Numerical approximation for a time optimal control problems governed by semilinear heat equations
 Guojie Zheng^{1, 2}Email author and
 Jingben Yin^{3}
https://doi.org/10.1186/16871847201494
© Zheng and Yin; licensee Springer. 2014
Received: 25 December 2013
Accepted: 4 March 2014
Published: 20 March 2014
Abstract
In this paper, we study the optimal time for a time optimal control problem $(\mathcal{P})$, governed by an internally controlled semilinear heat equation. By projecting the original problem via the finite element method, we obtain another time optimal control problem $({\mathcal{P}}_{h})$ governed by a semilinear system of ordinary differential equations. Here, h is the mesh sizes of the finite element spaces. The purpose of this study is to approach the optimal time for the problem $(\mathcal{P})$ through the optimal time for the problem $({\mathcal{P}}_{h})$. We obtain error estimates between the optimal times in terms of h.
MSC:35K05, 49J20.
Keywords
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 optimal time, is a very significant value. In this paper, we study numerical approximation for a time optimal control problems governed by semilinear 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.
In this problem, the number ${T}^{\ast}({y}_{0})={min}_{u\in {\mathcal{U}}_{ad}}\{T;y(T;{y}_{0},u)\in B(0,1)\}$ is called the optimal time, while a control ${u}^{\ast}$, in the set ${\mathcal{U}}_{ad}$, and holding the property that $y({T}^{\ast}({y}_{0});{y}_{0},{u}^{\ast})\in B(0,1)$, is called an optimal control. For each ${y}_{0}\in {L}^{2}(\mathrm{\Omega})$, we define ${T}^{\ast}({y}_{0})$ to be the optimal time for the problem $(\mathcal{P})$. Thus, ${T}^{\ast}(\cdot )$ is a function from ${L}^{2}(\mathrm{\Omega})$ to ${\mathbb{R}}^{+}$.
For each ${y}_{0}^{h}\in {V}_{0}^{h}$, we define ${T}_{h}^{\ast}({y}_{0}^{h})$ to be the optimal time for the problem $({\mathcal{P}}_{h})$ where the initial value ${P}_{h}{y}_{0}$ is replaced by ${y}_{0}^{h}$. Thus, ${T}_{h}^{\ast}(\cdot )$ is a function from ${V}_{0}^{h}$ to ${\mathbb{R}}^{+}$, and ${T}_{h}^{\ast}({P}_{h}{y}_{0})$ is the optimal time for $({\mathcal{P}}_{h})$.
In this study, we derive the error estimates between ${T}^{\ast}({y}_{0})$ and ${T}_{h}^{\ast}({P}_{h}{y}_{0})$, in terms of h. The main results of the paper are presented as follows.
Here and throughout the rest of the paper, ${\lambda}_{1}$ stand for the first eigenvalue of the operator −△, with the Dirichlet boundary condition, and C stands for a positive constant independent of h. This constant varies in different contexts.
Since $(\mathcal{P})$ is an optimal control problem governed by an infinite dimensional system, while $({\mathcal{P}}_{h})$ is an optimal control problem governed by a finite dimensional system, the study of ${T}^{\ast}({y}_{0})$ should be much more difficult than that of ${T}_{h}^{\ast}({P}_{h}{y}_{0})$. The main purpose of this paper is to study the approximation of ${T}^{\ast}({y}_{0})$ through ${T}_{h}^{\ast}({P}_{h}{y}_{0})$. This kind of problem has only been addressed in quite limited papers. To the best of our knowledge, the first study on this subject is the paper [3]. In this [3], the author was concerned with time optimal control problems for a class of boundary scalar controlled linear parabolic equations, 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 semilinear heat equation. The other important literature on this subject which we would like to mention is [4, 5].
The rest of the paper is structured as follows. In Section 2, we first construct finite element spaces ${V}_{0}^{h}$, then give certain properties for the functions ${T}^{\ast}(\cdot )$ and ${T}_{h}^{\ast}(\cdot )$. Section 3 presents the proof of Theorem 1.1.
2 Finite element spaces ${V}_{0}^{h}$ and preliminary results

(A_{1}) There exist two positive constants ρ and σ independent of h, such that $\rho (\tau )/\sigma (\tau )\le \sigma $ and $h/\rho (\tau )\le \rho $ for each element τ in ${\mathcal{T}}^{h}$. (The notations $\rho (\tau )$ and $\sigma (\tau )$ stand for the diameter of the set τ and the diameter of the greatest ball contained in τ, respectively.)

(A_{2}) ${\overline{\mathrm{\Omega}}}_{h}\equiv {\bigcup}_{\tau \in {\mathcal{T}}^{h}}\tau $ is a polygonal approximation of $\overline{\mathrm{\Omega}}$. The vertices of ${\mathcal{T}}^{h}$, which are on the boundary $\partial {\mathrm{\Omega}}_{h}$, belong to ∂ Ω. Furthermore, we see that the measure of $(\mathrm{\Omega}\setminus {\mathrm{\Omega}}_{h})\le C{h}^{2}$.
Now, we will present some lemmas, which will be used later.
From this, we can complete the proof of the lemma. □
We can deduce this lemma by classical finite element analysis; see [7] and [8].
Thus, we obtain the estimate (2.6). With the same argument, we can also obtain inequality (2.7). This completes the proof of the lemma. □
3 The proof of Theorem 1.1
Write ${z}_{h}={y}_{h}({T}^{\ast}({y}_{0});{P}_{h}{y}_{0},{u}^{\ast})$. There are only two possibilities: ${z}_{h}$ either belongs to ${B}_{h}(0,1)$ or is outside of ${B}_{h}(0,1)$.
In the first case, by the optimality of ${T}_{h}^{\ast}({P}_{h}{y}_{0})$ to the problem $({\mathcal{P}}_{h})$, we deduce that ${T}_{h}^{\ast}({P}_{h}{y}_{0})\le {T}^{\ast}({y}_{0})$. Therefore, the inequality (3.1) holds for this case.
This inequality, together with (3.4), yields the estimate (3.1) for the second case. In summary, we conclude that the estimate (3.1) stands.
Write $z=y({T}_{h}^{\ast}({P}_{h}{y}_{0});{y}_{0},{u}_{h}^{\ast})$. There are only two possibilities: z either belongs to $B(0,1)$ or is outside of $B(0,1)$.
In the first case, the solution $y({T}_{h}^{\ast}({P}_{h}{y}_{0});{y}_{0},{u}_{h}^{\ast})$ takes value in $B(0,1)$ at time ${T}_{h}^{\ast}({P}_{h}{y}_{0})$. This, together with the optimality of ${T}^{\ast}({y}_{0})$ to the problem $(\mathcal{P})$, indicates that ${T}^{\ast}({y}_{0})\le {T}_{h}^{\ast}({P}_{h}{y}_{0})$. Therefore, the inequality (3.2) stands in the first case.
In the second case, we let ${T}^{\ast}(z)$ and ${w}^{\ast}$ be the optimal time and an optimal control to the problem $(\mathcal{P})$, where ${y}_{0}$ is replaced by z.
This inequality, together with (3.6), gives the estimate (3.2) for the second case. In summary, we conclude that the estimate (3.2) stands, and we can complete the proof of this theorem.
Declarations
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.
Authors’ Affiliations
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