Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 1-26

Time-Optimal Control of Infinite Variables Parabolic Systems with
Time Lags Given in Integral Form

G. M. Bahaa1 and M. M. Tharwat 2

1 Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah,
Saudi Arabia

2 DepartmentofMathematics,UniversityCollege,UmmAl-QuraUniversity,Makkah,SaudiArabia
Emall: bahaa−gm@hotmail.com, zahraa26@yahoo.com

Abstract In this paper, the time-optimal control problem for second order parabolic system and also
for (n× n)– parabolic systems with infinite number of variables involving constant time lags appearing
in integral form in both the state equation and in the boundary condition is presented. Some specific
properties of the optimal control are discussed.

Keywords Time-optimal control (n × n) Parabolic systems Operator with an infinite number of

1. Introduction

Distributed parameters systems with delays can be used to describe many phenomena in
the real world. As is well known, heat conduction, properties of elastic-plastic material, fluid
dynamics, diffusion-reaction processes, the transmission of the signals at a certain distance by
using electric long lines, etc., all lie within this area. The object that we are studying (tempera-
ture, displacement, concentration, velocity, etc.) is usually referred to as the state.

The time-optimal control problems of distributed second order parabolic systems with fi-
nite number of variables involving time lags appearing in the boundary condition have been
widely discussed in many papers and monographs. A fundamental study of such problems
is given by (Wang, 1975) and was next developed by (Knowles, 1978) and (Wong, 1987). It
was also intensively investigated by (Kowalewski, 1988; 1990a; 1990b; 1993; 1998; 1999;
2009), (Kowalewski and Duda, 1992 ), (Kowalewski and Krakowiak, 1994; 2000; 2006; 2008),
(Kotarski, 1997), (Kotarski & El-Saify and Bahaa, 2002b), (Kotarski and Bahaa, 2007) and (El-
Saify, 2005; 2006) in which linear quadratic problem for parabolic systems with time delays
given in the different form (constant time delays, time-varying delays, time delays given in the
integral form, etc.) were presented.

The necessary and sufficient conditions of optimality for systems consists of only one equa-
tion and for (n × n) systems governed by different types of partial differential equations de-
fined on spaces of functions of infinitely many variables and also for infinite order systems
are discussed for example in ( Gali, I. M. & El-Saify, H. A. 1982; 1983), (El-Saify & Bahaa,
2001; 2003), (El-Saify, H. A., Serag, H. M, & Bahaa, G. M. 2000), (El-Saify, 2005; 2006),
(Kowalewski, 2009) and (Kowalewski and Krakowiak, 2008) in which the argument of (Lions,
1971 and Lions & Magenes, 1972) were used.

variables Time lag.variables Time lag

ISSN 1078-6236 International Institute for General Systems Studies, Inc.

  



2 Bahaa:Time-Optimal Control of Infinite Variables Parabolic Systems. . . . . .

Making use of the Dubovitskii-Milyutin Theorem in (Kotarski, El-Saify & Bahaa, 2002a,b),
(Bahaa, 2003; 2005a,b; 2008) and (Bahaa and Kotarski, 2008), the necessary and sufficient
conditions of optimality for similar systems governed by second order operator with an infinite
number of variables and also for infinite order systems were investigated. The interest in the
study of this class of operators is stimulated by problems in quantum field theory.

In particular, the papers of (Kowalewski & Krakowiak, 2006, 2008), the time-optimal
boundary control problem for a second order distributed parabolic systems with finite num-
ber of variables in which constant time lags appear in integral form in both the state equation
and the boundary condition is presented. Some particular properties of optimal control are
discussed.

In this paper we recall the problem in a more general formulation. We consider the time-
optimal distributed and boundary control problem for second order parabolic system and also
for (n×n) –second order parabolic systems with infinite number of variables involving constant
time lags appearing in integral form in both the state equation and in the boundary condition
simultaneously. Such an infinite variables parabolic systems can be treated as a generalization
of the mathematical model for a plasma control process. The quadratic performance functional
defined over a fixed time horizon are taken and some constraints are imposed on the boundary
control. Following a line of the Lions scheme (Lions, 1971) and (Lions & Magenes, 1972),
necessary and sufficient optimality conditions for the Neumann and Dirichlet problem applied to
the above systems were derived. The optimal control is characterized by the adjoint equations.

This paper is organized as follows. In section 2, we introduce Sobolev spaces with infinite
number of variables. In section 3, we formulate the mixed Neumann problem for infinite vari-
ables parabolic systems involving time lags. In section 4, the time-distributed control problem
for this case is formulated, then we give the necessary and sufficient conditions for the time
control to be an optimal. In section 5, we concluded and generalized our results.

2. Sobolev Spaces with Infinite Number of Variables

This section covers the basic notations, definitions and properties, which are necessary to
present this work (Berezanskii, 1975), ( Gali & El-Saify 1982; 1983), (El-Saify & Serag &
Bahaa, 2000) and (El-Saify & Bahaa, 2001).

Let (pk(t))
∞
k=1 be a sequence of weights, fixed in all that follows, such that;

0 < pk(t) ∈ C∞(R1),

∫
R1

pk(t)dt = 1,

with respect to it we introduce on the region R∞ = R1 × R1 × . . . , the measure dρ(x) by
setting,

dρ(x) = p1(x1)dx1 ⊗ p2(x2)dx2 ⊗ . . . , (R∞ 3 x = (xk)
∞
k=1, xk ∈ R1).

On R∞ we construct the space L2(R∞, dρ(x)) with respect to this measure i.e., L2(R∞, dρ(x))
is the space of quadratic integrable functions on R∞. We shall often set L2(R∞, dρ(x)) =
L2(R∞).



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 3

It is classical result that L2(R∞) is a Hilbert space for the scalar product

(φ, ψ)L2(R∞) =

∫
R∞

φ(x)ψ(x)dρ(x).

We next consider a Sobolev space in the case of an unbounded region. For functions which
are ` = 1, 2, . . . times continuously differentiable up to the boundary Γ of R∞ ( Γ is meant to
be the boundary of the support of the measure dρ(x)) and which vanish in a neighborhood of
∞, we introduce the scalar product

(φ, ψ)W `(R∞) =
∑
|α|≤`

(Dαφ,Dαψ)L2(R∞),

where Dα is defined by

Dα =
∂|α|

(∂x1)α1(∂x2)α2 · · ·
, |α| =

∞∑
i=1

αi,

and the differentiation is taken in the sense of generalized functions on R∞, and after the com-
pletion, we obtain the Sobolev space W `(R∞). So in short, Sobolev space W 1(R∞) is defined
by :

W 1(R∞) = {φ|φ,Dφ ∈ L2(R∞)}.

As in the case of a bounded region, the space W 1(R∞) form the space with positive
norm ||.||W 1(R∞). We can construct the space W−1(R∞) = (W 1(R∞))∗ with negative norm
||.||W−1(R∞) with respect to the space W 0(R∞) = L2(R∞) with zero norm ||.||L2(R∞), then
we have the following equipped,

W 1(R∞) ⊆ L2(R∞) ⊆W−1(R∞),

||φ||W 1(R∞) ≥ ||φ||L2(R∞) ≥ ||φ||W -1(R∞).

LetL2(0, T ;W 1(R∞)) be the space of square integrable measurable functions t→ φ(t) of ]0, T [→
W 1(R∞), where the variable t denotes the “ time ”; t ∈]0, T [, T <∞. This space is a Hilbert
space with respect to the scalar product

(φ, ψ)L2(0,T ;W 1(R∞)) =

∫ T

0
(φ(t), ψ(t))W 1(R∞)dt,

and its dual is the spaceL2(0, T ;W−1(R∞)), analogously, we can define the spacesL2(0, T ;L2(R∞))
which we shall denote by L2(Q).

Let Ω ⊂ R∞ is a bounded, open set with boundary Γ, which is aC∞ manifold of dimension
(n − 1). Locally, Ω is totally on one side of Γ and denote by W 1(Ω,R∞, dρ(x)) (briefly
W 1(Ω,R∞)) the Sobolev space of vector function y(x) defined on Ω.

