Numerical approximation for a time optimal control problems governed by semi-linear heat equations

In this paper, we study the optimal time for a time optimal control problem (\mathcal{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 ({\mathcal{P}}_h) governed by a semi-linear 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.

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 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 (\mathcal{P}) studied in this paper. We begin with introducing the controlled equation. Let Ω be a convex and bounded domain, with smooth boundary ∂ Ω, in {\mathbb{R}}^d (d=1,2,3). Let ω be an open and nonempty subset of Ω. In this paper, we consider the following semi-linear controlled heat equation:

where the initial value y_0 belongs to H_0^1(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega }), and u(\cdot ) is a control function taken from the space L^{\mathrm{\infty }}(0,+\mathrm{\infty };L^2(\mathrm{\Omega })), and f(\cdot ) is a C^1 function from ℝ to ℝ. We assume that

and

It is easy to see that under the present assumptions this semi-linear heat equations has a unique solution (see [1, 2]). Throughout this paper, we will treat the solutions of (1.1) as functions of the time variable t, from {\mathbb{R}}^{+}\equiv [0,+\mathrm{\infty }) to the space L^2(\mathrm{\Omega }), and denote y(\cdot ;y_0,u) the unique solution of (1.1) corresponding to the control u and the initial value y_0. We denote \parallel \cdot \parallel and 〈\cdot ,\cdot 〉 to the usual norm and the inner product of L^2(\mathrm{\Omega }) respectively. Besides, variables x and t for functions of (x,t) and variable x for functions of x will be omitted, provided that it is not going to cause any confusion. The constraint control set is taken as

while the target set is the closed ball B(0,1)\equiv \{w\in L^2(\mathrm{\Omega });\parallel w\parallel \le 1\}. The time optimal control problem reads as follows:

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}}^{+}.

We next build the approximate problem for (\mathcal{P}). We first build a finite element space V_0^h, which will be further discussed in the next section. Let P_h be the L^2-projection from L^2(\mathrm{\Omega }) to V_0^h, and we project the target set B(0,1) into

Now, we study the following semi-discrete system:

Here, the control u(\cdot ) is taken from the constraint control set {\mathcal{U}}_{ad}. We denote y_h(\cdot ;P_h{y}_0,u) the solution of (1.4) corresponding to the control u and the initial value P_h{y}_0. Consequently, we project the problem (\mathcal{P}) into the following time optimal control problem of ordinary differential equations:

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.

Theorem 1.1 Let y_0\in H_0^1(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega }). Equations (1.2) and (1.3) hold, and the constant L in (1.2) satisfies . Then there exists a positive number h_0 such that

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 semi-linear 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.

Since Ω is a convex set with a smooth boundary, there exists a positive number

having the property: corresponding to each h, with , one can construct such a family {\mathcal{T}}^h of regular triangulations in \overline{\mathrm{\Omega }} that satisfies the following conditions (see [6]):

• (A₁) 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.)

• (A₂) {\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.

For each \tau \in {\mathcal{T}}^h, we denote \mathcal{S}(\tau ) to the space of all polynomials of 1-order and defined on τ. Corresponding to the state space L^2(\mathrm{\Omega }), we build a finite element space as follows:

It is a subspace of H_0^1(\mathrm{\Omega }). Let P_h be the L^2-projection from L^2(\mathrm{\Omega }) to V_0^h, namely,

Now, we will present some lemmas, which will be used later.

Lemma 2.1 Suppose (1.2) and (1.3) hold, and y_0\in H_0^1(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega }). Then the corresponding solution y(\cdot ;y_0,u) of (1.1) is global and the following inequality holds:

Proof The proof for the existence of the global solution for (1.1) can be viewed in [1]. Now, we are going to prove inequality (2.2). According to (1.2) and (1.3), we get

Let F(t)={\int }_{\mathrm{\Omega }}{|y(t;y_0,0)|}^{2}dt, for t\in [0,+\mathrm{\infty }). Then,

From this, we can complete the proof of the lemma. □

Remark 2.1 With the same argument, we can also derive that the solution y_h(\cdot ;y_0^h,0) of (1.4) also satisfies the following inequality:

Lemma 2.2 Suppose y_0\in H_0^1(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega }) and u\in {\mathcal{U}}_{ad}. Then, for each T>0, there exists a constant C_T, which is independent of h but depends on T, such that

We can deduce this lemma by classical finite element analysis; see [7] and [8].

