Finite element approximation of time fractional optimal control problem with integral state constraint

1.   Introduction

This paper aims to develop a space-time finite element approximation of optimal control problem governed by time fractional diffusion equation and subjected to state constraint in integral form. Let $\Omega$ be a bounded domain of $R^{d}(d = 1, 2, 3)$ with sufficiently smooth boundary $\partial\Omega$. Set $\Omega_{T} = \Omega\times(0, T)$. We consider the following optimal control problem governed by time-fractional diffusion equations:

subject to

and

Here $y_{d}$ is the desired state. $\gamma > 0$ is the regularization constant. $f\in L^2(\Omega_T)$ is a given function.

The research of control problem governed by differential equation forms a hot topic in the past decades. Lots of literatures are devoted to developing theoretical analysis or numerical methods(^{[1,2,3,4,5,6,7]}). As we know, fractional calculus has a very long history and was applied into many fields of science and engineering. For instance, the fractional PDEs arise in many engineering applications such as anomalous diffusion on fractals and fractional random walk(^[8,9]), and contaminant transport in groundwater flow(^[10,11]). Over the past decades lots of numerical methods and algorithms are developed to discretize fractional derivative and solve the fractional differential equations, including finite difference methods ^[12,13], finite element methods ^{[14,15,16,17]}, spectral methods ^[18,19] and so on.

In recent years optimal control problem governed by fractional PDEs has attracted the researcher's attention with the rapid development of numerical methods for fractional PDEs. For optimal control problem governed by time fractional PDEs we refer to ^{[20,21,22,23,24]} for finite element approximation, ^[25,28] for spectral Galerkin discretization. For optimal control problem governed by space fractional PDEs we refer to ^[26] for spectral Galerkin discretization, and ^[27] for finite element approximation. As far as we know, the researches on optimal control problem governed by fractional PDEs mainly focus on control constrained problem. The study with respect to state constrained fractional optimal control problem is very limited. In ^[28] spectral Galerkin approximation of time fractional optimal control problem with integral state constraint was discussed. In ^[29] an optimal control problem governed by fractional elliptic equation with pointwise state constraint was investigated. In ^[30] the authors discussed finite element approximation of space fractional optimal control problem with integral state constraint.

In the present paper we consider an optimal control problem governed by time fractional diffusion equation with state constraint in integral form. The first order optimality condition is derived and the corresponding regularity is discussed. A space-time finite element discrete scheme is built up based on piecewise constant discontinuous Galerkin discretization for temporal discretization and conforming linear finite elements for spatial discretization. A priori error estimate for state, adjoint state and control variables is derived. Numerical example is given to verify the theoretical findings.

The structure of this paper is as follows. In Section 2, the preliminaries of fractional derivative and Sobolev space are introduced. In Section 3 the space-time finite element approximation of the control problem is discussed. In Section 4 a priori error estimates are proved. Numerical example is given to verify the theoretical results in Section 5.

2.   Preliminaries

In this section we introduce some basic knowledge of the fractional calculus and the Sobolev space.

For $0 < \alpha < 1,$ the left and right Riemann-Liouville fractional derivative are defined as follows

Let $H^{\delta}(I)$ and $H^{\delta}_0(I)$ denote the usual Sobolev space of order $\delta$ on $I$ with norm $\|\cdot \|_{H^{\delta}(I)}$ and semi-norm $|\cdot |_{H^{\delta}(I)}$. According to ^[14], we have the following properties.

Lemma 2.1. Assume that $-\infty < a < b < \infty$. If $v\in H^{\alpha}(a, b)$, then the following estimates hold

Further, if $v, w\in H^{\frac{\alpha}{2}}(a, b)$, then

We introduce the definition of space $\dot{H}^{s}(\Omega)$ following ^[31]. Let $\{\lambda_i\}_{i = 1}^{\infty}$ and $\{ \phi_i\}_{i = 1}^{\infty}$ denote eigenvalues and $L^{2}(\Omega)$-orthogonal eigenfunctions of $-\Delta$ operator on domain $\Omega$ with zero Dirichlet boundary condition. For every $v\in L^2(\Omega)$ we have the representation $v = \sum\limits_{i = 1}^{\infty}a_{i}\phi_{i}$, where $a_i = (v, \phi_i)$. For $s\geq0$, let $\dot{H}^{s}(\Omega)\subset L^{2}(\Omega)$ defined by

To define the weak formulation of state equation we also need to introduce the space

and endow this space with the norm

3.   Space-time finite element discretization

In this section we consider space-time finite element discretization of optimal control problem. To this end we begin with deriving the first order optimality condition and regularity of the solution.