The construction of the Cartesian product of n-times to the above Hilbert spaces can be
construct, for example

(W 1(Ω,R∞))n = W 1(Ω,R∞)×W 1(Ω,R∞)× · · · ×W 1(Ω,R∞)︸ ︷︷ ︸
n−times

=

n∏
i=1

(W 1(Ω,R∞))i,



4 Bahaa:Time-Optimal Control of Infinite Variables Parabolic Systems. . . . . .

with norm defined by:

||φ||(W 1(Ω,R∞))n =
n∑
i=1

||φi||W 1(Ω,R∞),

where φ = (φ1, φ2, ..., φn) = (φi)
n
i=1 is a vector function and φi ∈W 1(Ω,R∞).

Finally, we have the following chain:

(L2(0, T ;W 1(Ω,R∞)))n ⊆ (L2(Q))n ⊆ (L2(0, T ;W−1(Ω,R∞)))n,

where (L2(0, T ;W−1(Ω,R∞)))n are the dual spaces of (L2(0, T ;W 1(Ω,R∞)))n. The spaces
considered in this paper are assumed to be real.

3. Existence and Uniqueness of Solutions

Consider now the distributed-parameter system described by the following parabolic delay
equation:

∂y

∂t
+A(t)y +

∫ b

a
c(x, t)y(x, t− h) dh = u, x ∈ Ω, t ∈ (0, T ), h ∈ (a, b), (1)

y(x, t′) = Φ0(x, t′), x ∈ Ω, t′ ∈ [−b, 0), (2)

y(x, 0) = y0(x), x ∈ Ω, (3)

∂y(x, t)

∂ηA
=

∫ b

a
d(x, t)y(x, t− h) dh+ v, x ∈ Γ, t ∈ (0, T ), h ∈ (a, b), (4)

y(x, t′) = Ψ0(x, t′), x ∈ Γ, t′ ∈ [−b, 0), (5)

where Ω and Γ have the same properties as in Section 2. We have

y ≡ y(x, t;u), u ≡ u(x, t), v ≡ v(x, t),

Q ≡ Ω× (0, T ), Q ≡ Ω× [0, T ], Q0 ≡ Ω× [−b, 0)

Σ ≡ Γ× (0, T ), Σ0 ≡ Γ× [−b, 0),

T is a specified positive number representing a time horizon, c is a given real C∞ function
defined on Q, d is a given real C∞ function defined on Σ, h is a time lag such that h ∈ (a, b)
and a > 0, Φ0 and Ψ0 are initial functions defined on Q0 and Σ0, respectively.

The parabolic operator
∂

∂t
+ A(t) in the state equation (1) is a second order parabolic

operator with infinite number of variables andA(t) (Berezanskii, 1975), (Gali & El-Saify, 1982;
1983) and (Kotarski & El-Saify & Bahaa, 2002b ) is given by:

A(t)y(x) =

(
−
∞∑
k=1

1√
pk(xk, t)

∂2

∂x2
k

√
pk(xk, t) + q(x, t)

)
y(x)

= −
∞∑
k=1

D2
ky(x) + q(x, t)y(x),

(6)



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 5

where
Dky(x) =

1√
pk(xk, t)

∂

∂xk

√
pk(xk, t)y(x), (7)

and q(x, t) is a real-valued function in x which is a bounded and measurable on Ω ⊂ R∞, such
that q(x, t) ≥ ξ0 > 1, ξ0 is a constant. The operatorA(t) is a bounded second order self-adjoint
elliptic partial differential operator with an infinite number of variables maps W 1(Ω,R∞) onto
W−1(Ω,R∞).

For this operator we define the bilinear form as follows:

Definition 3.1. For each t ∈ (0, T ), we define a family of bilinear forms on W 1(Ω,R∞) by:

π(t; y, φ) = (A(t)y, φ)L2(Ω,R∞), y, φ ∈W 1(Ω,R∞), (8)

where A(t) maps W 1(Ω,R∞) onto W−1(Ω,R∞) and takes the above form. Then

π(t; y, φ) =

(
A(t)y, φ

)
L2(Ω,R∞)

=

(
−
∞∑
k=1

D2
ky(x) + q(x, t)y(x), φ(x)

)
L2(Ω,R∞)

=

∫
Ω

∞∑
k=1

Dky(x)Dkφ(x) dρ(x) +

∫
Ω
q(x, t)y(x)φ(x) dρ(x).

Lemma 3.1. The bilinear form π(t; y, φ) is coercive on W 1(Ω,R∞), that is

π(t; y, y) ≥ λ ||y||2W 1(Ω,R∞), λ > 0. (9)

Proof. It is well known that the ellipticity ofA(t) is sufficient for the coerciveness of π(t; y, φ)
on W 1(Ω,R∞).

π(t;φ, ψ) =

∫
Ω

∞∑
k=1

Dkφ(x)Dkψ(x) dρ+

∫
Ω
q(x, t)φ(x)ψ(x) dρ.

Then

π(t; y, y) =

∫
Ω

∞∑
k=1

|Dky(x)|2 dρ(x) +

∫
Ω
q(x, t)|y(x)|2 dρ(x)

≥
∞∑
k=1

||Dky(x)||2L2(Ω,R∞) + ξ0||y(x)||2L2(Ω,R∞)

= ||y(x)||2W 1(Ω,R∞) + ξ0||y(x)||2L2(Ω,R∞)

≥ ||y(x)||2W 1(Ω,R∞)

= λ||y||2W 1(Ω,R∞), λ > 0.



6 Bahaa:Time-Optimal Control of Infinite Variables Parabolic Systems. . . . . .

Also we have:

∀y, φ ∈W 1(Ω,R∞) the function t→ π(t; y, φ) is continuously differentiable in (0, T ) and
π(t; y, φ) = π(t;φ, y)

}
(10)

Equations (1)–(5) constitute a Neumann problem. Then the left-hand side of the boundary
condition (4) may be written in the following form:

∂y(u)

∂ηA
=
∞∑
k=1

(Dky(u)) cos(n, xk) = g(x, t), (11)

where
∂

∂ηA
is a normal derivative at Γ, directed towards the exterior of Ω, cos(n, xk) is the

k − th direction cosine of n, with n being the normal at Γ exterior to Ω, and

g(x, t) =

∫ b

a
d(x, t)y(x, t− h) dh+ v(x, t), x ∈ Γ, t ∈ (0, T ), h ∈ (a, b). (12)

First we shall prove sufficient conditions for the existence of a unique solution of the mixed
initial boundary value problem (1)–(5) for the cases where the control u or v belong to
L2(Q) or L2(Σ) respectively.

To this purpose, for any pair of real numbers r, s ≥ 0, we introduction the Sobolev space
W r,s(Q) (Lions and Magenes, 1972, Vol. 2, p. 6) defined by

W r,s(Q) = L2 (0, T ;W r(Ω,R∞)) ∩W s
(
0, T ;L2(Ω,R∞)

)
(13)

which is a Hilbert space normed by(∫ T

0
||y(t)||2W r(Ω,R∞)dt+ ||y||2W s(0,T ;L2(Ω,R∞))

)1/2

, (14)

where W s
(
0, T ;L2(Ω,R∞)

)
denotes the Sobolev space of order s of functions defined on

(0, T ) and taking values in L2(Ω,R∞).

The existence of a unique solution for the mixed initial-boundary value problem (1)–(5)
on the cylinder Q can be proved using a constructive method, i.e., first, solving (1)–(5) on
the sub-cylinder Q1 and in turn on Q2, and so on, until the procedure covers the whole cylinder
Q. In this way, the solution in the previous step determines the next one.

For simplicity, we introduce the following notation:

Ej , ((j − 1)a, ja), Qj = Ω× Ej , Σj = Γ× Ej , j = 1, 2, . . . . (15)

Case 1: u ∈ L2(Q)

Using Theorem 6.1 of Lions & Magenes (1972, vol. 2, p. 33), we can prove the following
lemma.



