Error estimates of variational discretization for semilinear parabolic optimal control problems

1.   Introduction

It is generally known that optimal control has an important role in this ever-changing society, such as engineering numerical simulation, economic, atmosphere, petroleum, architecture and scientific, and so on. Meanwhile, optimal control is an important knowledge point not only for science and engineering students, but also for non science and engineering students. for example, it is widely used in economic management disciplines, such as management accounting, dynamic optimization, actuarial, pricing analysis, etc, even in artistic modeling structure analysis. An effective numerical method is necessary for the successful application of optimal control, for optimal control problems (OCP), finite element approximation should be a powerful numerical method in the calculation, and there are many related literatures. Optimal control problems using finite element methods (FEM) for partial differential equations (PDEs) are introduced systematically, which can be found in ^[1,2,3,4] for elliptic optimal control problems in (see e.g., ^{[2,5,6,7,8,9,10]}), for parabolic optimal control problems in ^{[1,3,11,12,13]} and for Stokes optimal control problem in ^[14,15]. A priori error estimates of finite element method was established in ^[16,17,18], a posteriori error estimates of residual type has been established in ^[12,15,19], and a posteriori error estimates based on recovery techniques has been derived in ^[14,20,21]. Some error estimates and superconvergence results of mixed finite element method for optimal control problems can be found in ^[5,9].

For the optimal control problem with control constraints, the regularity of the general control method is lower than that of the state and the co-state. Therefore, the state and the co-state variables are approached by piecewise linear finite element functions, and the control variable is approached by piecewise constant function, and a projection gradient method with preset conditions is constructed. see ^{[1,3,12,14,15,19,20]}. In year 2012, Tang and Chen ^[22] studied variational discretization for the optimal control problem governed by parabolic equations with control constraints.

In this paper, we discuss a variational discretization for optimal control problems governed by nonlinear parabolic equations with control constraints, and we derive a priori error estimates and a posteriori error estimates of residual type. Some of techniques directly relevant to our work can be found in ^[16,23,24].

In this paper, the model problem that we shall investigate is the following two dimensional optimal control problem:

where $D$ is defined by

where $c$ and $d$ are two constants.

$\Omega$ in $\mathbb{R}^n(n \leq 3)$ is a bounded domain and $\partial\Omega$ is a Lipschitz boundary, $J = [0, T]$, $0 < T < +\infty$, $n$-matrix $A(x) = (a_{ij}(x)) \in (W^{1, \infty}(\bar{\Omega}))^{n\times n}$, and $(A(x)\xi)\cdot \xi\geq c\mid\xi\mid^2,$ $\forall\, \xi \in \mathbb{R}^n$. The function $\phi(\cdot)\in W^{2, \infty}$ for any $R > 0$, $\phi'(y)\in L^2(\Omega)$ for any $y\in H^1(\Omega)$, and $\phi'(y)\geq0$. Here $y_t(x, t)$ denotes the partial derivative of $y$ in time, $y_d(x, t), f(x, t)\in C(J; L^2(\Omega))$, $y_0(x)\in H^1_0(\Omega)$.

We use the standard notation $W^{m, p}(\Omega)$ for Sobolev spaces on $\Omega$ with norm $\|\cdot\|_{W^{m, q}(\Omega)}$ and seminorm $|\cdot|_{W{m, p}(\Omega)}$, when $p = 2$, $W^{m, 2}(\Omega)$ can be expressed as $H^m(\Omega)$, $W^{m, 2}_0(\Omega)$ can be expressed as $H^m_0(\Omega)$. Further more, $H_0^1(\Omega): = \left\{\; g \in H^1(\Omega): g|_{\partial \Omega} = 0 \; \right\}$. We denote by $L^k(J; W^{m, p}(\Omega))$ the Banach space of all $L^k$ integrable functions from $J$ into $W^{m, p}(\Omega)$ with norm $\|g\|_{L^k(J; W^{m, p}(\Omega))} = \left(\int_0^T\|g\|_{W^{m, p}(\Omega)}^kdt\right)^{\frac{1}{k}}$ for $k\in [1, \infty)$ and the standard modification for $k = \infty$. Similarly, the spaces $H^l(J; W^{m, p}(\Omega))$ and $C^l(J; W^{m, p}(\Omega))$ also can be defined, the details can be found in ^[17]. In addition, $c$ or $C$ denotes a ordinary positive constant.

