Superconvergence for optimal control problems governed by semilinear parabolic equations

1.   Introduction

As we know, optimal control theory is widely used in many subjects. In the past few decades, it has attracted the attention of more and more scholars, and is also related to some specific applications, from finance to aerospace industry, from biology to medicine and so on. For example, how a spacecraft to land on the moon surface at rest with minimal fuel consumption ^[1]? Under what circumstances the tumor can be eliminated ^[2]?

In fact, optimal control problem (OCP) for partial differential equations (PDEs) is a challenging research hotspot, and much has been done both on the mathematical analysis and on its numerical approximation. Among numerous numerical methods, finite element discretization of the state equation is widely applied. Finite element approximation of optimal control problems plays a great role in modern science, technology, engineering, etc. We can find systematic introduction of finite element methods and optimal control problems governed by PDEs, for example ^[6,7,28]. There have been extensive studies in error estimates, convergence of finite element approximation for OCP. Casas, Mateos and Raymond ^[3,4] have studied a priori error estimation of semilinear elliptic boundary control problems. Chen and Huang ^[5,9] have gained a priori error estimates of stochastic elliptic PDEs, both a prior and a posterior error estimates of stokes equations with $H^1$-norm state constraint. For a posteriori error estimates of quadratic OCP governed by linear parabolic equation, see Liu and Yan ^[24], for optimal rates of convergence with Ritz-Galerkin approximations and numerical approximation of a parabolic time OCP, see Lasiecka and Knowles ^[10,11]. In particular, Liu and Yan studied the posteriori error estimates for control problems gonverned by elliptic equations ^[12], and extended it to the OCP dominated by parabolic equation ^[13] and Stokes equation ^[14].

Furthermore, the superconvergence properties of OCP is a research focus in the field of optimal control problem, because superconvergence has always been an important tool to obtain high performance finite element discretization, which can provide high-precision approximate solutions. The research on superconvergence began in the late 1970s, and obtained fruitful results, see, e.g. ^{[8,15,21,22,23,27]}.

When the objective function in the OCP contains the gradient of scalar function, the mixed finite element method is an effective numerical method. In recent ten years, for the OCP of PDEs by the mixed finite element method, professor Chen's team has studied this aspect deeply, and has made a series of research achievements, such as a priori error estimation, a posteriori error estimation, $L^{\infty}$-error estimates and superconvergence etc ^{[16,17,18,19,20,25,26]}.

Among the numerous research, Chen and Dai in ^[27] showed the superconvergence for optimal control problems governed by semiliner elliptic equations. The purpose of this paper is to extend the superconvergence property of ^[27] to the semilinear parabolic control problems.

In this paper, given the state $y$ and the co-state $p$ variables together with their approximations $y_h$ and $p_h$, we say that the approx super converges if the state and co-state variables are approximated by the piecewise linear functions, the control variable is approximated by the piecewise constant functions, we can get the superconvergence properties for both the control variable and the state variables. We are interested in the following optimal control problem

where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with a Lipschitz boundary $\partial\Omega$, $0 < T < +\infty$, $J = [0, T]$, $y_t(x, t)$ denotes the partial derivative of $y$ in time, $A(x) = (a_{ij}(x))_{n\times n}\in (W^{1, \infty}(\bar{\Omega}))^{n\times n}$, such that $(A(x)\xi)\cdot \xi\geq c\mid\xi\mid^2,$ $\forall\, \xi \in \mathbb{R}^n$, $c > 0$. We assume that the function $\phi(\cdot) \in W^{2, \infty}(-R, R)$ for any $R > 0$, $\phi^\prime(y) \in {L^2(\Omega)}$ for any $y \in L^2(J; H^1(\Omega))$, and $\phi^\prime (y)\geq 0$. Moreover, we assume that $y_d(x, t)\in C(J; L^2(\Omega))$, $y_0(x)\in H^1_0(\Omega)$ and $K$ is a nonempty closed convex set in $L^2(J; L^2(\Omega))$, defined by