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 7

Lemma 3.2. Let
u ∈ L2(Q), (16)

fj(x, t) ∈ L2(Qj), (17)

where

fj = u(x, t)−
∫ b

a
c(x, t)yj−1(x, t− h) dh,

yj−1(·, (j − 1)a) ∈W 1(Ω,R∞), (18)

gj ∈W
1
2
, 1
4 (Σj), (19)

where

gj(x, t) =

∫ b

a
d(x, t)yj−1(x, t− h) dh+ v(x, t).

Then, there exists a unique solution yj ∈ W 2,1(Qj) for the mixed initial-boundary value prob-
lem (1), (4) and (18).

Proof. We observe that for j = 1, y0|Q0(x, t−h) = Φ0(x, t−h) and y0|Σ0(x, t−h) = Ψ0(x, t−
h). Then the assumptions (17)–(19) are fulfilled if we assume that Φ0 ∈ W 2,1(Q0), y0 ∈
W 1(Ω,R∞), v ∈ W

1
2
, 1
4 (Σ) and Ψ0 ∈ W

1
2
, 1
4 (Σ0). These assumptions are sufficient to ensure

the existence of a unique solution y1 ∈ W 2,1(Q1). In order to extend the result to Q2, we have
to prove that y1(·, a) ∈W 1(Ω,R∞), g2 ∈W

1
2
, 1
4 (Σ2) and f2 ∈ L2(Q2). Really, from Theorem

3.1, p.19 of Lions & Magenes vol.1, y1 ∈ W 2,1(Q1) implies that the mapping t → y1(·, t)
is continuous from [0, a] → W 1(Ω,R∞). Thus y1(·, a) ∈ W 1(Ω,R∞). Then using the trace
theorem of Lions & Magenes (1972, vol. 2, p. 9) we can verify that y1 ∈ W 2,1((Q1) implies
that y1 → y1|Σ1 is a linear, continuous mapping of W 2,1(Q1)→ W

1
2
, 1
4 (Σ1). Assuming that d

is a C∞ function and v ∈ W
1
2
, 1
4 (Σ), the condition g2 ∈ W

1
2
, 1
4 (Σ2) is fulfilled. Also it is easy

to notice that the assumption (17) follows from the fact that y1 ∈ W 2,1(Q1) and u ∈ L2(Q).
Then, there exists a unique solution y2 ∈ W 2,1(Q2). Finally, we can extend our result to any
Qj , j = 3, 4 . . ..

Theorem 3.3. Let y0, Φ0, Ψ0, v and u be given with y0 ∈ W 1(Ω,R∞), Φ0 ∈ W 2,1(Q0), v ∈
W

1
2
, 1
4 (Σ), Ψ0 ∈W

1
2
, 1
4 (Σ0) and u ∈ L2(Q). Then, there exists a unique solution y ∈W 2,1(Q)

for the mixed initial-boundary value problem (1)–(5). Moreover, y(·, ja) ∈W 1(Ω,R∞) for
j = 1, 2, . . ..

Case 2: v ∈ L2(Σ)

Using Theorem 15.2 of Lions & Magenes (1972, vol. 2, p. 81), we can prove the following
lemma.

Lemma 3.4. Let
u ∈W−

1
2
,− 1

4 (Q), v ∈ L2(Σ) (20)

fj ∈W−
1
2
,− 1

4 (Qj), (21)

yj−1(·, (j − 1)a) ∈W
1
2 (Ω,R∞), (22)



8 Bahaa:Time-Optimal Control of Infinite Variables Parabolic Systems. . . . . .

gj ∈ L2(Σj). (23)

Then, there exists a unique solution yj ∈ W
3
2
, 3
4 (Qj) for the mixed initial-boundary value

problem (1), (4) and (22).

Proof. For j = 1, the assumptions (21)–(23) are fulfilled if we assume that Φ0 ∈W
3
2
, 3
4 (Q0),

y0 ∈ W
1
2 (Ω,R∞) and Ψ0 ∈ L2(Σ0). These assumptions are sufficient to ensure the existence

of a unique solution y1 ∈ W
3
2
, 3
4 (Q1). In order to extend the result to Q2, we have to prove

that y1(·, a) ∈ W
1
2 (Ω,R∞), y1|Σ1 ∈ L2(Σ1) and f2 ∈ W−

1
2
,− 1

4 (Q2). First using Theorem
3.1 of Lions & Magenes (1972, vol. 1, p. 19) we can prove that y1 ∈ W

3
2
, 3
4 (Q1) implies that

the mapping t → y1(·, t) is continuous from [0, a] → W
3
4 (Ω,R∞) ⊂ W

1
2 (Ω,R∞). Hence

y1(·, a) ∈ W
1
2 (Ω,R∞). Again, from trace theorem of Lions & Magenes (1972, vol. 2, p. 9),

we can verify that y1 ∈ W
3
2
, 3
4 (Q1) implies that y1 → y1|Σ1 is a linear, continuous mapping

of W
3
2
, 3
4 (Q1) → W 1, 1

2 (Σ1). Thus y1|Σ1 ∈ L2(Σ1). Moreover, it is worth mentioning that the
assumption (21) follows from the fact that y1 ∈ W

3
2
, 3
4 (Q1) and u ∈ W−

1
2
,− 1

4 (Q). Then,
there exists a unique solution y2 ∈ W

3
2
, 3
4 (Q2). Finally, we can extend our result to any Qj ,

j = 3, 4 . . ..

Theorem 3.5. Let y0, Φ0, Ψ0, v and u be given with y0 ∈W
1
2 (Ω,R∞), Φ0 ∈W

3
2
, 3
4 (Q0), Ψ0 ∈

L2(Σ0), v ∈ L2(Σ) and u ∈ W−
1
2
,− 1

4 (Q). Then, there exists a unique solution y ∈ W
3
2
, 3
4 (Q)

for the mixed initial-boundary value problem (1)–(5). Moreover, y(·, ja) ∈ W
1
2 (Ω,R∞)

for j = 1, 2, . . ..

Now we shall verify the existence of a unique solution for the problem (1), (2), (3)
and (5) with the Dirichlet boundary condition involving a time lag

y(x, t) = g(x, t) (24)

where g is given by the formula (12).

Making use of the results of Lions & Magenes (1972, vol. 2, p. 33 and p. 81) we can prove
the following lemmas and theorems.

Case 3: u ∈ L2(Q)

Lemma 3.6. Let
u ∈ L2(Q), (25)

fj ∈ L2(Qj), (26)

yj−1(·, (j − 1)a) ∈W 1(Ω,R∞), (27)

gj ∈W
3
2
, 3
4 (Σj), (28)

and the following compatibility relation is fulfilled

yj−1(x, (j − 1)a) = gj(x, (j − 1)a), on Γ. (29)

Then, there exists a unique solution yj ∈ W 2,1(Qj) for the mixed initial-boundary value prob-
lem (1), (24) and (27).



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 9

Proof. For j = 1, the assumptions (26)–(28) can be satisfied if we assume that Φ0 ∈
W 2,1(Q0), v ∈ W

3
2
, 3
4 (Σ) and Ψ0 ∈ W

3
2
, 3
4 (Σ0). These assumptions are sufficient to ensure

the existence of a unique solution y1 ∈ W 2,1(Q1) if y0 ∈ W 1(Ω,R∞) and the following
compatibility relation is satisfied

y0(x, 0) = g1(x, 0), on Γ. (30)

In order to extend the result to Q2, we have to prove that y1 ∈ W 2,1(Q1) and it is necessary to
impose the compatibility relation

y1(x, a) = g2(x, a), on Γ (31)

and it is sufficient to verify that
f2 ∈ L2(Q2), (32)

y1(·, a) ∈W 1(Ω,R∞), (33)

g2 ∈W
3
2
, 3
4 (Σ2). (34)

First using the solution in the previous step and the condition (25) we can prove immediately
the condition (32). To verify (33), we use the fact that y1 ∈ W 2,1(Q1) implies that the
mapping t → y1(·, t) is continuous from [0, a] → W 1(Ω,R∞) (by Theorem 3.1 of Lions &
Magenes (1972, vol. 1, p. 19)), hence y1(·, a) ∈W 1(Ω,R∞). From the trace theorem of Lions
& Magenes (1972, vol. 2, p. 9) y1 ∈W 2,1((Q1) implies that y1 → y1|Σ1 is a linear, continuous
mapping of W 2,1(Q1) → W

3
2
, 3
4 (Σ1). Assuming that d is a C∞ function and v ∈ W

3
2
, 3
4 (Σ),

the condition (34) is fulfilled. Then, there exists a unique solution y2 ∈ W 2,1(Q2). Finally,
we can extend our result to any Qj , j = 3, 4 . . ..