The research ideas and concrete design of the paper is as follows: we give the backward Euler approximation and variational discretization approximation for our model in Section 2, then gain a prior error estimate in Section 3. After that, we get a posterior error estimate in Section 4. In Section 5, we give conclusion and future works.

2.   Approximation for the model problem of variational discretization

In this section, in the light of the model (1.1), we give variational discretization approximation using backward Euler method. We define $W = L^2(J; V)$ with $V = H_0^1(\Omega)$, $\|\cdot\|_V = \|\cdot\|_{H^1_0(\Omega)}$, and the control space $X = L^2(J; U)$ with $U = L^2(\Omega)$, $\|\cdot\| = \|\cdot\|_{L^2(\Omega)}$. Throughout the paper, $(\cdot \, \cdot)$ denotes the inner product in $L^2(\Omega)$.

Let

From Friedriechs' inequality, it produces

In order to keep things simple, we suppose the domain $\Omega$ is in $R^2$ and is a convex polygon.

Next, we introduce the co-state equation

Then a possible weak formula for the state equation reads: For given $u$, $y_0$, find $y(u)$ such that

It is well known (see ^[16]) that the above problem has a unique solution $y$. It follows from embedding that $y\in C(0, T;L^2(\Omega))$. So the above model control problem (1.1) can be rewritten as (QCP):

where $y\in H^1(0, T;U)\bigcap W$.

It is well known (see, e.g., ^[1]) that the convex control problem (QCP) has a unique solution $(y, u)$, and that a doublet $(y, u)$ is the solution of (QCP) if and only if there exists a co-state $p\in H^1(J; U)\bigcap W$, such that the triplet $(y, p, u)$ satisfies the following optimal conditions (QCP-OPT) for $t \in(0, T]$

Now, we introduce the pointwise projection operator as followed:

As in ^[16], it is easy to see that (2.9) can be equivalently read as:

Set $\Gamma^h$ be a regular triangulations of the polygonal domain $\Omega$, such that $\bar{\Omega} = \bigcup\limits_{\lambda\in \Gamma^h}\bar{\lambda}$. Let $h = \max\limits_{\lambda\in {\Gamma}^h}\{h_\lambda\}$, where $h_\lambda$ denotes the diameter of the element $\lambda$. Associated with ${\Gamma}^h$ is a finite dimensional subspace $W^h$ of $C(\bar{\Omega}),$ such that $w_h|_\lambda$ is the polynomial of total degree no more than $n$ $(n\geq 1)$, $\forall\, w_h\in W^h$. Let $V^h = W^h\cap V$. Then it is easy to see that $V^h\subset V$.

Suppose $N$ is a positive integer, and we set $\, 0 = t_0 < t_1 < \cdots < t_N = T$, $k_l = t_{l}-t_{l-1}, \, l = 1, 2, \cdot\cdot\cdot, N$, $k = \max\limits_{l\in [1, N]}\left\{k_l\right\}$. Set $q^l = q(x, t_l)$, we give the discrete time-dependent norms for $1\leq p < \infty$:

when $s = 0$ be the control variable and the state variable $u(x, t)$ and $y(x, t)$, when $s = 1$ be the co-state variable $p(x, t)$, with the standard modification for $p = \infty$.

A possible finite element approximation of (QCP) is to find $(y_h, u_h)\in U_h \times V_h$, which we shall label $({\rm QCP})^h$

where $y_h\in H^1(0;T; V_h)$, $y_0^h\in V_h$ is an approximation of $y_0$.