In this paper, we adopt the standard notation $W^{m, p}(\Omega)$ for Sobolev spaces on $\Omega$ with a norm $\|\cdot\|_{m, p}$ given by $\| v \|_{m, p}^{p} = \sum\limits_{|\alpha|\leq m}\| D^\alpha v\|_{L^{p}(\Omega)}^{p}$, a semi-norm $|\cdot|_{m, p}$ given by $| v|_{m, p}^{p} = \sum\limits_{|\alpha| = m}\| D^\alpha v\|_{L^{p}(\Omega)}^{p}$. We set $W_0^{m, p}(\Omega) = \{v\in W^{m, p}(\Omega): v|_{\partial \Omega} = 0\}$. For $p = 2$, we denote $H^m(\Omega) = W^{m, 2}(\Omega), \ H_0^m(\Omega) = W_0^{m, 2}(\Omega)$, and $\|\cdot\|_{m} = \|\cdot\|_{m, 2}, \ \|\cdot\| = \|\cdot\|_{0, 2}$. We denote by $L^s(0, T;W^{m, p}(\Omega))$ the Banach space of all $L^s$ integrable functions from $J$ into $W^{m, p}(\Omega)$ with norm $\| v\|_{L^s(J; W^{m, p}(\Omega))} = \Big(\int_0^T||v||_{W^{m, p}(\Omega)}^sdt\Big)^{\frac{1}{s}}\ \text{for}\ s\in [1, \infty),$ and the standard modification for $s = \infty$. Similarly, one can define the spaces $H^k(0, T;W^{m, q}(\Omega))$ and $C^k(0, T;W^{m, q}(\Omega))$.

The paper is organized as follows: in section 2, we briefly review the finite element method, and then the approximation schemes for the model optimal control problem will be constructed. In section 3, some intermediate error estimates which is the base of the result will be gained. In section 4, superconvergence properties for both control and state variables are derived.

2.   Finite element method for optimal control problems

In this section, we will discuss the finite element approximation of the quadratic optimal control problem governed by semilinear parabolic equations (1.1)–(1.4). We set $W = L^2(J; V)$ with $V = H_0^1(\Omega)$, $X = L^2(0, T;U)$ with $U = L^2(\Omega)$, $\|\cdot\|_V = \|\cdot\|_{H^1_0(\Omega)}$ and $\|\cdot\| = \|\cdot\|_{L^2(\Omega)}$. Let

It follows from Friedriechs' inequality that

We denote by $H^{-1}(\Omega)$ the dual space to $H_0^1(\Omega)$. If $f \in H^{-1}(\Omega)$, we note

Then the standard weak formula for the state equation reads: find $y(u)$ such that

Thus the above equation has a solution.

We recast (1.1)–(1.4) in the following weak form: find $(y, u)$ such that

It is well known (see, e.g., ^[28]) that the control problem (2.2)–(2.3) has a solution $(y, u)$, and that if a pair $(y, u)$ is the solution of (2.2)–(2.3), then there is a co-state $p \in {{H^1(J; L^2(\Omega))}\cap W}$ such that the triplet $(y, p, u)$ satisfies optimality conditions as follows:

Now we introduce the following significant result (see ^[29]).

Lemma 2.1. ^[29] A necessary and sufficient condition for the optimality of a control$u \in K$ with corresponding state $y(u)$ and co-state $p(u)$, respectively, is the following relation:

where $\bar{p} = \frac{\int_0^T\int_{\Omega}p dxdt}{\int_0^T\int_{\Omega}1 dxdt}$denotes the integral average on $\Omega\times J$ of the function p.

In the following, we will consider the semi-discrete finite element for the problem.

Let $\mathcal{T}^h$ be regular triangulations of $\Omega$, such that $\bar{\Omega} = \mathop{\cup}\limits_{\tau\in\mathcal{T}^h}{\bar{\tau}}$. Let $h = \mathop{\max}\limits_{\tau\in\mathcal{T}^h}\{h_\tau\}$, where $h_\tau$ denotes the diameter of the element $\tau$. Note two spaces as follows:

where $\mathcal{P}_{1}$ is the space of polynomials of degree less than or equal to 1. In addition, $c$ or $C$ denotes a general positive constant independent of $h$.

Now, the finite element approximation of the optimal control problem (2.2)-(2.3) is as follows:

The optimal control problem (2.12)–(2.13) has a solution $(y_h, u_h)$, and that if a pair $(y_h, u_h)$ is the solution of (2.12)–(2.13), then there is a co-state $p_h$ such that the triplet $(y_h, p_h, u_h)$ satisfying the following optimal conditions:

Similar to Lemma 2.1, we can get the relationship between the control approximation $u_h$ and the co-state approximation $p_h$, which satisfies

where $\bar{p_h} = \frac{\int_0^T\int_{\Omega}p_h dxdt}{\int_0^T\int_{\Omega}1 dxdt}$ denotes the integral average on $\Omega\times J$ of the function $p_h$.