3.1.   First order optimality and regularity

Theorem 3.1. Assume that $(y, u)$ is the solution of control problem (1.1) and (1.2). Then there exist a real number $\xi \geq 0$ and the adjoint state $p$ satisfying the following first order optimality system:

and

Proof. According to ^[17], for given $f +u \in L^{2}(\Omega_{T})$ the following weak formulation admits a unique solution $y\in V$

We introduce the operator $S$ mapping form $L^{2}(\Omega_{T})$ to $V$, and the operator $B$ mapping form $V$ to $L^{1}(\Omega_{T})$ $(V\hookrightarrow L^{1}(\Omega_{T})).$ We define $G:L^{2}(\Omega_{T}) \rightarrow L^{1}(\Omega_{T})$ by $G = S\circ B.$ Then we can rewrite the optimal control problem as the following optimization problem

subject to

In order to derive the first order optimality condition, we set $H(u) : = \int_{\Omega_{T}}y(u)dxdt-\delta.$ Here $y(u) : = Su$ is the solution of the state equation. According to ^[32,33], there exists a real number $\xi \geq0$ such that

and

Here $G(u, \xi) = \hat{J}(u)+\xi H(u)$ denotes the Lagrange functional with $\xi$ being the Lagrange multiplier.

By (3.6), we obtain

Then we have

By a simple calculation we derive

and

Thus we arrive at

By the state equation, we deduce

To simplify $(3.8)$, we introduce the adjoint state equation

By the integration by parts we deduce

Combining this with (3.8), we have

Thus we arrive at

i.e.

Remark 3.2. According to (3.3), we can deduce that

In the following we investigate the regularity of the solution to the optimal control problem. Assume that $\eta$ is the solution of the state equation with right hand term $g(x, t)$. Then according to ^[17] we have the following regularity estimates.

Lemma 3.3. Assume that $g\in L^{2}(0, T;L^{2}(\Omega)), 0 < \alpha < 1$. Then we have

Furthermore, for $g\in H^{1-\alpha}(0, T;L^{2}(\Omega)), \frac{1}{2} < \alpha < 1$, we have

Remark 3.4. Note that $\xi$ is a constant. Therefore the state and adjoint state have the same regularity to (3.9) for $f, u, y_d\in L^{2}(0, T;L^{2}(\Omega)), 0 < \alpha < 1$.

Further we restrict that $f, y_d\in H^{1-\alpha}(0, T;L^{2}(\Omega)), \frac{1}{2} < \alpha < 1$. Note that $\gamma u = -p$ and $\frac{1}{2} < \alpha < 1$. Then by (3.9) we have $u\in H^{\alpha}(0, T;L^{2}(\Omega))\subset H^{1-\alpha}(0, T;L^{2}(\Omega))$. Combining with $f\in H^{1-\alpha}(0, T;L^{2}(\Omega)), \frac{1}{2} < \alpha < 1$ leads to an improved regularity for the state variable $y$:

For $y_d\in H^{1-\alpha}(0, T;L^{2}(\Omega)), \frac{1}{2} < \alpha < 1$, above result further improves the regularity of the adjoint state variable $p$, i.e.,

3.2.   Space-time discrete scheme

The weak formulation of the control problem can be characterized as:

subject to

Then the corresponding first order optimality system reads

and

Let $0 = t_{0} < t_{1} < ... < t_{J} = T$ be a partition of [0, T] with $\tau = \frac{T}{J}$. Set $I_{j}: = (t_{j-1}, t_{j})$ for each $1\leq j\leq J.$ Let $\mathcal{T}_{h}$ be a quasi-uniform triangulation of $\Omega$. We denote by $h$ the maximum diameter of the elements in $\mathcal{T}_{h}$ and define