The control problem $({\rm QCP})^h$ again has a unique solution $(y_h, u_h)$ and a doublet $(y_h, u_h)\in V_h\times U_h$ is the solution of $({\rm QCP})^h$ if and only if there is a co-state $p_h$, such that the triplet $(y_h, p_h, u_h)$ satisfies the following optimality conditions $({\rm QCP-OPT})^h$ for $t\in (0, T]$:

Therefore, as we defined, the exact state solution and its approximation can be read as:

Next we consider the above discrete approximation using the backward Euler scheme to approximate the partial derivative in time $t$. Then the discrete approximation of (2.4)-(2.6) is the following form:

It follows that the control problem (2.18) has a unique solution $\left(y_h^l, u_h^l\right)$, $l = 1, 2, \cdot\cdot\cdot, N$, and $\left(y_h^l, u_h^l\right)\in V^h\times D$, $l = 1, 2, \cdot\cdot\cdot, N$, is the solution of (2.18) if and only if there is a co-state $p_h^{l-1}\in V^h$, $l = 1, 2, \cdot\cdot\cdot, N$, such that the triplet $\left(y_h^l, p_h^{l-1}, u_h^l\right)\in V^h\times V^h\times D$, $l = 1, 2, \cdot\cdot\cdot, N$, satisfies the following variational discretization optimality conditions:

where $w_h\in V^h$, $q_h\in V^h$ and $\tilde{u}_h\in D_h$. It is obvious that the variational inequality (2.21) satisfies the result as follows:

According to (2.22), we should solve the numerical solution of the control variable $u_h$ after the co-state variable $p_h$.

3.   A priori error estimates for OCP

In this section, we shall derive a priori error estimates for the backward Euler variational discretization approximation scheme. We will start with errors estimation of state and co-state variable. Now, we recall the elliptic projection $R_h$: $V\rightarrow V_h$, for any $v \in V$, which satisfies:

We have the approximation property (see ^[23])

Lemma 3.1. Let $(y, p, u)$ and $(y_h, p_h, u_h)$ be the solutions of (2.7)-(2.9) and (2.19)-(2.21), respectively. Assume $y\in H^1(J; H^2(\Omega))\cap H^2(J; U)$ and $y(x, 0)\in H^2(\Omega)$. Then there is a constant $C$ independent of $h$ and $k$ such that

Proof. We set

$a(R_h v, v_h) = a(v, v_h), \; \; \forall v_h\in V^h.$ It follows from Lemma 1.1 in ^[25], we have

It follows from (2.7) and (2.19), noting that $''I''$ denotes the unit operator, so we have

We select $w_h = \theta^l$,

Note that $0\leq c\left \|\theta^l\right\|_V^2\leq a\left(\theta^l, \theta^l\right),$ according to Hölder inequality, we obtain

Summing $l$ from $1$ to $N^*\; (1\leq N^*\leq N)$, note that $y\in H^1(J; H^2(\Omega))\cap H^2(J; U)$ and $y(x, 0)\in H^2(\Omega)$, we get

Then (3.3) follows from (3.5)-(3.7) and Triangle inequality.

Lemma 3.2. Suppose $(y, p, u)$ and $(y_h, p_h, u_h)$ be the solutions of (2.7)-(2.9) and (2.19)-(2.21), respectively. Let $p\in H^1(J; H^2(\Omega))\cap H^2(J; U)$, assume $y_d, y \in H^1(J; U)$, exist a constant $C$ independent of $k$ and $h$, so that

Proof. Similarly, we set

Note that $p^N = 0$, then

It follows from (2.8) and (2.20), we have

We select $q_h = \zeta^{l-1}$, note that $0\leq c\left\|\zeta^{l-1}\right\|_V^2\leq a\left(\zeta^{l-1}, \zeta^{l-1}\right)$, by using Hölder inequality, we obtain

Note that $\zeta^N = 0$ and $p\in H^1(J; H^2(\Omega))\cap H^2(J; U)$, summing $l$ from $M^*\; (0 \leq M^*\leq N)$ to $N$, we get

Then (3.8) follows from (3.9)-(3.10), (3.13) and the triangle inequality.

Now, we estimate the error of the control variable. We need give two intermediate variables $(y_h^l(u), p_h^l(u))\in V^h\times V^h, \; l = 1, 2, \cdots, N,$ satisfies the following system:

For ease of exposition, we set

It is clear that $\varphi^0 = 0$ and $\zeta^N = 0$.

