R E S E A R C H
Open Access
Numerical approximation for a time optimal
control problems governed by semi-linear
heat equations
Guojie Zheng
1,2*and Jingben Yin
3*Correspondence: available at the end of the article
Abstract
In this paper, we study the optimal time for a time optimal control problem (P), governed by an internally controlled semi-linear heat equation. By projecting the original problem via the finite element method, we obtain another time optimal control problem (Ph) governed by a semi-linear system of ordinary differential equations. Here,his the mesh sizes of the finite element spaces. The purpose of this study is to approach the optimal time for the problem (P) through the optimal time for the problem (Ph). We obtain error estimates between the optimal times in terms ofh.
MSC: 35K05; 49J20
Keywords: heat equation; time optimal control; finite element methods; numerical approximation
1 Introduction
One of the most important optimal control problems is how to drive the corresponding trajectory of the equation from an initial state to a given target set in the shortest time, through applying constrained controllers. With regard to this kind of problems, the opti-mal time, is a very significant value. In this paper, we study numerical approximation for a time optimal control problems governed by semi-linear heat equations. We first project the problem into another time optimal control problem of ordinary differential equations, via the finite element method. Then, we establish error estimates between the optimal times for the original problem and its projected problem.
Let us first state the time optimal control problem (P) studied in this paper. We begin with introducing the controlled equation. Letbe a convex and bounded domain, with smooth boundary∂, inRd(d= , , ). Letωbe an open and nonempty subset of. In this paper, we consider the following semi-linear controlled heat equation:
⎧
and
f() = . (.)
It is easy to see that under the present assumptions this semi-linear heat equations has a unique solution (see [, ]). Throughout this paper, we will treat the solutions of (.) as functions of the time variablet, fromR+≡[, +∞) to the space L(), and denote
y(·;y,u) the unique solution of (.) corresponding to the controluand the initial value
y. We denote · and·,·to the usual norm and the inner product ofL() respectively.
Besides, variablesxandtfor functions of (x,t) and variablexfor functions ofxwill be omitted, provided that it is not going to cause any confusion. The constraint control set is taken as
We next build the approximate problem for (P). We first build a finite element space
Vh
, which will be further discussed in the next section. LetPhbe theL-projection from
L() toVh
Now, we study the following semi-discrete system:
the solution of (.) corresponding to the control uand the initial value Phy.
Conse-quently, we project the problem (P) into the following time optimal control problem of ordinary differential equations:
Theorem . Let y∈H()∩L∞().Equations(.)and(.)hold,and the constant L
in(.)satisfies L<λ.Then there exists a positive number hsuch that
T∗
h(Phy) –T∗(y)≤Ch, when <h<h.
Here and throughout the rest of the paper,λstand for the first eigenvalue of the
op-erator –, with the Dirichlet boundary condition, andCstands for a positive constant independent ofh. This constant varies in different contexts.
Since (P) is an optimal control problem governed by an infinite dimensional system, while (Ph) is an optimal control problem governed by a finite dimensional system, the study ofT∗(y) should be much more difficult than that ofTh∗(Phy). The main purpose
of this paper is to study the approximation ofT∗(y) throughTh∗(Phy). This kind of
prob-lem has only been addressed in quite limited papers. To the best of our knowledge, the first study on this subject is the paper []. In this [], the author was concerned with time optimal control problems for a class of boundary scalar controlled linear parabolic equa-tions, obtained error estimates for optimal times, presented a full discretization of the original problem followed by numerical tests. In our paper, the problem which we study is governed by the internally controlled semi-linear heat equation. The other important literature on this subject which we would like to mention is [, ].
The rest of the paper is structured as follows. In Section , we first construct finite ele-ment spacesVh, then give certain properties for the functionsT∗(·) andTh∗(·). Section presents the proof of Theorem ..