Set $U_{h} = V_{h}\cap \mathcal{K}$. Then the space-time finite element approximation of control problem can be characterized as follows

subject to

In an analogous way to continuous case we can derive the discrete first order optimality system:

and

Remark 3.5. According to (3.23), the discrete Lagrange multipliers satisfies

4.   Error analysis

In this section we derive a priori error estimates for the space-time finite element discretization of the optimal control problem. For this purpose we need to introduce the following auxiliary problems for every $w\in L^{2}(0, T;\dot{H}^{1}(\Omega))$

For clarity we assume that $f, y_d\in L^{2}(0, T;L^{2}(\Omega))$ in the following analysis of Lemmas 4.1–4.3 and Theorems 4.4–4.5.

Lemma 4.1. Assume that $y$ and $y_{h}$ are the solutions of (3.15) and (3.21), respectively. Then we have

Proof. Combining (3.15) and (4.1) we have

By Lemma 3.3 and the embedding theorem we have

By (3.21) and (4.1) we get

Since $y_{h}$ is the finite element approximation of $y(u_{h})$, according to ^[17] we have

By the triangle inequality we obtain

Lemma 4.2. Assume that $(y, p, u, \xi)$ and $(y_{h}, p_{h}, u_{h}, \xi_{h})$ are the solutions of the optimality system (3.15)–(3.18) and the discrete counterpart, respectively. Then we have

Proof. By (3.16) and (4.3) we have

By Lemma 2.1 and $w = p-p(y_{h})$ we have

This implies

Combining (3.22) and (4.3) we obtain

Note that $p_{h}$ is the finite element approximation of $p(y_{h}).$ Then we can derive by ^[17]

Using the triangle inequality and Lemma 4.1 we obtain

Next we are going to derive the estimate of $|\xi-\xi_{h}|$ and $\| u-u_{h}\|_{L^{2}(0, T;L^{2}(\Omega))}$.

Lemma 4.3. Assume that $(y, p, u, \xi)$ and $(y_{h}, p_{h}, u_{h}, \xi_{h})$ are the solutions of the optimality system (3.15)–(3.18) and the discrete counterpart, respectively. Then we have

Proof. By (3.16) and (4.2) we have

Choosing $w = \psi\in V$ satisfy $\| \psi\|_V\leq C$ and $\frac{1}{| \Omega_{T}|}\int_{\Omega_{T}}\psi dxdt = 1$ and combining Lemma 3.3 we get

Setting $P = \frac{1}{|\Omega_{T}|}\int_{\Omega_{T}}(p-p(u_{h}))dxdt$ and choosing $w = p-p(u_{h})-P\psi$ in (4.7) leads to

Since $(\xi-\xi_{h}, p-p(u_{h})-P\psi)_{\Omega_{T}} = 0,$ then we can derive

Using Lemma 2.1 we have

Note that $\| p-p(u_{h})\|_{L^{2}(0, T;L^{2}(\Omega))}\leq C\| \nabla(p-p(u_{h}))\|_{L^{2}(0, T;L^{2}(\Omega))}.$ Further we have by Young inequality

Then we derive

By definition of $P$, (3.18), (3.24) and (4.5) we have

Combining (4.2) and (4.3) we get

By Lemma 2.1 and $w = p(y_{h})-p(u_{h})$ we have

Combined with the above inequality, we get

Then we derive

Inserting the above estimate into (4.8) we obtain

Finally we need to estimate $\| u-u_{h}\|_{L^{2}(0, T;L^{2}(\Omega))}$.

Theorem 4.4. Assume that $(y, p, u, \xi)$ and $(y_{h}, p_{h}, u_{h}, \xi_{h})$ be the solutions of (3.15)–(3.18) and the discrete counterpart, respectively. Then the following estimate holds

Proof. By (3.18) and (3.24) we have

Combining (3.15) and (4.1) and choosing $v = p(y_{h})-p$ leads to

Using (3.16) and (4.3), and setting $w = y-y(u_{h})$ yields

By using Lemma 2.1 we derive

Using above inequalities we further derive

Further we have

Indeed, we have

and

Then we have $(\xi-\xi_h, y-y_h)_{\Omega_T}\geq 0$. By Lemma 4.3 and Young inequality, we deduce

Using (4.4)–(4.6) we conclude

Based on above estimates, we can obtain the following result.