Lemma 3.3 Let $(y, p, u)$ and $(y_h(u), p_h(u))$ be the solutions of (2.7)-(2.9) and (3.14)-(3.15), respectively. Assume that $u, p\in H^1(J; U)$. Then there exists a positive constant $C$ independent of $k$ and $h$, so the following conclusion holds

Proof. It follows from variational inequalities (2.9), (2.21) and Hölder inequality that

Note that $\varphi^0 = 0$ and $\zeta^N = 0$, then it follows from (2.19)-(2.20) and (3.14)-(3.15) that

For the second term, note that $p\in H^1(J; U)$, according to Hölder inequality, we have

Let $\delta$ be small enough, then (3.16) follows from (3.17)-(3.19).

Lemma 3.4. Let $(y, p, u)$ be the solution of (2.7)-(2.9), $(y_h(u), p_h(u))$ is defined in (3.14)-(3.15), the conditions of obedience are the same as the three Lemmas 3.1-3.3, then

Proof. We set

where

From (2.7) and (2.19), for $\forall w_h \in V^h$ we have

We set $w_h = \theta^l_1$, so

Note that $0\leq c\left \|\theta_1^l\right\|_V^2\leq a\left(\theta_1^l, \theta_1^l\right),$ according to Hölder inequality, we obtain

Summing $i$ from $1$ to $N^*\; (1\leq N^*\leq N)$, noting $y\in H^1(J; H^2(\Omega))\cap H^2(J; U)$ and $y(x, 0)\in H^2(\Omega)$, we get

Then (3.20) follows from (3.21)-(3.22), (3.26) and Triangle inequality.

Lemma 3.5. Let $(y, p, u)$ be the solution of (2.7)-(2.9), $(y_h(u), y_h(u))$ is defined in (3.14)-(3.15). Then

Proof. By using Lemma 3.2 and Lemma 3.4, we get that

and

Then from (3.28)-(3.29) and embedding theorem, we have that

Thus,

We obtain (3.27).

Now we combine Lemmas 3.1-3.5 to come up with the following main result.

Theorem 3.1. Let $(y, p, u)$ and $(y_h, p_h, u_h)$ be the solutions of (2.7)-(2.9) and (2.19)-(2.21), respectively. Suppose all the conditions in Lemma 3.1-5 are valid. Then

Proof. It is easy to see that (3.32) follows from (3.3), (3.8), (3.16) and (3.27).

4.   A posteriori error estimates for OCP

In this section, we shall derive a posteriori error estimates for the backward Euler variational discretization approximation scheme.

First of all, we recall two important results:

Lemma 4.1. ^[6] Let $\pi_h$ be the standard $Lagrange$ interpolation operator. For $m = 0$ or $1$, $q > \frac{n}{2}$ and $\forall v\in W^{2, q}(\Omega)$, the conclusion is as follows

Lemma 4.2. ^[26] For all $v\in W^{1, q}(\Omega^h)$, $1\leq q < \infty$,

If we set

It can be shown that

where $y(u_h)$, $p(u_h)$ satisfies the system as follows:

For the needs of the paper, now we give two Lemmas and prove them.

Lemma 4.3. Let $(y, p, u)$ and $(y_h, p_h, u_h)$ be the solutions of (2.7)-(2.9) and (2.19)-(2.21), respectively. Then there is a constant $C$ independent of $k$ and $h$, it reduces that

Proof. It follows from $\delta$-Cauchy inequality, the variational inequality in (2.9) and (2.21), we have that

Let $\delta$ be small enough, then (4.3) follows from (4.4). For any function $w\in C(J; L^2(\Omega))$, we let

For $l = 1, 2, \cdot\cdot\cdot, N$, we set

Then the optimality conditions (2.19)-(2.21) can be rewritten as:

