Pontryagin’s Maximum Principle for a Advection-Diffusion-Reaction Equation ()
1. Introduction
Consider the following controlled advection convection diffusion equations:
(1.1)
where 
is a convex bounded domain with a smooth boundary
, the diffusity 
with
, the reaction 
with
, and the advective field
with 
and 
are assigned functions. Here
, with 
being a separable metric space. Function
, called a control, is taken from the set
Under some mild conditions, for any
, (1.1) admits a unique weak solution 
which is called the state(corresponding to the control
). The performance of the control is measured by the cost functional
(1.2)
for some given map
. Our optimal control problem can be stated as follows.
Problem (C). Find a 
such that
(1.3)
And the state constraint of form:
(1.4)
In this paper, we make the following assumptions.
(H1) Set 
is a convex bounded domain with a smooth boundary
.
(H2) Set 
is a separable metric space.
(H3) The function 
has the following properties: 
is measurable on
, and 
continuous on 
and for any
, a constant
, such that 
(H4) Function
is measurable in 
and continuous in
for almost all
. Moreover, for any
, there exists a 
such that
(1.5)
(H5) 
is a Banach space with strictly convex dual
, 
is continuously Fréchet differentiable, and 
is closed and convex set.
(H6) 
has finite condimensionality in 
for some
, where
.
Definition 1.1 (see [1]) Let 
is a Banach space and 
is a subspace of
. We say that 
is finite codimensional in 
if there exists 
such that
A subset 
of 
is said to be finite codimensional in 
if for some
, 
the closed subspace spanned by 
is a finite codimensional subspace of 
and 
the closed convex hull of 
has a nonempty interior in this subspace.
Lemma 1.2. Let (H1) - (H3) hold. Then, for any
, (1.1) admits a unique weak solution
.
Furthermore, there exists a constant
, independent of
(1.6)
The weak solution 
of the state Equation (1.1) is determined by
using the bilinear form 
given by
Existence and uniqueness of the solution to (1.1) follow from the above hypotheses on the problem data (see [2]). Let 
be the set of all pairs 
satisfying (1.1) and (1.4) is called an admissible set. Any 
is called an admissible pair. The pair
, moveover satisfies
for all 
is called an optimal pair. If it exists, refer to 
and 
as an optimal state and control, respectively.
Now, let 
be an optimal pair of Problem (C).
Let 
be the unique solution of the following problem:
(1.7)
And define the reachable set of variational system (1.7)
(1.8)
Now, let us state the first order necessary conditions of an optimal control to Problem (C) as follows.
Theorem 1.3. (Pontryagin’s maximum principle) Let (H1) - (H6) hold. Let 
be an optimal pair of Problem (C). Then there exists a triplet
such that
(19)
(1.10)
(1.11)
where
(1.9), (1.10), and (1.11) are called the transversality condition, the adjoint system(along the given optimal pair), and the maximum condition, respectively.
Many authors (Dede [3], Yan [4], Becker [5], Stefano [6], Collis [7]) have already considered control problems for convection-diffusion equations from theoretical or numerical point of view. In the work mentioned above, the control set is convex. However, in many practical cases, the control set can not convex. This stimulates us to study Problem (C). To get Pontryagin’s Principle, we use a method based on penalization of state constraints, and Ekeland’s principle combined with diffuse perturbations [8].
In the next section, we will prove Pontryagin’s maximum principle of optimal control of Problem (C).
2. Proof of the Maximum Principle
This section is devoted to the proof of the maximum principle.
Proof of Theorem 1.3. Firstly, let
where 
is the Lebesgue measure of
. We can easily prove that 
is a complete metric space. Let 
be anoptimal pair of Problem (C). For any 
be the corresponding state, emphasizing the dependence on the control. Without loss of generality, we may assume that
. For any 
define
(2.1)
where
, and 
is an optimal control.
Clearly, this function is continuous on the (complete) metric space
. Also, we have
(2.2)
Hence, by Ekeland’s variational principle, we can find a
, such that
(2.3)
Let 
and 
be fixed and let
, we know that for any
, there exists a measurable set 
with the property 
such that if we define
and let 
be the corresponding state, then
(2.4)
where 
and 
satisfying the following
(2.5)
(2.6)
with
(2.7)
We take
. It follows that
(2.8)
where
denotes the subdifferential of
.
Next, we define 
as follows:
(2.9)
By (2.1) and chapter 4 of [8], (2.8) becomes
(2.10)
(2.11)
On the other hand, by the definition of the subdifferential, we have
(2.12)
Next, from the first relation in (2.3) and by some calculations, we have
(2.13)
Consequently,
(2.14)
From (2.5) and (2.6), we have
(2.15)
where 
is the solution of system (1.7) and
(2.16)
From (2.10), (2.12) and (2.15), we have
(2.17)
with 
Because 
has finite condimensionality in
, we can extract some subsequence, still denoted by itself, such that
From (2.17), we have
(2.18)
Now, let
.
Then
.
Then we have
(2.19)
Take
, we obtain (1.9).
Next, we let 
to get
(2.20)
Because
, for the given
, there exists a unique solution 
of the adjoint Equation (1.10). Then, from (1.6), (2.16), and (2.2), we have
(2.21)
There, (1.11) follows. Finally, by (1.10), if 
, then
. Thus, in the case where
we must have
, because
.
3. Conclusion
We have already attained Pontryagin’s Maximum Principle for the advection-diffusion-reaction equation. It seems to us that this method can be used in treating many other relevant problems.
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