓN ))
2 Ω×{T }
ε
ε
2
ε
≤ c8 kuε kL2 (0,T ;H 1 (Ω)) ·
where k := 1 − c16 > 0. Collecting the estimates and using the pde (4.3) to get the
estimate of the time derivative quotient, gives
uε − u
≤ c,
L2 (0,T ;H01 (Ω))
ε
uε − u
≤ c,
L2 (0,T ;L2 (ΓN ))
ε
uεt − ut
≤ c,
L2 (0,T ;H −1 (Ω))
ε
ε
ϕ −ϕ
≤ c,
L2 (0,T ;H 1 (Ω))
ε
where the generic constant c > 0 does not depend on ε. Claim: we can derive a
better estimate on (ϕε − ϕ)/ε than we have. This better estimate is needed to pass
to the limit as ε → 0 in the derivation of the sensitivity equations. Indeed, we can
write for a.e. t ∈ (0, T )
∇ · (σ(uε )∇ϕε ) = 0
∇ · (σ(u)∇ϕ) = 0
in Ω,
in Ω.
Subtract these equations as well as appropriate boundary conditions to obtain
i
h σ(uε ) − σ(u))
ϕε − ϕ ∇ · σ(u)∇
∇ϕε
in Ω,
=∇·
ε
ε
(4.10)
ε
ϕ −ϕ
= 0 on ∂Ω.
ε
By Lemma 2.1, for a.e. t ∈ (0, T ), we have k∇ϕε k2r ≤ c, where r > 2 and ϕε solves
∇ · (σ(uε )∇ϕε ) = 0
ε
ϕ = ϕ0
in Ω,
on ∂Ω.
Also, since Ω ⊂ R2 it follows that H 1 (Ω) ⊂ Lq (Ω) for any q ∈ [1, ∞). Thus, we can
write for a.e. t ∈ (0, T ),
uε − u
uε − u
≤
≤ c.
(4.11)
2r
H 1 (Ω)
ε
ε
Therefore, returning to (4.10), we obtain
σ(uε ) − σ(u)
ϕε − ϕ ϕε − ϕ ≤ c∗
∇ϕ r + ∇
.
(4.12)
∇
r
2
ε
ε
ε
We can estimate the first term on the right hand side of (4.12),
Z
σ(uε ) − σ(u)
σ(uε ) − σ(u) r
r
∇ϕ r =
· |∇ϕε |r dx
ε
ε
Ω
Z
uε − u r
· |ϕε |r dx
≤C
ε
Ω
14
V. HRYNKIV
≤C
EJDE-2009/83
uε − u
ε
2r
2r
+ k∇ϕε k2r
2r ≤ c.
Therefore,
ϕε − ϕ ϕε − ϕ
≤ C,
≤ C,
r
r
ε
ε
where a generic constant C > 0 is independent of ε. This implies that
∇
s
ϕε → ϕ
s
∇ϕε → ∇ϕ
(4.13)
in Lr (Ω),
in Lr (Ω)
as ε → 0. Moreover, by reiterating this argument we can show that for a.e. t ∈
(0, T )
ϕε − ϕ
ϕε − ϕ ≤
C,
≤ C.
(4.14)
∇
2r
2r
ε
ε
This proves the claim. Now we can proceed to the derivation of sensitivity equations, starting with
Z T ε
Z
Z
ut − ut
uε − u uε − u , v dt +
∇
∇v dx dt +
β
v ds dt
ε
ε
ε
0
Q
ΓT
N
Z
Z h
i
1
ε
`u v ds dt +
=−
(ϕ0 − ϕε )σ(uε )∇ϕε − (ϕ0 − ϕ)σ(u)∇ϕ · ∇v dx dt
ε Q
ΓT
ZN h
i
1
+
σ(uε )∇ϕε − σ(u)∇ϕ · ∇ϕ0 v dx dt,
ε Q
Z h
i
1
σ(uε )∇ϕε − σ(u)∇ϕ · ∇w dx dt = 0.