2 Finite element spacesVh
0and preliminary results
Sinceis a convex set with a smooth boundary, there exists a positive number
h≡h() (depending only on) (.)
having the property: corresponding to eachh, with <h<h, one can construct such a
familyThof regular triangulations inthat satisfies the following conditions (see []):
(A) There exist two positive constantsρandσindependent ofh, such thatρ(τ)/σ(τ)≤σ
andh/ρ(τ)≤ρfor each elementτ inTh. (The notationsρ(τ)andσ(τ)stand for the
diameter of the setτand the diameter of the greatest ball contained inτ, respectively.) (A) h≡
τ∈Thτis a polygonal approximation of. The vertices ofTh, which are on the boundary∂h, belong to∂. Furthermore, we see that the measure of(\h)≤Ch.
For eachτ ∈Th, we denoteS(τ) to the space of all polynomials of -order and defined onτ. Corresponding to the state spaceL(), we build a finite element space as follows:
Vh=vh∈C();vh|τ∈S(τ) for everyτ∈Thandvh|\h=
.
It is a subspace ofH
(). LetPhbe theL-projection fromL() toVh, namely,
Phv,vh=v,vh, ∀v∈L(),vh∈Vh.
Lemma . Suppose(.)and(.)hold,and y∈H()∩L∞().Then the corresponding
solution y(·;y,u)of(.)is global and the following inequality holds:
y(t;y, )≤ y e–(λ–L)t for t≥. (.)
Proof The proof for the existence of the global solution for (.) can be viewed in []. Now, we are going to prove inequality (.). According to (.) and (.), we get
f(y)≤L|y|. (.)
LetF(t) =|y(t;y, )|dt, fort∈[, +∞). Then,
F(t)≤–λF(t) +
y(t;y, )f
y(t;y, ) dx
≤–λF(t) + L
y(t;y, )
dx
≤–(λ–L)F(t).
From this, we can complete the proof of the lemma.
Remark . With the same argument, we can also derive that the solutionyh(·;yh, ) of
(.) also satisfies the following inequality:
yh
t;yh, ≤yhe–(λ–L)t fort≥. (.)
Lemma . Suppose y∈H()∩L∞()and u∈Uad.Then,for each T> ,there exists
a constant CT,which is independent of h but depends on T,such that
y(·;y,u) –yh(·;Phy,u)C([,T];L())≤CTh
y +u(t)L(,T;L()) . (.)
We can deduce this lemma by classical finite element analysis; see [] and [].
Lemma . Suppose that L<λ,and let g be the function fromR+toR+defined by
g(s) =
, s∈[, ],
λ–Llns, s∈(, +∞).
Then,we have
T∗(y)≤g
y for all y∈L() (.)
and
Th∗yh ≤gyh for all yh∈Vh. (.)
(.) withu= , has the estimate
y(t;y, )≤e–(λ–L)t y for eacht≥.
Combined withL<λ, we see that whent=λ–Lln y ,
y(t;y, )≤e–(λ–L)λ
–Lln y y
= .
Namely,y(·;y, ) have entered into the ballB(, ) at timet=λ–Lln y . This fact, to-gether with the optimality ofT∗(y) to the problem (P), yields the inequality:
T∗(y)≤
λ–L
ln y .
Thus, we obtain the estimate (.). With the same argument, we can also obtain inequality
(.). This completes the proof of the lemma.
3 The proof of Theorem 1.1
Leth be the positive number given in (.). It suffices to show that the following two
inequalities hold for anyhwith <h<h:
Th∗(Phy) –T∗(y)≤Ch (.)
and
T∗(y) –Th∗(Phy)≤Ch. (.)
We first prove the inequality (.). It is well knows that there exist optimal controls for problem (P) and (Ph), respectively (see [, ] and []). Letu∗be the optimal control to the problem (P). Then, by (.) we obtain
yT∗(y);y,u∗ –yh
T∗(y);Phy,u∗ ≤Ch.
From the optimality ofT∗(y) andu∗to the problem (P), it follows that
yT∗(y);y,u∗ = .