Theorem 4.5. Assume that $(y, p, u, \xi)$ and $(y_{h}, p_{h}, u_{h}, \xi_{h})$ are the solution of (3.15)–(3.18) and (3.21)–(3.24), respectively. Then the following estimate holds

Proof. Combining Lemma 4.1–4.4 leads to the theorem results.

Remark 4.6. If $f, y_d\in H^{1-\alpha}(0, T;L^{2}(\Omega))$, according to ^[17], we get $\|y(u_{h})-y_{h}\|_{L^{2}(0, T;L^{2}(\Omega))}\leq C(h^{2}+\tau)$ and $\|p(y_{h})-p_{h}\|_{L^{2}(0, T;L^{2}(\Omega))}\leq C(h^{2}+\tau)$. By analogy with the above error estimate derivation process, we get the following error estimates

5.   Numerical experiments

In this section, we use the projected gradient algorithm (^[5]) to effectively solve the optimal control problem.

5.1.   Projected Gradient Algorithm

Set the objective gradient function $d^{k} = \gamma u_{h}^{k}+p_{h}^{k}.$ Then we have

Here

and

We set $p^{k}_{h} = \hat{p}^{k}_{h}+\xi^{k}_{h}\psi_{h}$ with

and

Further, we define $\hat{y}^{k+1}_{h}$ by

Here $\hat{u}^{k+1}_{h} = u^{k}_{h}-\rho(\gamma u^{k}_{h}+\hat{p}^{k}_{h})$. Note that

Then we have

Here $\varphi_{h}$ satisfies

In order to guarantee the state constraints, we choose $\xi_{h}^{k}$ as follows

It is easy to check that $\int_{\Omega_{T}}y_{h}^{k+1}dxdt\leq \delta$. The projected gradient algorithm is described in detail.

5.2.   Numerical Examples

Example 5.1. In this example, we consider the optimal control problem with $\Omega = [0, 1], T = 1.$ The exact solution are given by

In this example we consider a uniform partition for space with $h = \frac{1}{M}$. We first consider the case with smooth solution. Tables 1–4 present the errors and convergence orders of state, control and Lagrange multiplier for $\alpha = \frac{1}{3}$ and $\frac{2}{3}$ with ${s} = 2$. Tables 5 and 6 present the errors and convergence orders of state, control and Lagrange multiplier for $\alpha = 0.4$ with ${s} = 0.51$. We can observe that the convergence rates with respect to time and space variable are in agreement with the theoretical findings predicted in Remark 4.6, i.e., first order convergence in time and second order convergence in space.

Secondly we consider the nonsmooth case about time variable with ${s} = \alpha-0.49$. In this case the right hand term $f$ and $y_d$ with respect to time belong to $L^2(0, T)$. The errors and convergence rates of state, control and Lagrange multiplier are listed in Tables 7 and 8 with $\alpha = 2/3$. In this case we can observe that the convergence rate for space variable and the convergence rate for time variable approach to $2$ and $\alpha$, which are in agreement with the theoretical findings given in Theorem 4.5.

6.   Conclusion

In this paper we discussed a space-time finite element approximation of time fractional optimal control problem with integral state constraint. A priori error estimate for the discrete scheme is derived. Numerical examples are presented to illustrate the theoretical findings.

In our future work we are going to investigate the finite element approximation of time fractional optimal control problem with integral state constraint taking the form $\int_{\Omega}y(x, t)\leq\delta, t\in [0, T]$.

Acknowledgments

The researth was supported by National Natural Science Foundation of China (No. 11971276) and National Natural Science Foundation of Shandong Province (No. ZR2016JL004, ZR2017MA020).

Conflict of interest

The authors declare there is no conflict of interests.