CUBO A Mathematical Journal
Vol.15, No¯ 02, (111–119). June 2013

Existence and uniqueness solution of a class of quasilinear
parabolic boundary control problems

M. H. Farag1, T. A. Talaat2 and E. M. Kamal3

Minia University,

Department of Mathematics, Faculty of Science, Minia, Egypt.

farag5358@yahoo.com1

talaat.2008@yahoo.com2

esamkamal55@yahoo.com3

ABSTRACT

This paper presents an optimal control of processes described by a quasilinear parabolic

systems with controls in the coefficients of equation, in the boundary condition and in

the right side of this equation. Theorems concarning the existence and uniqueness for

the solution of the cosidering problem are invistigated.

RESUMEN

Este art́ıculo presenta un control óptimo de procesos descritos por un sistema parabólico

cuasilineal con control en los coeficientes de la ecuación, en la condición de frontera y

en el lado derecho de esta ecuación. Se investigan los teoremas relacionados con la

existencia y unicidad para la solución del problema considerado

Keywords and Phrases: Optimal control, Quasilinear Parabolic Equation, Existence and Uniquness

Theorems.

2010 AMS Mathematics Subject Classification: 49J20, 49K20, 49M29, 49M30



112 M. H. Farag, T. A. Talaat and E. M. Kamal CUBO
15, 2 (2013)

1 Introduction

Optimal control problems for partial differential equations are currently of much interest. A larage

amount of the theoretical concept which governed by quasilinear parabolic equations [1-5] has been

investigated in the field of optimal control problems. These problems have dealt with the processes

of hydro- and gasdynamics, heatphysics, filtration, the physics of plasma and others [6-8]. The

study and determination of the optimal regimes of heat conduction processes at a long interval

of the change of temperture gives rise to optimal control problems with respect to a quasilinear

equation of parabolic type. In this work, we consider a constrained optimal control problem with

respect to a quasilinear parabolic equation with controls in the coefficients of the equation. The

existence and uniqueness of the optimal control problem is proved.

2 Statement of the problem

Let D is a bounded domain of the N-dimensional Euclidean space EN; Γ be the boundary of D,

assumed to be sufficiently smooth; ν is the exterior unit normal of Γ; T > 0 be a fixed time ;

Ω = D × (0, T] ; S = Γ × (0, T].

Now we consider a class of optimal control problems governed by the following quasilinear

parabolic system.

L(v)y(x, t) = f(x, t, v2), (x, t) ∈ Ω,

y(x, 0) = φ(x), x ∈ D,

∑n
i=1

λi(y, v0)
∂y
∂xi

cos(ν, xi)|S = g(ζ, t), (x, t) ∈ S (1)

where φ ∈ L2(D), g(ζ, t) ∈ L2(S) are given functions and the differential operator L takes the

following form:

L(v)z(x, t) =
∂z

∂t
−

n∑

i=1

∂

∂xi
[λi(z, v0)

∂z

∂xi
] +

n∑

i=1

Bi(z, v1)
∂z

∂xi
(2)

y(x, t), v = (v0, v1, v2) are the state and the controls rspectively for the system (1).

Furthermore, we consider the functional of the form

Jβ(v) =

∫

S

[y(ζ, t) − f0(ζ, t)]
2dζdt + β

2∑

m=0

‖vm − ωm‖
2
l2
, (3)

which is to minimized under condition (1) and additional restricitions

ν0 ≤ λi(y, v0) ≤ µ0, ν1 ≤ Bi(y, v1) ≤ µ1, r1 ≤ y(x, t) ≤ r2, i = 1, n (4)



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Existence and uniqueness solution of a class of quasilinear . . . 113

over the class

V = {v = (v0, v1, v2) : vm = (v0m, v1m, · · · , vim, · · · ) ∈ l2, ‖vm‖l2 ≤ Rm, m = 0, 2}

and f0(ζ, t) ∈ L2(S) is a given function and β ≥ 0, νj, µj, j = 1, 2, r1, r2,Rm > 0 are positive num-

bers, ωm = (ω0m, ω1m, · · · , ωim, · · · ) ∈ l2, m = 0, 2 are given numbers.

Throughout this paper, we adopt the following assumptions.

Assumption 2.1: V is closed and bonded subset of l2.