ε Ω
(4.15)
The following convergence are immediate
Z T ε
Z T
E
ut − ut
, v dt →
(ψ1 )t , v dt,
ε
Z 0 ε
Z0
u − u
∇
∇v dx dt →
∇ψ1 · ∇v dx dt,
ε
Q
Q
Z
Z
uε − u β
v ds dt →
βψ1 v ds dt,
ε
ΓT
ΓT
N
N
Z
Z
ε
`u v ds dt →
`uv ds dt.
ΓT
N
ΓT
N
We write the second term on the right hand side of (4.15) in the form
Z h
i
1
(ϕ0 − ϕε )σ(uε )∇ϕε · ∇v − (ϕ0 − ϕ)σ(u)∇ϕ · ∇v dx dt
ε Q
Z
h
i
1
(ϕ0 − ϕε ) σ(uε )∇ϕε − σ(u)∇ϕ · ∇v dx dt
=
ε Q
Z
1
+
(ϕ − ϕε )σ(u)∇ϕ · ∇v dx dt
ε Q
= G + F.
EJDE-2009/83
OPTIMAL CONTROL FOR A THERMISTOR PROBLEM
15
As far as the term F is concerned, we have that σ(u)∇ϕ · ∇v ∈ L2 (Q) and (ϕ −
w
ϕε )/ε * −ψ2 in L2 (Q), and therefore
Z
F →−
ψ2 σ(u)∇ϕ · ∇v dx dt
Q
as ε → 0. Now we consider terms coming from G. We write
Z
h
i
1
G=
(ϕ0 − ϕε ) σ(uε )∇ϕε − σ(u)∇ϕ · ∇v dx dt
ε Q
Z
h
i
1
(ϕ0 − ϕε ) σ(uε ) − σ(u) ∇ϕε · ∇v dx dt
=
ε Q
Z
h
i
1
+
(ϕ0 − ϕε )σ(u) ∇ϕε − ∇ϕ · ∇v dx dt = G1 + G2
ε Q
w
Using (4.14) and the fact that ∇(ϕε − ϕ)/ε * ∇ψ2 in L2 (Q), one can show
Z
G2 →
(ϕ0 − ϕ)σ(u)∇ψ2 · ∇v dx dt as ε → 0.
(4.16)
Q
To illustrate terms from G1 , observe that
σ(uε ) − σ(u) s 0
→ σ (u)ψ1 in L2 (0, T ; Ls (Ω)) as ε → 0.
ε
Indeed, since N = 2 we have H01 (Ω) ⊂⊂ Lq (Ω) for any q ∈ [1, ∞), L2 (0, T ; H01 (Ω)) ⊂
L2 (0, T ; Ls (Ω)), and
uε − u
ε
L2 (0,T ;Ls (Ω))
≤c
uε − u
ε
L2 (0,T ;H01 (Ω))
≤ C.
(4.17)
Therefore,
uεt − ut
≤ C,
(4.18)
L2 (0,T ;H −1 (Ω))
ε
where (4.18) can be derived from the PDE. Since 1s + 1r + 12 = 1 and r > 2,
2r
we have that s = r−2
> 2. If we define W̃ := {v : v ∈ L2 (0, T ; Ls (Ω)), vt ∈
2
−1
L (0, T ; H (Ω))}, then, because H01 (Ω) ⊂⊂ Ls (Ω) ⊂ H −1 (Ω), it follows from
[16] that W̃ ⊂⊂ L2 (0, T ; Ls (Ω)). Hence, the estimates (4.18) imply that on a
subsequence
uε − u
− ψ1 L2 (0,T ;Ls (Ω)) → 0 as ε → 0,
ε
and therefore,
σ(uε ) − σ(u)
− σ 0 (u)ψ1 L2 (0,T ;Ls (Ω)) → 0 as ε → 0.
ε
Now, we are ready show
Z
σ(uε ) − σ(u) ε
def
G1 =
(ϕ0 − ϕε )
∇ϕ · ∇v dx dt
ε
Q
Z
→
(ϕ0 − ϕ)σ 0 (u)ψ1 ∇ϕ · ∇v dx dt.