Lemma 4.4. Let $(y_h, p_h, u_h)$ be the solution of (4.5)-(4.7), and $(y(u_h), p(u_h))$ as defined in (4.1)-(4.2), so there is a constant $C$ independent of $k$ and $h$, it holds that

where

where $h_s$ is the size of the face $s = \bar{\lambda}_s^1\cap\bar{\lambda}_s^2$, with $\lambda_s^1$, $\lambda_s^2$ are two neighboring elements in ${\Gamma}^h$, $\left[A\nabla\hat{y}_h\cdot n\right]_s$ and $\left[A^*\nabla\tilde{p}_h\cdot n\right]_s$ are the $A-normal$ and $A^*-normal$ derivative jumps over the interior face $s$, respectively, defined by

where $n$ is the normal vector on $s = \lambda_s^1\cap\lambda_s^2$ outwards $\lambda_s^1$. Then we define $\left[A\nabla\hat{y}_h\cdot n\right]_s = 0$ and $\left[A^*\nabla\tilde{p}_h\cdot n\right]_s = 0$ when $s\subset\partial\Omega$.

Proof. Let $e^p = p(u_h)-u_h$, and $e_I^p = \hat{\pi}_he^p$, where $\hat{\pi}_h$ is the interpolation defined in Lemma 4.1. Note that $p(u_h)(x, T) = 0$ and $p_h(x, T) = 0$, using integration by parts, we have

It follows from $a(u, u)\geq c\|u\|_V^2, \, \forall v \in V$, combining equations (4.2) and (4.6), we have

For the first term, by using $\delta$-Cauchy inequality, Lemma 4.1 and Lemma 4.2, we get

From $\delta$-Cauchy inequality, Lemma 4.1 and Lemma 4.2, we obtain

Using $\delta$-Cauchy inequality, we get

It yields from Friedriechs inequality and $\delta$-Cauchy inequality, so we have

Let $\delta$ be small enough, we obtain

Similarly, assume $e^y = y(u_h)-y_h$, and its average interpolation is $e_I^y$. It holds from integration by parts that

It follows from $\delta$-Cauchy inequality, Lemmas 4.1, Lemma 4.2, (4.1) and (4.5), we obtain

Let $\delta$ be small enough, we have

It follows from the assumptions on $A(x)$ and Friedriechs inequality, we obtain

From (4.9)-(4.16), (4.8) is derived.

Theorem 4.1. Let $(y, p, u)$ and $(y_h, p_h, u_h)$ be the solutions of (2.7)-(2.9) and (2.19)-(2.21), respectively. Assume that all the conditions in Lemmas 4.1-4.4 are valid. Then there exists a constant $C$ independent $h$ and $k$, it yields the result as follows

where $\eta_1, \ \eta_2\, \cdot\cdot\cdot, \, \eta_8$ are defined in Lemma 4.4.

Proof. It follows from Friedriechs inequality and the conditions on $A(x)$, we have

Note that

From the regularity estimation of (2.7)-(2.8) minus (4.1)-(4.2), we have

Then (4.17) follows from (4.3), (4.8) and (4.18)-(4.21).

5.   Conclusion and future works

In this paper we discuss the variational discretization for the nonlinear parabolic OCP. We derive a priori error estimates where $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h^2+k)$ and a posteriori error estimates of residual type. The results for these error estimates by variational discretization be an extension of the linear parabolic problems.

In our future work, we shall use this method to deal with fourth order parabolic optimal control problems, including linear and nonlinear styles.

Acknowledgments

This work is supported by National Science Foundation of China (11201510), China Postdoctoral Science Foundation (2017T100155, 2015M580197), Youth Innovative Talents Project (Natural Science) of Research on Humanities and Social Sciences in Guangdong Normal University (2017KQNCX265), General Scientific Research Project of "Innovation and Strengthening School Engineering" of Guangdong Education Department (2016GXJK227), Innovation Team Building at Institutions of Higher Education in Chongqing (CXTDX201601035), and Chongqing Research Program of Basic Research and Frontier Technology (cstc2019jcyj-msxmX0280), and School projects of Huashang College Guangdong University of Finance and Economics(2020HSDS02).

Conflict of interest

The authors declare no conflict of interest in this paper.