Assumption 2.2: The functions Bi(y, v1), λi(y, v0), i = 1, n are continuous on (y, v) ∈

[r1, r2] × l2 have continuous derivatives in y at ∀(y, v) ∈ [r1, r2] × l2 and
∂Bi
∂y

, ∂λi
∂y

, i = 1, nare

bounded.

Assumption 2.3: The function f(x, t, v2) is given function continuous in v2 on l2 for almost

all (x, t) ∈ Ω, bounded and measurable in x, t on Ω ∀v2 ∈ l2.

Assumption 2.4:The functions Bi(y, v1), λi(y, v0), i = 1, n, f(x, t, v2) satisfy a Lipschitz con-

dition for v1, v0, v2 ,then

|Bi(y(x, t), v1 + δv1) − Bi(y(x, t), v1)| ≤ S0(x, t)‖δv1‖l2, i = 1, n

|λi(y(x, t), v0 + δv0) − λi(y(x, t), v0)| ≤ S1(x, t)‖δv0‖l2, i = 1, n

|f(x, t, v2 + δv2) − f(x, t, v2)| ≤ S2(x, t)‖δv2‖l2

for almost all (x, t) ∈ Ω, ∀y ∈ [r1, r2], ∀vm, vm + δvm ∈ l2 such that ‖vm‖l2, ‖vm + δvm‖l2 ≤ Rm
where Sm(x, t) ∈ L∞, m = 0, 2.

Assumption 2.5: The first derivatives of the functions Bi(y, v0), λi(y, v0), i = 1, n and

f(x, t, v2) with respect to v are continuous functions in [r1, r2] × l2 and for any vm ∈ l2 such that

‖vm‖l2 ≤ Rm, m = 0, 2.

Definition 2.1: The problem of finding the function y = y(x, t) ∈ V0,12 (Ω) from condition

(1)-(2) at given v ∈ V is called the reduced problem.

Definition 2.2: A function y = y(x, t) ∈ V1,02 (Ω) is said to be a solution of the problem



114 M. H. Farag, T. A. Talaat and E. M. Kamal CUBO
15, 2 (2013)

(1)-(2), if for all η = η(x, t) ∈ W1,12 (Ω) the equation

∫
Ω
[−y

∂η
∂t

+
∑n

i=1
λi(y, v0)

∂y
∂xi

∂η
∂xi

−
∑n

i=1
Bi(y, v1)(

∂y
∂xi

)η(x, t)

−f(x, t, v2)η(x, t)]dxdt =
∫
D
φ(x)η(x, 0)dx +

∫
S
g(ζ, t)η(ζ, t)dζdt, (5)

is valid and η(x, T) = 0.

It is proved in [8] that, under the foregoing assumptions, a reduced problem (1)-(2) has a

unique solution and | ∂y
∂xi

| ≤ C1, i = 1, n almost at all (x, t) ∈ Ω, ∀v ∈ V, where C1 is a certain

constant.

3 The Existence Theorem

Optimal control problems of the coefficients of differential equations do not always have solution

[9]. Examples in [10] and elswhere of problems of the type (1)-(4) having no solution for β = 0.

A problem of minimization of a functional is said to be unstable, when a minimizing sequare does

not converge to an element minimizing the functional [6].

To begin with, we need

Theorem 3.1 Under the above assumptions for every solution of the reduced problem (1)-(2)

the following estimate is valid:

‖δy‖V1,0
2

(Ω) ≤ C2[‖

√

√

√

√

n∑

i=1

(∆λi
∂y

∂xi
)2‖L2(Ω) + ‖∆f −

n∑

i=1

∆Bi
∂y

∂xi
‖L2(Ω)], (6)

where δy(x, t) = y(x, t; v + δv) − y(x, t; v), δy(x, t) ∈ W1,12 (Ω), ∆λi = λi(u, v0 + δv0) − λi(u, v0)

,∆Bi = Bi(u, v1 + δv1) − Bi(u, v1), ∆f = f(x, t, v2 + δv2) − f(x, t, v2) and C2 ≥ 0 is a constant not

dependent on δv = (δv0, δv1, δv2), δvm ∈ l2, m = 0, 2.

ptoof

Set δy(x, t) = y(x, t, v + δv) − y(x, t; v), y = y(x, t; v), y = y(x, t; v + δv). From (5) it follows that