3.   Intermediate error estimates

First of all, we will introduce some intermediate variables. For any $\tilde{u} \in K$, let $(y(\tilde{u}), p(\tilde{u}))$ be the solution of the following equations:

Then, for any $\tilde{u}\in K$, let $(y_h(\tilde{u}), p_h(\tilde{u}))$ be the solution of the following equations:

Note that $(y, p) = (y(u), p(u))$, $(y_h, p_h) = (y_h(u_h), p_h(u_h))$.

Now we give the standard $L^2(\Omega)-$orthogonal projection $Q_h: U \rightarrow U^h,$ for $U = L^2(\Omega)$, which satisfies: for any $\psi \in U$

and the elliptic projection $R_h: V \rightarrow V^h$, which satisfies: for all $v\in V$

We have the following approximation properties (see e.g., ^[27] and ^[30]):

Lemma 3.1. Let $u\in L^2(J; H^1(\Omega))$, for h sufficiently small, thereexists a positive constant C such that

Proof. Choose $\tilde{u} = Q_hu$ and $\tilde{u} = u$ in (3.1)–(3.2), respectively, then we have the following error equations

for any $w \in V$ and $q \in V$.

First, choose $w = y(Q_hu)-y(u)$ in (3.11), we have

Now, we estimate the right hand side of (3.13). Using (3.7), we have

From (3.13) and (3.14), using $\epsilon$-Cauchy inequlity and the assumption of $A$ and $\phi(\cdot)$, we have

Note that

next, integrating the both sides of (3.15) in time from 0 to $t$, we get

which implies (3.9).

Choose $q = p(Q_hu)-p(u)$ in (3.11), we have

namely,

Notice that

Using the assumption for $\phi(\cdot)$ and (3.9), we have

where we used the embedding $\|v\|_{0, 4} \leq C\|v\|_1$. Then, using (3.17), (3.18), (3.19) and the assumption for $\phi(\cdot)$, we have

Next, we consider the given condition

then, we integrate in time from $t$ to $T$ in (3.11) and use Gronwall's inequality, we have

which implies (3.10).

Lemma 3.2. For any $\tilde{u} \in K$, if the intermediate solution satisfies

and $\Omega$ is convex, then we have

Proof. From (3.1)–(3.2) and (3.3)–(3.4), we have the following error equations:

for any $w_h \in V_h$ and $q_h \in V_h$. Using the definition of $R_h$, the above equation can be restated as

First, let $w_h = y_h(\tilde{u})-R_hy(\tilde{u})$ in (3.28), using the $\epsilon$-Cauchy inequality and the assumptions for $A$ and $\phi(\cdot)$, we have

It is known that

then integrating in time for (3.30) and using Gronwall's inequality, we have

which implies (3.24).

Then, let $q_h = p_h(\tilde{u})-R_hp(\tilde{u})$ in (3.29). Note that

and

Using the assumption for $\phi(\cdot)$, we get

where we used the embedding $\| v\|_{0, 4} \leq C \| v \|_1$. Then, using the definition of $R_h$ and the assumption for $\phi(\cdot)$, we get

From (3.29) and (3.32)–(3.35), we have

Note that

then combining (3.32)–(3.36), and using the $\epsilon$-Cauchy inequality and the assumptions for $A$ and $\phi(\cdot)$, (3.29) can be rewritten as

Integrating the above inequality in time and using Gronwall's inequality, we have

which implies (3.25).

Lemma 3.3. For $\tilde{u}\in L^2(J; H^1(\Omega))$, assume$p(\tilde{u}), p_t(\tilde{u}), y(\tilde{u}), y_t(\tilde{u})\in L^2(J; H^1(\Omega))\cap L^2(J; H^2(\Omega))$, then we have theestimate

Proof. After rewriting

from Lemma 3.1 and assumption of $p$, it is known that

and from Lemma 3.2, we get

so we have

Choose $\tilde{u} = Q_h{\tilde{u}}$, $w_h = y_h(Q_h{\tilde{u}})-y_h(\tilde{u})$ in (3.3), and let $q_h = p_h(Q_h{\tilde{u}})-p_h(\tilde{u})$ in (3.4), then we obtain the following error equations

Then from equality (3.41), using $\rm\epsilon$-Cauchy inequality and (3.7), we derive

Note that

next, integrating both sides of (3.43) in time, we obtain

and obviously

Next, we consider the equality (3.42) similar to the above idea.

Note that

Using the assumption for $\phi(\cdot)$ and (3.9), we get

where we used the embedding $\|v\|_{0, 4} \leq C\|v\|_1$. Then, using (3.45), (3.46), (3.47) and the assumption for $\phi(\cdot)$, we have

Next, we consider the given condition

then, we integrate in time for (3.48) and use Gronwall's inequality and (3.44), we have

which implies (3.39).