Theorem 3.7. Let y0, Φ0, Ψ0, v and u be given with y0 ∈ W 1(Ω,R∞), Φ0 ∈ W 2,1(Q0),
v ∈ W

3
2
, 3
4 (Σ), Ψ0 ∈ W

3
2
, 3
4 (Σ0), u ∈ L2(Q) and the compatibility relation (29) is fulfilled.

Then, there exists a unique solution y ∈W 2,1(Q) for the mixed initial-boundary value problem
(1), (2), (3) (5) and (24) with y(·, ja) ∈W 1(Ω,R∞) for j = 1, 2, . . ..

Case 4: v ∈ L2(Σ)

Lemma 3.8. Let
u ∈W−

3
2
,− 3

4 (Q), v ∈ L2(Σ) (35)

fj ∈W−
3
2
,− 3

4 (Qj), (36)

yj−1(·, (j − 1)a) ∈W−
1
2 (Ω,R∞), (37)

gj ∈ L2(Σj). (38)

Then, there exists a unique solution yj ∈ W
1
2
, 1
4 (Qj) for the mixed initial-boundary value

problem (1), (24) and (37).



10 Bahaa:Time-Optimal Control of Infinite Variables Parabolic Systems. . . . . .

Proof. We observe that for j = 1, the assumptions (36)–(38) are satisfied if we assume that
Φ0 ∈ W

1
2
, 1
4 (Q0), y0 ∈ W−

1
2 (Ω,R∞) and Ψ0 ∈ L2(Σ0). These assumptions are sufficient to

ensure the existence of a unique solution y1 ∈ W
1
2
, 1
4 (Q1). Next for j = 2, using the solution

in the first step, it is sufficient to verify that f2 ∈ W−
3
2
,− 3

4 (Q2), y1(·, a) ∈ W−
1
2 (Ω,R∞) and

y1|Σ1 ∈ L2(Σ1). Then it is worth mentioning that the condition f2 ∈ W−
3
2
,− 3

4 (Q2) follows
from the fact that y1 ∈ W

1
2
, 1
4 (Q1) and u ∈ W−

3
2
,− 3

4 (Q). Since y1 ∈ W
1
2
, 1
4 (Q1) implies

that the mapping t → y1(·, t) is continuous from [0, a] → W
1
4 (Ω,R∞) (by Theorem 3.1 of

Lions & Magenes (1972, vol. 1, p. 19)), hence y1(·, a) ∈ W
1
4 (Ω,R∞) ⊂ L2(Ω,R∞) ⊂

W−
1
4 (Ω,R∞) ⊂ W−

1
2 (Ω,R∞). We shall prove that y1|Σ1 ∈ L2(Σ1). We must notice that for

proving y1|Σ1 ∈ L2(Σ1) we cannot use the trace theorem of Lions & Magenes (1972, vol. 2,
p. 9), since y1 ∈W

1
2
, 1
4 (Q1). It is worth mentioning that this difficulty can be avoided by using

the condition y1|Σ1 = g1 ∈ L2(Σ1). This implies that y1|Σ1 ∈ L2(Σ1). Then, there exists a
unique solution y2 ∈W

1
2
, 1
4 (Q2). Finally, we can extend our result to any Qj , j = 3, 4 . . ..

Theorem 3.9. Let y0, Φ0, Ψ0, v and u be given with y0 ∈ W−
1
2 (Ω,R∞), Φ0 ∈ W

1
2
, 1
4 (Q0),

Ψ0 ∈ L2(Σ0), v ∈ L2(Σ) and u ∈ W−
3
2
,− 3

4 (Q). Then, there exists a unique solution y ∈
W

1
2
, 1
4 (Q) for the mixed initial-boundary value problem (1), (2), (3), (5) and (24).

Moreover, y(·, ja) ∈W−
1
2 (Ω,R∞) for j = 1, 2, . . ..

4. Optimal Distributed Control

Now, we shall restrict our considerations to the case of the distribute control for the Neu-
mann problem. Therefore, we shall formulate the minimum-time problem for (1)–(5) in the
context of the Theorem 3.3, i.e.,

u ∈ U = {u ∈ L2(Q) : |u(x, t)| ≤ 1}. (39)

We shall define the reachable set H such that

H = {y ∈ L2(Ω,R∞) : ||y − zd||L2(Ω,R∞) ≤ ε} (40)

where zd ∈ L2(Ω,R∞) and ε > 0.

Solving the stated minimum-time problem is equivalent to hitting the target set H in min-
imum time, that is, minimizing the time t, for which y(t;u) ∈ H and u ∈ U . Moreover, we
assume that

there exists a T > 0 and u ∈ U with y(T ;u) ∈ H (41)

then we have the following theorem

Theorem 4.1. If the assumption (41) holds, then the setH is reached in minimum time t∗ by
an admissible control u∗ ∈ U . Moreover∫

Ω
[zd − y(t∗;u∗)] [y(t∗;u)− y(t∗;u∗)] dρ ≤ 0, ∀u ∈ U. (42)



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 11

Proof. Let us define the following set

t∗ := inf{t : y(t;u) ∈ H for some u ∈ U} (43)

The minimum is well defined, as (41) guarantees that this set is nonempty. By definition, we
can choose tn ↓ t∗ and admissible controls {un} such that

y(tn;un) ∈ H, n = 1, 2, 3, . . . . (44)

Each un is defined on Ω × (0, tn) ⊃ Ω × (0, t∗). To simplify the notation, we denote the
restriction of un to Ω× (0, t∗) again by un. The set of admissible controls then forms a weakly
compact, convex set in L2(Ω×(0, t∗)), and so we can extract a weakly convergent subset {um},
which converges weakly to some admissible control u∗.

Consequently, Theorem 3.3 implies that y(t;u) ∈ W 1(Ω,R∞) ⊂ L2(Ω,R∞) for each
u ∈ L2(Q) and t > 0. Then using Theorem 1.2 of (Lions, 1971, p. 102) and Theorem
3.3 it is easy to verify that the mapping u → y(t∗;u) from L2(Ω × (0, t∗)) into L2(Ω,R∞),
is continuous. Since any continuous linear mapping between Banach spaces is also weakly
continuous (Dunford and Schwartz, 1958), Theorem V. 3.15, the affine mapping u → y(t∗;u)
must also be weakly continuous. Hence,

y(t∗;um)→ y(t∗;u∗) weakly in L2(Ω,R∞). (45)

Moreover,
dy(u)

dt
∈ L2

(
[0, t∗];L2(Ω,R∞)

)
, (46)

for each u ∈ U , by definition of W 2,1(Ω× (0, t∗)) and

||y(tm;um)− y(t∗;um)||L2(Ω,R∞) =

∣∣∣∣∣∣∣∣∫ tm

t∗
ẏ(σ;um) dσ

∣∣∣∣∣∣∣∣
L2(Ω,R∞)

(47)

≤
√
tm − t∗

(∫ tm

t∗
||ẏ(σ;um)||2L2(Ω,R∞) dσ

)1/2

.(48)

Applying Theorem 1.2 of (Lions, 1971) and Theorem 3.3 again, the set {ẏ(um)} must be
bounded in L2(0, t∗;L2(Ω,R∞)), and so

||y(tm;um)− y(t∗;um)||L2(Ω,R∞) ≤M
√
tm − t∗. (49)

Combining (45) and (49) shows that

y(tm;um)− y(t∗;u∗) = (y(tm;um)− y(t∗;um)) + (y(t∗;um)− y(t∗;u∗)), (50)

converges weakly to zero in L2(Ω,R∞), and therefore y(t∗;u∗) ∈ H as H is closed and
convex, hence weakly closed. This shows that H is reached in time t∗ by an admissible control
accordingly, t∗ must be the minimum time and u∗ an optimal control.