(4.19)
Q
To show this convergence, we estimate
Z
Z
σ(uε ) − σ(u) ε
(ϕ0 − ϕε )
∇ϕ · ∇v dx dt − (ϕ0 − ϕ)σ 0 (u)ψ1 ∇ϕ · ∇v dx dt
ε
Q
Q
16
V. HRYNKIV
EJDE-2009/83
n σ(uε ) − σ(u) o
(ϕ0 − ϕ)
∇ϕε − σ 0 (u)ψ1 ∇ϕ · ∇v dx dt
ε
Q
Z
ε
σ(u ) − σ(u)
(ϕ − ϕε )
+
− σ 0 (u)ψ1 + σ 0 (u)ψ1 ∇ϕε · ∇v dx dt
ε
Q
Z n
h σ(uε ) − σ(u)
i
(ϕ0 − ϕ)
≤
− σ 0 (u)ψ1 ∇ϕε · ∇v
ε
Q
o
+ (ϕ0 − ϕ)σ 0 (u)ψ1 ∇(ϕε − ϕ) · ∇v dx dt
Z
σ(uε ) − σ(u)
(ϕ − ϕε )
− σ 0 (u)ψ1 ∇ϕε · ∇v
+
ε
Q
Z
≤
+ (ϕ − ϕε )σ 0 (u)ψ1 ∇ϕε · ∇v dx dt
Z
n σ(uε ) − σ(u) o
(ϕ0 − ϕ)
− σ 0 (u)ψ1 ∇ϕε · ∇v dx dt
≤
ε
Q
Z
+
(ϕ0 − ϕ)σ 0 (u)ψ1 ∇(ϕε − ϕ) · ∇v dx dt
Q
Z
+
(ϕ − ϕε )
Q
Z
+
σ(uε ) − σ(u)
− σ 0 (u)ψ1 ∇ϕε · ∇v dx dt
ε
(ϕ − ϕε )σ 0 (u)ψ1 ∇ϕε · ∇v dx dt := I + II + III + IV.
Q
We deal with the term I first. We have
Z TZ
σ(uε ) − σ(u)
I ≤ kϕ − ϕ0 kL∞ (Q)
− σ 0 (u)ψ1 · |∇ϕε | · |∇v| dx dt
ε
0
Ω
Z T
σ(uε ) − σ(u)
− σ 0 (u)ψ1 s · k∇ϕε kr · k∇vk2 dt
≤ kϕ − ϕ0 kL∞ (Q)
ε
0
Z T
σ(uε ) − σ(u)
≤ kϕ − ϕ0 kL∞ (Q) sup k∇ϕε kr
− σ 0 (u)ψ1 s · k∇vk2 dt
ε
t
0
σ(uε ) − σ(u)
≤C
− σ 0 (u)ψ1 L2 (0,T ;Ls (Ω)) · k∇vkL2 (Q) → 0 as ε → 0,
ε
by (4.19). Similarly, we obtain
Z TZ
II :=
(ϕ0 − ϕ)σ 0 (u)ψ1 ∇(ϕε − ϕ) · ∇v dx dt
0
Z
≤C
Ω
T
kψ1 ks · k∇(ϕε − ϕ)kr · k∇vk2 dt
0
≤ Ckψ1 kL2 (0,T ;Ls (Ω)) · k∇vkL2 (Q) · sup k∇(ϕε − ϕ)kr → 0
as ε → 0,
t
where we used (4.13). Analogously, it can be shown that III, IV → 0 as ε → 0.
One can show that the third term on the right hand side of (4.15) satisfies
Z h
i
1
σ(uε )∇ϕε − σ(u)∇ϕ · ∇ϕ0 v dx dt
ε Q
Z
Z
→
σ 0 (u)ψ1 ∇ϕ · ∇ϕ0 v dx dt +
σ(u)∇ψ2 · ∇ϕ0 v dx dt as ε → 0.