Let $y(u)$ and $y_h(u_h)$ are the solutions of (2.3) and (2.13), respectively. Let

Then, the simplified problems of (2.2) and (2.12) read as

and

respectively. It can be shown that

where $p(u_h)$ and $p(Q_hu)$ are solutions of (3.1)–(3.2) for $\tilde{u} = u_h$ and $\tilde{u} = Q_hu$, respectively.

In many application, $J(\cdot)$ is uniform convex near the solution $u$. The convexity of $J(\cdot)$ is bound up with the second order sufficient conditions of the control problem, which are supposed in many studies on numerical methods of the problem. Next, there is a constant $c > 0$, independent of $h$, such that

where $u$ and $u_h$ are solutions of (3.51) and (3.52) respectively, $Q_hu$ is the orthogonal projection of $u$ which is introduced in (3.5). From beginning to end, we will use the above inequality in this paper. More discussion of this can be found in ^[3,4].

4.   Superconvergence

In this section, superconvergence for both the control variable and the state variables will be discussed. Let $\pi^c$ defined in ^[31] is the average operator such that $\pi^cu = Q_hu$. Let

In this paper, we assume that $u$ and $T_{h}$ are regular such that meas$(\Omega^{-})$ = meas$(\Omega^{b})\leq Ch.$

Theorem 4.1. Let $u$ be the solution of (2.4)–(2.6) and$u_h$ be the solution of (2.14)–(2.16). Weassume that the exact control and state solution satisfy

and

Then, we have

Proof. Set $v = u_h$ in (2.6) and $v_h = Q_hu$ in (2.16), and add the two inequalities, then we get

By using the definition of $Q_h$ and (4.2), we get

For the first term of (4.3), we separate it into three parts,

from (4.3)–(4.4), we get that

We can estimate the following by $\epsilon$-Cauchy inequality

and

For the second term of (4.3)

Obviously, $(Q_hu-u)|_{\Omega ^0} = 0.$ From (2.6), we have pointwise a.e. $(u+p)\geq 0$, we set $\tilde{u}|_{\Omega ^+} = 0$ and $\tilde{u}|_{\Omega \backslash \Omega ^+} = u$, so that $(u+p, u)|_{\Omega^{+}}\leq 0$. So, $(u+p)|_{\Omega ^+} = 0$. Then

According to (3.53), the left hand of (4.5) can be restated as:

Then, combining (3.10), (3.39) and (4.5)–(4.9), we have

which completes the proof of Theorem 4.1.

Theorem 4.2. Let $u$ be the solution of (2.4)–(2.6), $u_h$be the solution of (2.14)–(2.16) and $\Omega$is convex. We assume that the exact control and state solutionsatisfy

and

Then, we have

Proof. From (2.4)–(2.5) and (2.14)–(2.15), We have the following error equations

Using the definition of $R_h$, we have

for any $w_h$ and $q_h \in V_h$.

First, taking $w_h = y_h-R_hy$ in (4.14) and using the assumption of $\phi(\cdot)$, we have

Note that

integrating in time and using Gronwall's inequality, we estimate

which implies (4.10).

Then, we take $q_h = p_h-R_hp$ in (4.15). Notice that

and

Using the definition of $R_h$ and the assumption for $\phi(\cdot)$, we have

and

From (4.15) and (4.18)–(4.20), we have

Note that

integrating in time and using Gronwall's inequality and (4.10), we estimate

which implies (4.11).

5.   Conclusions

In this paper, we present finite element approximation method for solving semilinear parabolic OCP. When the state and co-state variables are approximated by the piecewise linear functions, the control variable is approximated by the piecewise constant functions, superconvergence properties for both the control variable and the state variables are discussed. In our future work, we shall use this method to deal with hyperbolic optimal control problems, including linear and nonlinear styles.

Acknowledgments

This work is supported by Guangdong Basic and Applied Basic Research Foundation of Joint Fund Project (2021A1515111048), Science Research Team Project in Guangzhou Huashang College (2021HSKT01), National Science Foundation of China (11201510), National Social Science Fund of China (19BGL190), China Postdoctoral Science Foundation (2017T100155, 2015M580197), Innovation Team Building at Institutions of Higher Education in Chongqing (CXTDX201601035), and Chongqing Research Program of Basic Research and Frontier Technology (cstc2019jcyj-msxmX0280) and Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJZD-K202001201).

Conflict of interest

The authors declare no conflict of interest.