We shall now prove the second part of our theorem. Indeed, from Theorem 3.1 (Lions
and Magenes, 1972, Vol. 1, p. 19) y(u) ∈ W 2,1(Q) implies that the mapping t → y(t;u) is



12 Bahaa:Time-Optimal Control of Infinite Variables Parabolic Systems. . . . . .

continuous from [0, T ] → W 1(Ω,R∞) ⊂ L2(Ω,R∞) is continuous for each fixed u, and so
y(t∗;u) 6∈ intH , for any u ∈ U , by the minimality of t∗.

From our earlier remarks, the set

A(t∗) = {y(t∗;ux) : ux ∈ U}, (51)

is weakly compact and convex in L2(Ω,R∞). Applying Theorem 21.11 of (Choquet, 1969) to
the setsA(t∗) and H shows that there exists a nontrivial hyperplane z ∈ L2(Ω,R∞) separating
these sets, that is, ∫

Ω
zy(t∗;u) dρ ≤

∫
Ω
zy(t∗;u∗) dρ ≤

∫
Ω
zy dρ (52)

for all u ∈ U and y ∈ L2(Ω,R∞) with ||y − zd||L2(Ω,R∞) ≤ ε.
From the second inequality in (52), z must support the setH at y(t∗;u∗). SinceL2(Ω,R∞)

is a Hilbert space, z must be of the form

z = µ(zd − y(t∗;u∗)) for some µ > 0. (53)

Subsequently, dividing (52) by µ gives the desired result (42).

We shall apply Theorem 4.1 to the control problem of (1)–(5).

To simplify (42), we introduce the adjoint equation, and for every u ∈ U we define the
adjoint variable p = p(u) = p(x, t;u) as the solution of the following system

−∂p(u)

∂t
+A∗(t)p(u) +

∫ b

a
c(x, t+h)p(x, t+h;u) dh = 0, x ∈ Ω, t ∈ (0, t∗− b), (54)

−∂p(u)

∂t
+A∗(t)p(u) +

∫ t∗−t

a
c(x, t+ h)p(x, t+ h;u) dh = 0, x ∈ Ω, t ∈ (t∗− b, t∗− a),

(55)

−∂p(u)

∂t
+A∗(t)p(u) = 0, x ∈ Ω, t ∈ (t∗ − a, t∗), (56)

p(x, t∗;u) = zd(x)− y(x, t∗;u), x ∈ Ω, (57)

∂p(u)

∂ηA∗
(x, t) =

∫ b

a
d(x, t+ h)p(x, t+ h;u) dh, x ∈ Γ, t ∈ (0, t∗ − b), (58)

∂p(u)

∂ηA∗
(x, t) =

∫ t∗−t

a
d(x, t+ h)p(x, t+ h;u) dh, x ∈ Γ, t ∈ (t∗ − b, t∗ − a), (59)

∂p(u)

∂ηA∗
(x, t) = 0, x ∈ Γ, t ∈ (t∗ − a, t∗), (60)

where
∂p(u)

∂ηA∗
(x, t) =

∞∑
k=1

(Dkp(u)) cos(n, xk), (61)

A∗(t)p(u) =

(
−
∞∑
k=1

D2
k + q(x, t)

)
p(u). (62)



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 13

Remark 4.2. If t∗ < b, then we consider (55) and (59) on Ω×(0, t∗−a) and Γ×(0, t∗−a),
respectively.

The existence of a unique solution to the problem (54)–(60) on the cylinder Ω× (0, t∗)
can be proved using a constructive method. It is easy to notice that for given zd and u, the
problem (54)–(60) can be solved backwards in time starting from t = t∗, i.e., first, solving
(54)–(60) on the sub-cylinder Qk and in turn onQk−1, and so on, until the procedure covers
the whole cylinder Ω × (0, t∗). For this purpose, we may apply Theorem 3.3 (with an obvious
change of variables).

Hence, using Theorem 3.3, the following result can be proved.

Theorem 4.3. Let the hypothesis of Theorem 3.3 be satisfied. Then for given zd ∈ L2(Ω,R∞)
and any u ∈ L2(Q), there exists a unique solution p(u) ∈ W 2,1(Ω × (0, t∗)) for the adjoint
problem (54)–(60).

Now, we have the main result.

Theorem 4.4. If the assumptions concerning system (1)–(5) and controllability condition
(41) are satisfied, then the time-optimal control u∗ exists and is characterized by the following
condition ∫ t∗

0

∫
Ω
p(u∗)(u− u∗) dρ dt ≤ 0, ∀u ∈ U, (63)

where p(u∗) is the solution of the adjoint system (54)–(60).

Proof. We simplify the left-hand side of the inequality (42) using the adjoint equation (54)–
(60). For this purpose, setting u = u∗ in (54)–(60), multiplying both sides of (54),
(55) and (56) by y(u) − y(u∗), then integrating over Ω × (0, t∗ − b), Ω × (t∗ − b, t∗ − a)
and Ω× (t∗− a, t∗) respectively and then adding both sides of (54), (55) and (56), we get∫ t∗

0

∫
Ω

(
−∂p(u

∗)

∂t
+A∗(t)p(u∗)

)
(y(u)− y(u∗)) dρ dt

+

∫ t∗−b

0

∫
Ω

(∫ b

a
c(x, t+ h)p(x, t+ h;u∗) dh

)
× (y(x, t;u)− y(x, t;u∗)) dρ dt

+

∫ t∗−a

t∗−b

∫
Ω

(∫ t∗−t

a
c(x, t+ h)p(x, t+ h;u∗) dh

)
(y(x, t;u)− y(x, t;u∗)) dρ dt

=−
∫

Ω
p(x, t∗;u∗)(y(x, t∗;u)− y(x, t∗;u∗)) dρ+

∫ t∗

0

∫
Ω
p(u∗)

∂

∂t
(y(u)− y(u∗)) dρ dt

+

∫ t∗

0

∫
Ω
A∗p(u∗)(y(u)− y(u∗)) dρ dt

+

∫ t∗−b

0

∫
Ω

∫ b

a
c(x, t+ h)p(x, t+ h;u∗) (y(x, t;u)− y(x, t;u∗)) dh dρ dt

+

∫ t∗−a

t∗−b

∫
Ω

∫ t∗−t

a
c(x, t+ h)p(x, t+ h;u∗) (y(x, t;u)− y(x, t;u∗)) dh dρ dt = 0.

(64)



14 Bahaa:Time-Optimal Control of Infinite Variables Parabolic Systems. . . . . .

Then, applying (57), the equation (64) can be expressed as

∫
Ω

(zd − y(t∗;u∗))(y(x, t∗;u)− y(x, t∗;u∗)) dρ

=

∫ t∗

0

∫
Ω

p(u∗)
∂

∂t
(y(u)− y(u∗)) dρ dt+

∫ t∗

0

∫
Ω

A∗p(u∗)(y(u)− y(u∗)) dρ dt

+

∫ b

a

∫
Ω

∫ t∗−b

0

c(x, t+ h)p(x, t+ h;u∗) (y(x, t;u)− y(x, t;u∗)) dt dρ dh

+

∫ t∗−t

a

∫
Ω

∫ t∗−a

t∗−b
c(x, t+ h)p(x, t+ h;u∗) (y(x, t;u)− y(x, t;u∗)) dt dρ dh.