∫
Ω
[−δy

∂η
∂t

+
∑n

i=1
λi

∂δy
∂xi

∂η
∂xi

+
∑n

i=1
∂λi(y+θ1i,v0+δv0)

∂y
∂y
∂xi

∂η
∂xi

δy

+
∑n

i=1
∆λi

∂y
∂xi

∂η
∂xi

+
∑n

i=1
Bi

∂δy
∂xi

η +
∑n

i=1
∆Bi(

∂y
∂xi

)η

−
∑n

i=1
∂Bi(y+θ2i,v1+δv1)

∂y
∂y
∂xi

δyη − ∆fη]dxdt = 0 (7)



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Existence and uniqueness solution of a class of quasilinear . . . 115

for all η = η(x, t) ∈ W1,12 (Ω) and η(x, T) = 0.

Here θ1i, θ2i ∈ (0, 1), i = 1, n is some number, λi ≡ λi(y + δy, v0 + δv0) ,∆λi ≡ λi(y, v0 +

δv0) − λi(y, v0), Bi ≡ Bi(y + δy, v1 + δv1) ,∆Bi ≡ Bi(y, v1 + δv1) − λi(y, v1), i = 1, n, i = 1, n,

∆f ≡ f(x, t, v2 + δv2) − f(x, t, v2).

Let ηh(x, t) =
1
h

∫t
t−h

η(x, τ)dτ, 0 < h < τ where η = δy(x, t) at (x, t) ∈ Ωt1, zero at

t > t1(t1 ≤ T − h) and Ωt1 = D × (0, t1]. In identity (5) put η(x, t) instead of ηh(x, t), and

following the method in [11,p. 166-168] we obtain

1
2

∫
D
(δy)2dx +

∫
Ωt1

[
∑n

i=1
λi(

∂δy
∂xi

)2 +
∑n

i=1
∂λi(y+θ1i,v0+δv0)

∂y
∂y
∂xi

∂δy
∂xi

δy]dxdt

+
∫
Ωt1

∑n
i=1

∆λi
∂y
∂xi

∂δ
∂xi

dxdt +
∑n

i=1
∂Bi(y+θ2i,v1+δv1)

∂y
∂y
∂xi

(δy)2dxdt

+
∫
Ωt1

∑n
i=1

Bi
∂δy
∂xi

δy +
∫
Ωt1

∑n
i=1

∆Bi(
∂y
∂xi

)δydxdt −
∫
Ωt1

∆fδydxdt = 0 (8)

Hence,from the above assumptions and applying Cauchy Bunyakoviskii inequality, we obtain

1
2

∫
D
(δy(x, t1)

2dx + ν0
∫
Ωt1

∑n
i=1

|
∂δy
∂xi

|2dxdt

≤ (C3 + C4)(
∫
Ωt1

∑n
i=1

|
∂δy
∂xi

|2dxdt)
1
2 (
∫
Ωt1

(δy(x, t))2dxdt)
1
2

+{
∫
Ωt1

∑n
i=1

|∆λi
∂y
∂xi

|2dxdt}
1
2 (
∫
Ωt1

∑n
i=1

|
∂δy
∂xi

|2dxdt)
1
2 + C5

∫
Ωt1

(δy(x, t))2dxdt

−
∫t1
0
{
∫
D
|∆f −

∑n
i=1

∆Bi(
∂y
∂xi

)|
1
2 dx(

∫
D
(δy)2dx)

1
2 }dt, (9)

where C3, C4, C5 are positive constants not depending on δv.

Applying Cauchy’s inequality with ε and combine similar terms, then multiply both sides by

two, we obtain

‖δy(x, t1)‖
2
L2(D)

+ ν0
2
‖
∑n

i=1
∂δy
∂xi

‖2
L2(Ωt1 )

≤ C6‖δy(x, t)‖
2
L2(Ωt1 )

+2{
∫
Ωt1

∑n
i=1

|∆λi
∂y
∂xi

|2dxdt}
1
2 ‖

∑n
i=1

∂δy
∂xi

‖2
L2(Ωt1 )

+2 max0≤τ≤t1 ‖δy(x, τ‖L2(D)
∫t1
0
{
∫
D
|∆f −

∑n
i=1

∆Bi(
∂y
∂xi

)|2dx}
1
2 dt (10)

Now we replace

y(t1) = max
0≤τ≤t1

‖δy(x, τ‖L2(D), ‖δy(x, t)‖
2
L2(Ωt1 )

= t1(y(t1))
2.