Q
Q
EJDE-2009/83
OPTIMAL CONTROL FOR A THERMISTOR PROBLEM
17
Finally, the equation for ϕ in (4.15) can be shown to satisfy
Z h
i
1
σ(uε )∇ϕε · ∇w − σ(u)∇ϕ · ∇w dx dt
ε Q
Z
Z
→
σ 0 (u)ψ1 ∇ϕ · ∇w dx dt +
σ(u)∇ψ2 · ∇w dx dt as ε → 0.
Q
Q
Letting ε → 0 in (4.15), we obtain
Z
Z
Z T
(ψ1 )t , v dt +
∇ψ1 ∇v dx dt +
0
Z
=
(βψ1 + `u)v ds dt
ΓT
N
Q
Z
0
(ϕ0 − ϕ)σ (u)ψ1 ∇ϕ · ∇v dx dt + (ϕ0 − ϕ)σ(u)∇ψ2 · ∇v dx dt
Q
Z
Z
−
ψ2 σ(u)∇ϕ · ∇v dx dt +
σ 0 (u)ψ1 ∇ϕ · ∇ϕ0 v dx dt
Q
Q
Z
+
σ(u)∇ψ2 · ∇ϕ0 v dx dt,
Q
Z
(σ 0 (u)ψ1 ∇ϕ + σ(u)∇ψ2 ) · ∇w dx dt = 0
Q
(4.20)
Q
for all v ∈ L (0, T ; H 1 (Ω)) ∩ L∞ (Q) and w ∈ L2 (0, T ; H01 (Ω)). This leads to the
sensitivity system (4.2).
The weak formulation for (4.2) is given by
Z T
Z
Z
− (ψ1 )t , v dt +
∇ψ1 · ∇v dx dt +
(βψ1 + `u)v ds dt
0
Q
ΓT
N
(4.21)
Z
Z
0
2
=
σ (u)ψ1 |∇ϕ| v dx dt + 2
σ(u)∇ϕ · ∇ψ2 v dx dt,
Q
Q
Z
(σ 0 (u)ψ1 ∇ϕ + σ(u)∇ψ2 ) · ∇w dx dt = 0
(4.22)
2
Q
for all v ∈ L (0, T ; H (Ω)) ∩ L∞ (Q) and w ∈ L2 (0, T ; H01 (Ω)) which can be shown
to be equivalent to (4.20).
2
1
To characterize the optimal control, we need to introduce adjoint functions and
∂p
+ β ∗ p = 0 on ΓN × (0, T ),
the adjoint operator associated with ψ1 and ψ2 . Using ∂n
q = ψ2 = 0 on ∂Ω × (0, T ), p = 0 on ΓD × (0, T ), p(x, T ) = 0, and the notation L
for the operator in the sensitivity system gives:
Z
ψ1
p q L
dx dt
ψ2
Q
Z
Z
Z
=
p (−ψ1 )t dx dt +
p ∆ψ1 dx dt +
pσ 0 (u)|∇ϕ|2 ψ1 dx dt
Q
Q
Q
Z
Z
Z
+2
pσ(u)∇ϕ · ∇ψ2 dx dt +
q∇ · [σ(u)∇ψ2 ] dx dt +
q∇ · [σ 0 (u)ψ1 ∇ϕ] dx dt
Q
Q
Q
Z
Z
Z
0
2
=
ψ1 pt dx dt +
ψ1 ∆p dx dt +
ψ1 σ (u)|∇ϕ| p dx dt
Q
Q
Q
Z
Z
−2
ψ2 ∇ · [pσ(u)∇ϕ] dx dt −
ψ1 σ 0 (u)∇ϕ · ∇q dx dt
Q
Q
18
V. HRYNKIV
EJDE-2009/83
Z
+
ψ2 ∇ · [σ(u)∇q] dx dt
p
ψ1 ψ2 L∗
dx dt,
q
Q
Z
=
Q
where
L∗
p
pt + ∆p + σ 0 (u)|∇ϕ|2 p − σ 0 (u)∇ϕ · ∇q
=
.