(65)

Using (1), the first integral on the right-hand side of (65) can be rewritten as

∫ t∗

0

∫
Ω
p(u∗)

∂

∂t
(y(u)− y(u∗)) dρ dt

=−
∫ t∗

0

∫
Ω
p(u∗)A(y(u)− y(u∗)) dρ dt

−
∫ t∗

0

∫
Ω
p(x, t;u∗)

(∫ b

a
c(x, t)(y(x, t− h;u)− y(x, t− h;u∗)) dh

)
dρ dt

+

∫ t∗

0

∫
Ω
p(x, t;u∗)(u− u∗) dρ dt



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 15

=−
∫ t∗

0

∫
Ω
p(u∗)A(y(u)− y(u∗)) dρ dt

−
∫ t∗

0

∫
Ω

∫ b

a
p(x, t;u∗)c(x, t)(y(x, t− h;u)− y(x, t− h;u∗)) dh dρ dt

+

∫ t∗

0

∫
Ω
p(x, t;u∗)(u− u∗) dρ dt

=−
∫ t∗

0

∫
Ω
p(u∗)A(y(u)− y(u∗)) dρ dt

−
∫ b

a

∫
Ω

∫ t∗

0
p(x, t;u∗)c(x, t)(y(x, t− h;u)− y(x, t− h;u∗)) dt dρ dh

+

∫ t∗

0

∫
Ω
p(x, t;u∗)(u− u∗) dρ dt

=−
∫ t∗

0

∫
Ω
p(u∗)A(y(u)− y(u∗)) dρ dt

−
∫ b

a

∫
Ω

∫ t∗−h

−h
p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh

+

∫ t∗

0

∫
Ω
p(x, t;u∗)(u− u∗) dρ dt

=−
∫ t∗

0

∫
Ω
p(u∗)A(y(u)− y(u∗)) dρ dt

−
∫ b

a

∫
Ω

∫ 0

−h
p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh

−
∫ b

a

∫
Ω

∫ t∗−b

0
p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh

−
∫ b

a

∫
Ω

∫ t∗−h

t∗−b
p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh

+

∫ t∗

0

∫
Ω
p(x, t;u∗)(u− u∗) dρ dt

=−
∫ t∗

0

∫
Ω
p(u∗)A(y(u)− y(u∗)) dρ dt

−
∫ b

a

∫
Ω

∫ 0

−h
p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh

−
∫ b

a

∫
Ω

∫ t∗−b

0
p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh

−
∫ t∗−t

a

∫
Ω

∫ t∗−a

t∗−b
p(x, t′ + h;u∗)c(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dρ dh

+

∫ t∗

0

∫
Ω
p(x, t;u∗)(u− u∗) dρ dt.

(66)

The second integral on the right-hand side of (65), in view of Green formula, can be



16 Bahaa:Time-Optimal Control of Infinite Variables Parabolic Systems. . . . . .

expressed as

∫ t∗

0

∫
Ω
A∗p(u∗)(y(u)− y(u∗)) dρ dt =

∫ t∗

0

∫
Ω
p(u∗)A(y(u)− y(u∗)) dρ dt

+

∫ t∗

0

∫
Γ
p(u∗)

(
∂y(u)

∂ηA
− ∂y(u∗)

∂ηA

)
dΓ dt−

∫ t∗

0

∫
Γ

∂p(u∗)

∂ηA∗
(y(u)− y(u∗)) dΓ dt.

(67)

Using the boundary condition (4), the second component on the right-hand side of (67) can
be written as

∫ t∗

0

∫
Γ

p(u∗)

(
∂y(u)

∂ηA
− ∂y(u∗)

∂ηA

)
dΓ dt

=

∫ t∗

0

∫
Γ

p(x, t;u∗)

(∫ b

a

d(x, t)(y(x, t− h;u)− y(x, t− h;u∗)) dh

)
dΓ dt

=

∫ t∗

0

∫
Γ

∫ b

a

p(x, t;u∗)d(x, t)(y(x, t− h;u)− y(x, t− h;u∗)) dh dΓ dt

=

∫ b

a

∫
Γ

∫ t∗

0

p(x, t;u∗)d(x, t)(y(x, t− h;u)− y(x, t− h;u∗)) dt dΓ dh

=

∫ b

a

∫
Γ

∫ t∗−h

−h
p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dΓ dh

=

∫ b

a

∫
Γ

∫ 0

−h
p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dΓ dh

+

∫ b

a

∫
Γ

∫ t∗−b

0

p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dΓ dh

+

∫ b

a

∫
Γ

∫ t∗−h

t∗−b
p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dΓ dh

=

∫ b

a

∫
Γ

∫ 0

−h
p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dΓ dh

+

∫ b

a

∫
Γ

∫ t∗−b

0

p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dΓ dh

+

∫ t∗−t

a

∫
Γ

∫ t∗−a

t∗−b
p(x, t′ + h;u∗)d(x, t′ + h)(y(x, t′;u)− y(x, t′;u∗)) dt′ dΓ dh.

(68)

The last component in (67) can be rewritten as

∫ t∗

0

∫
Γ

∂p(u∗)

∂ηA∗
(y(u)− y(u∗)) dΓ dt =

∫ t∗−b

0

∫
Γ

∂p(u∗)

∂ηA∗
(y(u)− y(u∗)) dΓ dt

+

∫ t∗−a

t∗−b

∫
Γ

∂p(u∗)

∂ηA∗
(y(u)− y(u∗)) dΓ dt+

∫ t∗

t∗−a

∫
Γ

∂p(u∗)

∂ηA∗
(y(u)− y(u∗)) dΓ dt.

(69)



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 17

Substituting (68) and (69) into (67) and then (66) and (67) into (65), we obtain∫
Ω

(zd − y(t∗;u∗))(y(x, t∗;u)− y(x, t∗;u∗)) dρ

=−
∫ t∗

0

∫
Ω

p(u∗)A(y(u)− y(u∗)) dρ dt

−
∫ b

a

∫
Ω

∫ 0

−h
c(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dρ dh

−
∫ b

a

∫
Ω

∫ t∗−b

0

c(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dρ dh

−
∫ t∗−t

a

∫
Ω

∫ t∗−a

t∗−b
c(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dρ dh

+

∫ t∗

0

∫
Ω

p(u∗)A(y(u)− y(u∗)) dρ dt

+

∫ b

a

∫
Γ

∫ 0

−h
d(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dΓ dh

+

∫ b

a

∫
Γ

∫ t∗−b

0

d(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dΓ dh

+

∫ t∗−t

a

∫
Γ

∫ t∗−a

t∗−b
d(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dΓ dh

+

∫ t∗

0

∫
Ω

p(x, t;u∗)(u− u∗) dρ dt

−
∫ t∗−b

0

∫
Γ

∂p(u∗)

∂ηA∗
(y(u)− y(u∗)) dΓ dt

−
∫ t∗−a

t∗−b

∫
Γ

∂p(u∗)

∂ηA∗
(y(u)− y(u∗)) dΓ dt−

∫ t∗

t∗−a

∫
Γ

∂p(u∗)

∂ηA∗
(y(u)− y(u∗)) dΓ dt

+

∫ b

a

∫
Ω

∫ t∗−b

0

c(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dρ dh

+

∫ t∗−t

a

∫
Ω

∫ t∗−a

t∗−b
c(x, t+ h)p(x, t+ h;u∗)(y(x, t;u)− y(x, t;u∗)) dt dρ dh.

(70)

Then, using the fact that y(x, t;u) = y(x, t;u∗) = Φ0(x, t) for x ∈ Ω and t ∈ [−b, 0), and
y(x, t;u) = y(x, t;u∗) = Ψ0(x, t) for x ∈ Γ and t ∈ [−b, 0), we obtain∫

Ω

(zd − y(t∗;u∗))(y(x, t∗;u)− y(x, t∗;u∗)) dρ =

∫ t∗

0

∫
Ω

p(x, t;u∗)(u− u∗) dρ dt, (71)

then, substituting (71) into (42), we get (63) and this finishes proof of the theorem.

5. Generalization

Time-optimal control problem presented her can be extended to certain different two cases.
Case 1: Time-optimal control problem for (2 × 2) coupled system of parabolic equations with
infinite number of variables, in which time lags appear in integral form in both the state equation



18 Bahaa:Time-Optimal Control of Infinite Variables Parabolic Systems. . . . . .

and the boundary condition. Case 2: Time-optimal control problem for (n×n) coupled system
of parabolic equations with infinite number of variables, in which time lags appear in integral
form in both the state equation and the boundary condition.

5.1 Time-optimal Control Problem for (2 × 2) Coupled System of Parabolic Equations with
Infinite Number of Variables.

We can extend the discussions to study the time-optimal control problem for 2×2 coupled
system of parabolic equations with infinite number of variables, in which time lags appear in
integral form in both the state equation and the boundary condition.