Along with the above-mentioned inequality, this indicates that
yh
T∗(y);Phy,u∗ ≤ +Ch. (.)
Write zh =yh(T∗(y);Phy,u∗). There are only two possibilities: zh either belongs to
Bh(, ) or is outside ofBh(, ).
In the first case, by the optimality of Th∗(Phy) to the problem (Ph), we deduce that
Th∗(Phy)≤T∗(y). Therefore, the inequality (.) holds for this case.
such an optimal control can be verified easily.) Then, the solutionyh(·;zh,w∗h) takes value inBh(, ) at timeTh∗(zh). One can utilize Lemma . and (.) to deduce that
Th∗(zh)≤
λ–L
ln zh ≤
λ–L
ln( +Ch)≤Ch. (.)
Now we construct another controluhby setting
uh(t) =
u∗(t), t∈[,T∗(y)],
w∗h(t–T∗(y)), t∈(T∗(y), +∞).
Clearly,uh∈Uad, and the solutionyh(·;Phy,uh) takes value inBh(, ) at timeT∗(y) +
Th∗(zh). Combined with the optimality ofTh∗(Phy) to the problem (Ph), these indicate that
Th∗(Phy)≤T∗(y) +Th∗(zh).
This inequality, together with (.), yields the estimate (.) for the second case. In sum-mary, we conclude that the estimate (.) stands.
Next, we are in the position to prove (.). Letu∗hbe the optimal control to the problem (Ph). Then it follows from (.) that
yTh∗(Phy);y,u∗h –yh
Th∗(Phy);Phy,u∗h ≤Ch.
By the optimality ofTh∗(Phy) andu∗hto the problem (Ph), we get
yh
Th∗(Phy);Phy,u∗h = .
Therefore, we have
yTh∗(Phy);y,u∗h ≤ +Ch. (.)
Writez=y(Th∗(Phy);y,u∗h). There are only two possibilities:zeither belongs toB(, ) or
is outside ofB(, ).
In the first case, the solutiony(Th∗(Phy);y,u∗h) takes value inB(, ) at timeTh∗(Phy).
This, together with the optimality ofT∗(y) to the problem (P), indicates thatT∗(y)≤
Th∗(Phy). Therefore, the inequality (.) stands in the first case.
In the second case, we letT∗(z) andw∗ be the optimal time and an optimal control to the problem (P), whereyis replaced byz.
Then, the solutiony(·;z,w∗) takes value in the target setB(, ) at timeT∗(z). Further-more, it follows from Lemma . and (.) that
T∗(z)≤
λ–L
ln z ≤
λ–L
ln( +Ch)≤Ch. (.)
Now we construct another controluby setting
u(t) =
u∗h(t), t∈[,Th∗(Phy)],
Clearly,u∈Uad, and the solutiony(·;y,u) takes value inB(, ) at timeTh∗(Phy) +T∗(z).
Combined with the optimality ofT∗(y) to the problem (P), these indicate that
T∗(y)≤Th∗(Phy) +T∗(z).
This inequality, together with (.), gives the estimate (.) for the second case. In sum-mary, we conclude that the estimate (.) stands, and we can complete the proof of this theorem.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
GZ provided the question. GZ and JY gave the proof for the main result together. All authors read and approved the final manuscript.
Author details
1College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, P.R. China.2School of
Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa. 3Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, 453003, P.R. China.
Acknowledgements
The authors would like to express their sincere thanks to the referees for their providing several important references and for their valuable suggestions. This work was partially supported by the National Natural Science Foundation of China under Grants (U1204105, 61203293), the Natural Science Foundation of Zhejiang (Y6110751), the Natural Science Foundation of Ningbo (2010A610096), the Key Foundation of Henan Educational Committee (13A120524, 12B120006), and the National Research Foundation of South Africa.
Received: 25 December 2013 Accepted: 4 March 2014 Published:20 Mar 2014
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