Lemma 2.3 Suppose that , and let g be the function from {\mathbb{R}}^{+} to {\mathbb{R}}^{+} defined by

Then, we have

and

Proof Clearly, it suffices to show that the desired inequality in this lemma stands in the case that y_0\notin B(0,1). According to Lemma 2.1, we observe that the solution y(\cdot ;y_0,0) of (1.1) with u=0, has the estimate

Combined with , we see that when t=\frac{1}{{\lambda }_1-L}ln\parallel y_0\parallel,

Namely, y(\cdot ;y_0,0) have entered into the ball B(0,1) at time t=\frac{1}{{\lambda }_1-L}ln\parallel y_0\parallel. This fact, together with the optimality of T^{\ast }(y_0) to the problem (\mathcal{P}), yields the inequality:

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. □

Let h_0 be the positive number given in (2.1). It suffices to show that the following two inequalities hold for any h with :

and

We first prove the inequality (3.1). It is well knows that there exist optimal controls for problem (\mathcal{P}) and ({\mathcal{P}}_h), respectively (see [9, 10] and [11]). Let u^{\ast } be the optimal control to the problem (\mathcal{P}). Then, by (2.5) we obtain

From the optimality of T^{\ast }(y_0) and u^{\ast } to the problem (\mathcal{P}), it follows that

Along with the above-mentioned inequality, this indicates that

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.

In the second case, we let T_h^{\ast }(z_h) and w_h^{\ast } be the optimal time and an optimal control to the problem ({\mathcal{P}}_h), where the initial state P_h{y}_0 is replaced by the state z_h. (The existence of such an optimal control can be verified easily.) Then, the solution y_h(\cdot ;z_h,w_h^{\ast }) takes value in B_h(0,1) at time T_h^{\ast }(z_h). One can utilize Lemma 2.3 and (3.3) to deduce that

Now we construct another control {\overline{u}}_h by setting

Clearly, {\overline{u}}_h\in {\mathcal{U}}_{ad}, and the solution y_h(\cdot ;P_h{y}_0,{\overline{u}}_h) takes value in B_h(0,1) at time T^{\ast }(y_0)+T_h^{\ast }(z_h). Combined with the optimality of T_h^{\ast }(P_h{y}_0) to the problem ({\mathcal{P}}_h), these indicate that

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.

Next, we are in the position to prove (3.2). Let u_h^{\ast } be the optimal control to the problem ({\mathcal{P}}_h). Then it follows from (2.5) that

By the optimality of T_h^{\ast }(P_h{y}_0) and u_h^{\ast } to the problem ({\mathcal{P}}_h), we get

Therefore, we have

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.

Then, the solution y(\cdot ;z,w^{\ast }) takes value in the target set B(0,1) at time T^{\ast }(z). Furthermore, it follows from Lemma 2.3 and (3.5) that

Now we construct another control \overline{u} by setting

Clearly, \overline{u}\in {\mathcal{U}}_{ad}, and the solution y(\cdot ;y_0,\overline{u}) takes value in B(0,1) at time T_h^{\ast }(P_h{y}_0)+T^{\ast }(z). Combined with the optimality of T^{\ast }(y_0) to the problem (\mathcal{P}), these indicate that

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.

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.

Affiliations

1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, P.R. China

   • Guojie Zheng

2. School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa

   • Guojie Zheng

3. Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, 453003, P.R. China

   • Jingben Yin

Corresponding author

Correspondence to Guojie Zheng.

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.

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