Consider now the distributed-parameter system described by the following (2× 2) coupled
system of parabolic equations with infinite number of variables, for i = 1, 2,

∂yi
∂t

+ A(t)yi +

∫ b

a
ci(x, t)y(x, t− h) dh = ui, x ∈ Ω, t ∈ (0, T ), h ∈ (a, b), (72)

yi(x, t
′) = Φi,0(x, t′), x ∈ Ω, t′ ∈ [−b, 0), (73)

yi(x, 0) = yi,0(x), x ∈ Ω, (74)

∂yi(x, t)

∂ηA
=

∫ b

a
di(x, t)yi(x, t− h) dh+ vi, x ∈ Γ, t ∈ (0, T ), h ∈ (a, b), (75)

yi(x, t
′) = Ψi,0(x, t′), x ∈ Γ, t′ ∈ [−b, 0), (76)

where

A(t)yi(x) =

(
−
∞∑
k=1

D2
k + q(x, t)

)
yi(x) +

2∑
j=1

aijyj(x) ∀ i = 1, 2, (77)

aij =

{
1, i ≥ j;
−1, i < j.

(78)

It is easy to see that A(t) is (2× 2) matrix which takes the form

A(t) =


−
∞∑
k=1

D2
k + q + 1 −1

1 −
∞∑
k=1

D2
k + q + 1


2×2

. (79)

Also we have

yi ≡ yi(x, t;u), ui ≡ ui(x, t), vi ≡ vi(x, t), u ≡ (u1, u2),

ci and di, i = 1, 2, are real C∞ functions defined on Q and Σ, respectively, Φi,0 and Ψi,0,
i = 1, 2, are initial functions defined on Q0 and Σ0, respectively.

Now we discuss the case of the distributed control for the Neumann problem. Then, as
Section 3, for u = (u1, u2) ∈ (L2(Q))2, we can obtain the following results.



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 19

Lemma 5.1. Let
u ∈

(
L2(Q)

)2
, (80)

fj = (f1,j , f2,j) ∈
(
L2(Qj)

)2
, (81)

where

fi,j(x, t) = ui(x, t)−
∫ b

a
ci(x, t)yi,j−1(x, t− h) dh, i = 1, 2,

yj−1(·, (j − 1)a) = (y1,j−1(·, (j − 1)a), y2,j−1(·, (j − 1)a)) ∈
(
W 1(Ω,R∞)

)2
, (82)

gj = (g1,j , g2,j) ∈
(
W

1
2
, 1
4 (Σj)

)2
, (83)

where

gi,j(x, t) =

∫ b

a
di(x, t)yi,j−1(x, t− h) dh+ vi(x, t), i = 1, 2.

Then, there exists a unique solution yj ∈
(
W 2,1(Qj)

)2 for the mixed initial-boundary value
problem (72), (75) and (82).

Theorem 5.2. Let yi,0, Φi,0, Ψi,0, v and u be given with yi,0 ∈W 1(Ω,R∞), Φi,0 ∈W 2,1(Q0),
vi ∈ W

1
2
, 1
4 (Σ), Ψi,0 ∈ W

1
2
, 1
4 (Σ0) and ui ∈ L2(Q), i = 1, 2. Then, there exists a unique

solution y ∈
(
W 2,1(Q)

)2 for the mixed initial-boundary value problem (72)–(76). Moreover,
y(·, ja) ∈

(
W 1(Ω,R∞)

)2 for j = 1, 2, . . ..

Now, we shall formulate the minimum-time problem for (72)–(76) in the context of the
Theorem 5.2, i.e.,

u ∈ U = {u ∈
(
L2(Q)

)2
: |ui(x, t)| ≤ 1, i = 1, 2}. (84)

We shall define the reachable set H such that

H = {y ∈
(
L2(Ω,R∞)

)2
:

2∑
i=1

||yi − zi,d||L2(Ω,R∞) ≤ ε} (85)

where zi,d ∈ L2(Ω,R∞), i = 1, 2, and ε > 0.

Solving the stated minimum-time problem is equivalent to hitting the target set H in min-
imum time, that is, minimizing the time t, for which y(t;u) ∈ H and u ∈ U. Moreover, we
assume that

there exists a T > 0 and u ∈ U with y(T ;u) ∈ H (86)

then, as Section 4, we can prove the following theorem

Theorem 5.3. If the assumption (86) holds, then the set H is reached in minimum time t∗ by
an admissible control u∗ ∈ U. Moreover

2∑
i=1

∫
Ω

[zi,d − yi(t∗;u∗)] [yi(t
∗;u)− yi(t∗;u∗)] dρ ≤ 0, ∀u ∈ U. (87)



20 Bahaa:Time-Optimal Control of Infinite Variables Parabolic Systems. . . . . .

Now, we introduce the adjoint equation, and for every u ∈ U we define the adjoint variable
p = p(u) = p(x, t;u) as the solution of the following (2× 2) system

−∂pi(u)

∂t
+A∗(t)pi(u)+

∫ b

a
ci(x, t+h)pi(x, t+h;u) dh = 0, x ∈ Ω, t ∈ (0, t∗−b), (88)

−∂pi(u)

∂t
+A∗(t)pi(u)+

∫ t∗−t

a
ci(x, t+h)pi(x, t+h;u) dh = 0, x ∈ Ω, t ∈ (t∗−b, t∗−a),

(89)

−∂pi(u)

∂t
+ A∗(t)pi(u) = 0, x ∈ Ω, t ∈ (t∗ − a, t∗), (90)

pi(x, t
∗;u) = zi,d(x)− yi(x, t∗;u), x ∈ Ω, (91)

∂pi(u)

∂ηA∗
(x, t) =

∫ b

a
di(x, t+ h)pi(x, t+ h;u) dh, x ∈ Γ, t ∈ (0, t∗ − b), (92)

∂pi(u)

∂ηA∗
(x, t) =

∫ t∗−t

a
di(x, t+ h)pi(x, t+ h;u) dh, x ∈ Γ, t ∈ (t∗ − b, t∗ − a), (93)

∂pi(u)

∂ηA∗
(x, t) = 0, x ∈ Γ, t ∈ (t∗ − a, t∗), (94)

where
∂pi(u)

∂ηA∗
(x, t) =

∞∑
k=1

(Dkpi(u)) cos(n, xk), (95)

A∗(t)pi(u) =

(
−
∞∑
k=1

D2
k + q(x, t)

)
pi(u) +

2∑
j=1

ajipj(u), (96)

and aji is the transpose of aij .

Hence, as the proof of Theorem 4.4 in Section 4, we can prove the following theorem.

Theorem 5.4. If the assumptions concerning system (72)–(76) and controllability condition
(86) are satisfied, then the time-optimal control u∗ = (u∗1, u

∗
2) exists and is characterized by

the following condition

2∑
i=1

∫ t∗

0

∫
Ω
pi(u

∗)(ui − u∗i ) dρ dt ≤ 0, ∀u ∈ U, (97)

where p(u∗) is the solution of the adjoint system (88)–(94).

5.2 Time-optimal Control Problem for (n × n )Coupled System of Parabolic Equations with
Infinite Number of Variables.

We can extend the discussions to study the time-optimal control problem for n×n coupled
system of parabolic equations with infinite number of variables, in which time lags appear in
integral form in both the state equation and the boundary condition.



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 21

Consider now the distributed-parameter system described by the following (n×n) coupled
system of parabolic equations with infinite number of variables, for i = 1, 2, . . . , n,

∂yi
∂t

+A(t)yi +

∫ b

a
ci(x, t)y(x, t− h) dh = ui, x ∈ Ω, t ∈ (0, T ), h ∈ (a, b), (98)

yi(x, t
′) = Φi,0(x, t′), x ∈ Ω, t′ ∈ [−b, 0), (99)

yi(x, 0) = yi,0(x), x ∈ Ω, (100)

∂yi(x, t)

∂ηA
=

∫ b

a
di(x, t)yi(x, t− h) dh+ vi, x ∈ Γ, t ∈ (0, T ), h ∈ (a, b), (101)

yi(x, t
′) = Ψi,0(x, t′), x ∈ Γ, t′ ∈ [−b, 0), (102)

where

A(t)yi(x) =

(
−
∞∑
k=1

D2
k + q(x, t)

)
yi(x) +

n∑
j=1

aijyj(x) ∀ i = 1, 2, . . . , n, (103)

aij =

{
1, i ≥ j;
−1, i < j.