q
∇ · [−2pσ(u)∇ϕ + σ(u)∇q]
Thus, the adjoint system is given by
pt + ∆p + σ 0 (u)|∇ϕ|2 p − σ 0 (u)∇ϕ · ∇q = 1 in Ω × (0, T ),
∇ · [−2pσ(u)∇ϕ + σ(u)∇q] = 0
in Ω × (0, T ),
∂p
+ β ∗ p = 0 on ΓN × (0, T ),
∂n
p = 0 on ΓD × (0, T ),
(4.23)
q = 0 on ∂Ω × (0, T ),
p=0
on Ω × {T },
where the nonhomogeneous term “1” comes from differentiating the integrand of
J(β) with respect to the state u.
Theorem 4.2. Let kϕ0 kW 1,∞ (Q) be sufficiently small. Then, given an optimal
control β ∗ ∈ UM and corresponding states u, ϕ, there exists a solution (p, q) ∈
L2 (0, T ; H01 (Ω)) × L2 (0, T ; H01 (Ω)) to the adjoint system (4.23). Furthermore, β ∗
can be explicitly characterized as:
up ,λ ,M .
(4.24)
β ∗ (x, t) = min max −
2
Proof. The weak formulation of (4.23) is
Z T
Z
Z
hpt , vi dt −
∇p · ∇v dx dt −
β ∗ p v ds dt
0
Q
ΓN ×(0,T )
Z
Z
Z
0
2
0
+
σ (u)|∇ϕ| p v dx dt −
σ (u)(∇ϕ · ∇q)v dx dt =
v dx dt,
Q
Q
Q
Z
Z
2
pσ(u)∇ϕ · ∇w dx dt −
σ(u)∇q · ∇w dx dt = 0
Q
2
(4.25)
Q
1
∞
for all v ∈ L (0, T ; H (Ω)) ∩ L (Q) and w ∈ L2 (0, T ; H01 (Ω)).
We show that the solution to the adjoint system (4.23) exists. Define the map
F : L2 (0, T ; H 1 (Ω)) × L2 (0, T ; H01 (Ω)) 7→ L2 (0, T ; H 1 (Ω)) × L2 (0, T ; H01 (Ω)) with
F (V, W ) = (p, q) as follows:
pt + ∆p + σ 0 (u)|∇ϕ|2 V − σ 0 (u)∇ϕ · ∇W = 1
∇ · [−2V σ(u)∇ϕ + σ(u)∇q] = 0
in Ω × (0, T ),
∂p
+ β ∗ p = 0 on ΓN × (0, T ),
∂n
p = 0 on ΓD × (0, T ),
q = 0 on ∂Ω × (0, T ),
p=0
on Ω × {T },
in Ω × (0, T ),
(4.26)
EJDE-2009/83
OPTIMAL CONTROL FOR A THERMISTOR PROBLEM
The weak formulation of (4.26) is
Z T
Z
Z
hpt , Θi dt −
∇p · ∇Θ dx dt −
0
Z
β ∗ pΘ ds dt
ΓT
N
Q
0
19
Z
2
Z
0
σ (u)|∇ϕ| V Θ dx dt −
σ (u)(∇ϕ · ∇W )Θ dx dt =
Θ dx dt,
Q
Q
Z
Z
2
V σ(u)∇ϕ · ∇Ψ dx dt =
σ(u)∇q · ∇Ψ dx dt
+
Q
Q
2
Q
1
∞
for all Θ ∈ L (0, T ; H (Ω)) ∩ L (Q) and Ψ ∈ L2 (0, T ; H01 (Ω)). We use the Banach
fixed point theorem. Let F (V1 , W1 ) = (p1 , q1 ) and F (V2 , W2 ) = (p2 , q2 ). We show