116 M. H. Farag, T. A. Talaat and E. M. Kamal CUBO
15, 2 (2013)

This gives us the inequality

‖δy(x, t1)‖
2
L2(D)

+ ν0
2
‖
∑n

i=1
∂δy
∂xi

‖2
L2(Ωt1 )

≤ C6t1(y(t1))
2

+2{
∫
Ωt1

∑n
i=1

|∆λi
∂y
∂xi

|2dxdt}
1
2 ‖

∑n
i=1

∂δy
∂xi

‖2
L2(Ωt1 )

+2y(t1)
∫t1
0
{
∫
D
|∆f −

∑n
i=1

∆Bi(
∂y
∂xi

)|2dx}
1
2 dt ≡ j(t1). (11)

From this follows the two inequalities

(y(t1))
2 ≤ j(t1) (12)

and

‖

n∑

i=1

∂δy

∂xi
‖2L2(Ωt1 )

≤
2

ν0
j(t1) (13)

We take the square root of both sides of (12) and (13), add together the resulting inequalities

and then majorize the right-hand side in the same way in [12] (pp. 117-118) and this proves the

estimate (6). This completes the proof of the theorm.

Corollary 3.1 Under the above assumptions, the right part of estimate (6) converges to zero

at
∑2

m=0
‖δvm‖l2 → 0, therefore

‖δy‖V1,0
2

(Ω) → 0 at

2∑

m=0

‖δvm‖l2 → 0. (14)

Hence from the theorem on trace [13] we get

‖δy‖L2(Ω) → 0, ‖δy‖L2(S) → 0 at

2∑

m=0

‖δvm‖l2 → 0. (15)

Now we consider the functional J0(v) =
∫
S
[y(ζ, t) − f0(ζ, t)]

2dζdt.

Theorem 3.2 The functional J0(v) is continuous on V.

proof

Let δv = (δv0, δv1, δv2), δvm ∈ l2, m = 0, 2 be an increment of control on an element v ∈ V such

that v + δv ∈ V. For the increment of J0(v) we have

∆J0(v) = J0(v + δv) − J0(v) = 2

∫

S

[y(ζ, t) − f0(ζ, t)]δy(ζ, t)dζdt +

∫

S

[δy(ζ, t)]
2
dζdt (16)



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Existence and uniqueness solution of a class of quasilinear . . . 117

Applying the Cauchy-Bunyakovskii inequality, we obtain

|∆J0(v)| ≤ 2‖y(ζ, t) − f0(ζ, t)‖L2(S)‖δy(ζ, t)‖L2(S) + ‖δy(ζ, t)‖
2
L2(S)

(17)

An application of the Corollary 3.1 completes the proof.

Theorem 3.3 For any β ≥ 0 the problem (1)-(4) has a least one solution.

proof

The set of V is closed and bounded in l2. Since J0(v) is continuous on V by Theorem 3.2, so is

Jβ(v) = J0(v) + β

2∑

m=0

‖vm − wm‖
2
l2
. (18)

Then from the Weierstrass theorem [14] it follows that the problem (1)-(4) has a least one

solution. This completes the proof of the theorm.

4 The Uniqueness Theorem

According to the above discussions, we ca easily obtain a theorem concerning solution uniqueness

for the considering optimal control problem (1)-(4).

Theorem 4.1 There exists a dense set K of l2 such that for any ωm ∈ K, m = 0, 2 the problem

(1)-(4) for β > 0 has a unique solution.

proof The functional J0(v) is bounded below, and the foreging establishes that it is continues

on V. Furthermore, l2 is uniformaly convex [12]. It thus follows from a theorm in [16] that the

space l2 contains an everywhere-dense subset K such that the problem (1)-(4) has a unque solution

when ωm ∈ K, m = 0, 2 and β > 0. This completes the proof of the theorm.

5 Conclusion

We have investigated a constrained optimal control problems governed by quasilinear parabolic

equations with controls in the coefficients of the equation. The existence and uniqueness of the

optimal control problem is proved.



118 M. H. Farag, T. A. Talaat and E. M. Kamal CUBO
15, 2 (2013)

6 Acknowledgment

The authers gratefully acknowledgment the referee, who made useful suggestions and remarks

which helped to improve the paper.

Received: September 2011. Accepted: September 2012.

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