Applied Mathematics
Vol.3 No.12(2012), Article ID:25444,5 pages DOI:10.4236/am.2012.312258
Pontryagin’s Maximum Principle for a Advection-Diffusion-Reaction Equation*
1School of Mathematics and Physics, University of South China, Hengyang, China
2School of Mathematical Sciences, Fudan University, Shanghai, China
3Department of Mathematics and Computation Sciences, Hunan City University, Yiyang, China
Email: youjunxu@163.com, xiaocuie@163.com
Received July 2, 2012; revised November 19, 2012; accepted November 26, 2012
Keywords: Optimal Control; Pontryagin’s Maximum Principle; State Constraint
ABSTRACT
In this paper we investigate optimal control problems governed by a advection-diffusion-reaction equation. We present a method for deriving conditions in the form of Pontryagin’s principle. The main tools used are the Ekeland’s variational principle combined with penalization and spike variation techniques.
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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NOTES
*This work was partially supported by National Natural Science Foundation of China (Grant No. 11126170) and Hunan Provincial Natural Science Foundation of China (Grant No. 11JJ4006) and Ph.D startup Foundation of University of South China (Grant No 5-2011-XQD- 008).