the contraction property
(4.27)
kp1 − p2 kH 1 + kq1 − q2 kH01 ≤ δ kV1 − V2 kH 1 + kW1 − W2 kH01
for some 0 < δ < 1. Indeed, for F (V1 , W1 ) = (p1 , q1 ), we have
Z T
Z
Z
β ∗ p1 Θ ds dt
h(p1 )t , Θi dt −
∇p1 · ∇Θ dx dt −
0
Z
0
+
Q
ΓT
N
Q
Z
2
Z
0
σ (u)|∇ϕ| V1 Θ dx dt −
σ (u)(∇ϕ · ∇W1 )Θ dx dt =
Θ dx dt,
Q
Q
Z
Z
2
V1 σ(u)∇ϕ · ∇Ψ dx dt =
σ(u)∇q1 · ∇Ψ dx dt,
Q
Q
and similarly, for F (V2 , W2 ) = (p2 , q2 ). Let us denote p̄ := p1 − p2 , q̄ := q1 − q2 ,
W̄ := W1 − W2 , and V̄ := V1 − V2 . Take Θ = p̄, then equations for p1 and p2 give
Z T
Z
Z
h(p̄)t , p̄i dt +
σ 0 (u)|∇ϕ|2 V̄ p̄ dx dt −
σ 0 (u)∇ϕ · ∇W̄ p̄ dx dt
0
Q
Q
Z
Z
2
∗ 2
=
|∇p̄| dx dt +
β p̄ ds dt.
ΓT
N
Q
Thus, we obtain
Z
Z
Z
1
p̄2 dx +
|∇p̄|2 dx dt + λ
p̄2 ds dt
2 Ω×{0}
Q
ΓT
N
Z
Z
0
2
0
≤
σ (u)|∇ϕ| V̄ p̄ dx dt +
σ (u)∇ϕ∇W̄ p̄ dx dt .
Q
Q
We have
Z
σ 0 (u)|∇ϕ|2 V̄ p̄ dx dt ≤ C sup k|∇ϕ|2 kr
Q
t
Z
T
kV̄ ks · kp̄k2 dt
0
≤ Cϕ kV̄ kL2 (0,T ;Ls (Ω)) kp̄kL2 (Q)
≤ Cϕ kV̄ kL2 (0,T ;Ls (Ω)) kp̄kL2 (0,T ;H 1 (Ω)) ,
and
Z
Q
σ 0 (u)∇ϕ · ∇W̄ p̄ dx dt ≤ C̃ϕ k∇W̄ kL2 (Q) kp̄kL2 (0,T ;H 1 (Ω)) .
20
V. HRYNKIV
EJDE-2009/83
The above three inequalities give
kp̄kL2 (0,T ;H 1 (Ω)) ≤ Cϕ kV̄ kL2 (0,T ;H 1 (Ω)) + C̃ϕ kW̄ kL2 (0,T ;H 1 (Ω)) ,
(4.28)
where the constants Cϕ and C̃ϕ depend on the boundary data ϕ0 . A similar estimate
can be derived for q̄
kq̄kL2 (0,T ;H 1 (Ω)) ≤ Cϕ0 kV̄ kL2 (0,T ;H 1 (Ω)) .
(4.29)
Adding (4.28) and (4.29) we get
kp̄kL2 (0,T ;H 1 (Ω)) + kq̄kL2 (0,T ;H 1 (Ω)) ≤ cϕ (kV̄ kL2 (0,T ;H 1 (Ω)) + kW̄ kL2 (0,T ;H 1 (Ω)) ).
Now, choosing ϕ0 so that kϕ0 kW 1,∞ (Q) is small, we obtain δ < 1. This proves
(4.27), the contraction property, which gives the desired fixed point of the map and
therefore the existence of solutions to the adjoint system.
Recall that for variation ` ∈ L∞ (ΓTN ), with β ∗ + ε` ∈ UM for ε small, the weak
formulation of the sensitivity system (4.2) is given by equations (4.21) and (4.22).