(104)

It is easy to see thatA(t) is (n× n) matrix which takes the form

A(t) =



−
∞∑
k=1

D2
k + q + 1 −1 · · · −1

1 −
∞∑
k=1

D2
k + q + 1 · · · −1

...
...

...
...

1 1 · · · −
∞∑
k=1

D2
k + q + 1


n×n

. (105)

Also we have

yi ≡ yi(x, t;u), ui ≡ ui(x, t), vi ≡ vi(x, t), u ≡ (u1, u2, . . . , un),

ci and di, i = 1, 2, . . . , n, are real C∞ functions defined on Q and Σ, respectively, Φi,0 and
Ψi,0, i = 1, 2, . . . , n, are initial functions defined on Q0 and Σ0, respectively.

Now we discuss the case of the distributed control for the Neumann problem. Then, as
Section 3, for u = (u1, u2, . . . , un) ∈

(
L2(Q)

)n, we can obtain the following results.

Lemma 5.5. Let
u ∈

(
L2(Q)

)n
, (106)

fj = (f1,j , f2,j , . . . , fn,j) ∈
(
L2(Qj)

)n
, (107)

where

fi,j(x, t) = ui(x, t)−
∫ b

a
ci(x, t)yi,j−1(x, t− h) dh, i = 1, 2, . . . , n,



22 Bahaa:Time-Optimal Control of Infinite Variables Parabolic Systems. . . . . .

yj−1(·, (j − 1)a) =

(y1,j−1(·, (j − 1)a), y2,j−1(·, (j − 1)a), . . . , yn,j−1(·, (j − 1)a)) ∈
(
W 1(Ω,R∞)

)n
, (108)

gj = (g1,j , g2,j , . . . , gn,j) ∈
(
W

1
2
, 1
4 (Σj)

)n
, (109)

where

gi,j(x, t) =

∫ b

a
di(x, t)yi,j−1(x, t− h) dh+ vi(x, t), i = 1, 2, . . . n.

Then, there exists a unique solution yj ∈
(
W 2,1(Qj)

)n for the mixed initial-boundary value
problem (98), (101) and (108).

Theorem 5.6. Let yi,0, Φi,0, Ψi,0, v and u be given with yi,0 ∈W 1(Ω,R∞), Φi,0 ∈W 2,1(Q0),
vi ∈ W

1
2
, 1
4 (Σ), Ψi,0 ∈ W

1
2
, 1
4 (Σ0) and ui ∈ L2(Q), i = 1, 2, . . . , n. Then, there exists a

unique solution y ∈
(
W 2,1(Q)

)n for the mixed initial-boundary value problem (98)–(102).
Moreover, y(·, ja) ∈

(
W 1(Ω,R∞)

)n for j = 1, 2, . . ..

Now, we shall formulate the minimum-time problem for (98)–(102) in the context of the
Theorem 5.6, i.e.,

u ∈ U = {u ∈
(
L2(Q)

)n
: |ui(x, t)| ≤ 1, i = 1, 2, . . . , n}. (110)

We shall define the reachable setH such that

H = {y ∈
(
L2(Ω,R∞)

)n
:

n∑
i=1

||yi − zi,d||L2(Ω,R∞) ≤ ε} (111)

where zi,d ∈ L2(Ω,R∞), i = 1, 2, . . . , n, and ε > 0.

Solving the stated minimum-time problem is equivalent to hitting the target setH in min-
imum time, that is, minimizing the time t, for which y(t;u) ∈ H and u ∈ U . Moreover, we
assume that

there exists a T > 0 and u ∈ U with y(T ;u) ∈H (112)

then, as Section 4, we can prove the following theorem

Theorem 5.7. If the assumption (112) holds, then the setH is reached in minimum time t∗ by
an admissible control u∗ ∈ U . Moreover

n∑
i=1

∫
Ω

[zi,d − yi(t∗;u∗)] [yi(t
∗;u)− yi(t∗;u∗)] dρ ≤ 0, ∀u ∈ U . (113)

Now, we introduce the adjoint equation, and for every u ∈ U we define the adjoint variable
p = p(u) = p(x, t;u) as the solution of the following (n× n) system, i = 1, 2, . . . , n,

−∂pi(u)

∂t
+A∗(t)pi(u) +

∫ b

a
ci(x, t+ h)pi(x, t+ h;u) dh = 0, x ∈ Ω, t ∈ (0, t∗ − b),

(114)



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 23

−∂pi(u)

∂t
+A∗(t)pi(u)+

∫ t∗−t

a
ci(x, t+h)pi(x, t+h;u) dh = 0, x ∈ Ω, t ∈ (t∗−b, t∗−a),

(115)

−∂pi(u)

∂t
+A∗(t)pi(u) = 0, x ∈ Ω, t ∈ (t∗ − a, t∗), (116)

pi(x, t
∗;u) = zi,d(x)− yi(x, t∗;u), x ∈ Ω, (117)

∂pi(u)

∂ηA∗
(x, t) =

∫ b

a
di(x, t+ h)pi(x, t+ h;u) dh, x ∈ Γ, t ∈ (0, t∗ − b), (118)

∂pi(u)

∂ηA∗
(x, t) =

∫ t∗−t

a
di(x, t+ h)pi(x, t+ h;u) dh, x ∈ Γ, t ∈ (t∗ − b, t∗ − a), (119)

∂pi(u)

∂ηA∗
(x, t) = 0, x ∈ Γ, t ∈ (t∗ − a, t∗), (120)

where
∂pi(u)

∂ηA∗
(x, t) =

∞∑
k=1

(Dkpi(u)) cos(n, xk), (121)

A∗(t)pi(u) =

(
−
∞∑
k=1

D2
k + q(x, t)

)
pi(u) +

n∑
j=1

ajipj(u), (122)

and aji is the transpose of aij .

Hence, as the proof of Theorem 4.4 in Section 4, we can prove the following theorem.

Theorem 5.8. If the assumptions concerning system (98)–(102) and controllability condition
(112) are satisfied, then the time-optimal control u∗ = (u∗1, u

∗
2, . . . , u

∗
n) exists and is charac-

terized by the following condition

n∑
i=1

∫ t∗

0

∫
Ω
pi(u

∗)(ui − u∗i ) dρ dt ≤ 0, ∀u ∈ U , (123)

where p(u∗) is the solution of the adjoint system (114)–(120).

6. Conclusions and Perspectives

The results presented in the paper can be treated as a generalization of the results obtained
by Knowles (1978), and Kowalewski and Krakowiak (2006; 2008) onto the case of time opti-
mal distributed and boundary control of second order infinite variables parabolic systems with
deviating arguments appearing in the integral form both in state equations and in boundary
conditions. We considered a different type of control, namely, the control function defined in
the distributed and boundary of the spatial domain. Sufficient conditions for the existence of
a unique solution of such parabolic equations with Neumann boundary conditions are proved
(Lemmas 3.2; 3.4; 3.6; 3.8; 5.1 and 5.5) and (Theorems 3.3; 3.5; 3.7; 3.9; 5.2 and 5.6). The
optimal control is characterized by using the adjoint equations (Theorems 4.3; 4.4; 5.4 and
5.8). The conditions (41; 86 and; 112) plays a fundamental role in controllability problems



24 Bahaa:Time-Optimal Control of Infinite Variables Parabolic Systems. . . . . .

for time-delay parabolic systems. With regard to the controllability assumption (41; 86 and
112), we can investigate the exact controllability problem for the parabolic system (1)-(5).

In this paper, we considered the time-optimal distributed and boundary control problem for
infinite variables parabolic systems with non-homogeneous Neumann and Dirichlet boundary
conditions. We can also consider an analogous minimum time problem for hyperbolic systems
with non-homogeneous Neumann and Dirichlet boundary conditions. Finally, we can consider
the time-optimal control problem for discrete time delay distributed and boundary parameter
systems. The ideas mentioned above will be developed in forthcoming papers.

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