Now we are ready to characterize the optimal control. Since the minimum of J is
achieved at β ∗ and for small ε > 0, β ∗ +ε` ∈ UM , and denoting uε = u(β ∗ +ε`), ϕε =
ϕ(β ∗ + ε`), we obtain
J(β ∗ + ε`) − J(β ∗ )
0 ≤ lim
ε
ε→0+
Z
Z
=
ψ1 dx dt +
2β ∗ ` ds dt
ΓT
Q
N
Z
1
ψ1 ψ 2
=
dx dt +
2β ∗ ` ds dt
0
T
Q
ΓN
Z
Z
Z
= − h(ψ1 )t , pi dt −
∇p∇ψ1 dx dt −
Z
Q
Z
0
β ∗ p ψ1 ds dt
ΓT
N
Q
Z
ψ1 σ 0 (u)∇ϕ · ∇q dx dt
Z
Z
Z
+2
pσ(u)∇ϕ · ∇ψ2 dx dt −
σ(u)∇q · ∇ψ2 dx dt +
2
ψ1 σ (u)|∇ϕ| p dx dt −
+
Q
Q
Q
=
n
Z
−
Z
∇p · ∇ψ1 dx dt −
2β ∗ ` ds dt
ΓT
N
Q
β ∗ p ψ1 ds dt
ΓT
N
Q
Z
Z
o
ψ1 σ 0 (u)|∇ϕ|2 p dx dt + 2
pσ(u)∇ϕ · ∇ψ2 dx dt
Q
Q
Z
h Z
i Z
0
+ −
ψ1 σ (u)∇ϕ · ∇q dx dt −
σ(u)∇q · ∇ψ2 dx dt +
+
Q
Z
=
Q
Z
` up ds dt +
ΓT
N
2β ∗ ` ds dt
ΓT
N
2β ∗ ` ds dt,
ΓT
N
where we integrated by parts and used the sensitivity system (4.21), (4.22) with
test functions p, q. Thus, we obtain
Z
`(2β ∗ + up) ds ≥ 0.
(4.30)
ΓT
N
We use this inequality with three cases to characterize β ∗ :
EJDE-2009/83
OPTIMAL CONTROL FOR A THERMISTOR PROBLEM
21
(i) Take the variation ` to have support on the set {x ∈ ΓTN : λ < β ∗ (x, t) < M }.
The variation `(x, t) can be of any sign, therefore we obtain 2β ∗ + up = 0 whence
β ∗ = − up
2 .
(ii) On the set {(x, t) ∈ ΓN × (0, T ) : β ∗ (x, t) = M }, the variation must satisfy
`(x, t) ≤ 0 and therefore we get 2β ∗ + up ≤ 0 implying M = β ∗ (x, t) ≤ − up
2 .
(iii) On the set {(x, t) ∈ Γ × (0, T ) : β ∗ (x, t) = λ}, the variation must satisfy
`(x, t) ≥ 0. This implies 2β ∗ + up ≤ 0 and hence λ = β ∗ (x) ≥ − up
2 . Combining
cases (i), (ii), and (iii) gives us the explicit characterization (4.24) of the optimal
control β.
Substituting (4.24) into the state system (1.1) and the adjoint equations (4.23)
we obtain the optimality system:
ut − ∆u − σ(u)|∇ϕ|2 = 0 in Ω × (0, T ),
∇ · (σ(u)∇ϕ) = 0
0
in Ω × (0, T ),
0
2
pt + ∆p + σ (u)|∇ϕ| p − σ (u)∇ϕ · ∇q = 1 in Ω × (0, T ),
∂p
∂n
∂u
∂n
∇ · [−2pσ(u)∇ϕ + σ(u)∇q] = 0 in Ω × (0, T ),
up + min max −
, λ , M p = 0 on ΓN × (0, T ),
2 up
+ min max −
, λ , M u = 0 on ΓN × (0, T ),
2
u = p = 0 on ΓD × (0, T ),
u = u0
ϕ = ϕ0
on Ω × {0},
on ∂Ω × (0, T ),
q = 0 on ∂Ω × (0, T ),
p=0
on Ω × {T },
Note that existence of solution to the above optimality system follows from the
existence of solution to the state system (1.1) and Theorem 4.2.
References
[1] S. N. Antontsev and M. Chipot, The thermistor problem: existence, smoothness, uniqueness,
and blow up, SIAM J. Math. Anal., 25 (1994), pp. 1128–1156.
[2] G. Cimatti, Existence of weak solutions for the nonstationary problem of the Joule heating
of a conductor, Ann. Mat. Pura Appl. (4), 162 (1992), pp. 33–42.
[3] G. Cimatti and G. Prodi, Existence results for a nonlinear elliptic system modeling a temperature dependent electrical resistor, Ann. Mat. Pura Appl. (4), 63 (1989), pp. 227–236.
[4] G. Cimatti, A bound for the temperature in the thermistor’s problem, IMA J. Appl. Math.,
40 (1988), pp. 15–22.
[5] G. Cimatti, Optimal control for the thermistor problem with a current limiting device, IMA
J. Math. Control Inform., 24 (2007), pp. 339–345.
[6] G. Cimatti, Existence of periodic solutionsfor a thermally dependent resistor in a R-C circuit,
Int. J. Engng Sci., vol. 32, no. 1 (1994), pp. 69–78.
[7] A. C. Fowler, I. Frigaard, and S. D. Howison, Temperature surges in current-limiting circuit
devices, SIAM J. Appl. Math., 52 (1992), pp. 998–1011.
[8] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd
edition, Springer-Verlag, Berlin, 1983.
[9] S. D. Howison, J. F. Rodrigues, and M. Shillor, Stationary solutions to the thermistor problem, J. Math. Anal. Appl., 174 (1993), pp. 573–588.
[10] V. Hrynkiv, S. Lenhart, and V. Protopopescu, Optimal control of a convective boundary
condition in a thermistor problem, SIAM J. Control Optim., 47 (2008), pp. 20-39
22
V. HRYNKIV
EJDE-2009/83
[11] K. Kwok, Complete guide to semiconductor devices, McGraw-Hill, New York, 1995.
[12] H-C. Lee and T. Shilkin, Analysis of optimal control problem for the two-dimensional thermistor system, SIAM J. Control Optim., 44 (2005), pp. 268–282.
[13] E. D. Maclen, Thermistors, Electrochemical Publications, Glasgow, 1979.
[14] J. F. Rodriguez, Obstacle problems in mathematical physics, North-Holland Math Studies
#134, Elsevier Science Publishers B.V., Amsterdam, 1987.
[15] J. F. Rodriguez, A nonlinear parabolic system arising in thermomechanics and in thermomagnetism, Math. Models Methods Appl. Sci., vol. 2, no. 3, (1992), pp. 271-281.
[16] J. Simon, Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl. (4) , 146 (1987),
pp. 65–96.
[17] P. Shi, M. Shillor, and X. Xu, Existence of a solution to the Stefan problem with Joule’s
heating, J. Differential Equations, 105 (1993), pp. 239–263.
[18] X. Xu, Local regularity theorems for the stationary thermistor problem, Proc. Roy. Soc.
Edinburgh, 134A (2004), pp. 773–782.
[19] X. Xu, Exponential integrability of temperature in the thermistor problem, Differential Integral Equations, 17 (2004), pp. 571–582.
[20] X. Xu, Local and global existence of continuous temperature in the electrical heating of concuctors, Houston Journal of Mathematics, 22 (1996), pp. 435–455.
[21] X. Xu, A degenerate Stefan-like problem with Joule’s heating, SIAM J. Math. Anal., 23
(1992), pp. 1417–1438.
[22] G. Yuan, Regularity of solutions of the nonstationary thermistor problem, Appl. Anal., 53
(1994), pp. 149–164.
[23] S. Zhou and D. R. Westbrook, Numerical solution of the thermistor equations, J. Comput.
Appl. Math., 79 (1997), pp. 101–118.
Volodymyr Hrynkiv
Department of Computer and Mathematical Sciences, University of Houston-Downtown, One Main St, Houston, TX 77002, USA
E-mail address: HrynkivV@uhd.edu