Open Access

A posteriori error estimates for the fractional optimal control problems

Journal of Inequalities and Applications20152015:141

https://doi.org/10.1186/s13660-015-0662-z

Received: 11 November 2014

Accepted: 9 April 2015

Published: 22 April 2015

Abstract

In this paper, we study the spectral approximation for a constrained optimal control problem governed by the time fractional diffusion equation. A posteriori error estimates are obtained for both the state and the control approximations. Some numerical experiments are carried out to show that the obtained a posteriori error estimates are reliable.

Keywords

fractional optimal control problemspace-time spectral method a posteriori error

1 Introduction

Optimal control problems have been subject of many research works in scientific and engineering computing. The literature on this field is huge, and it is impossible to give even a very brief review. It has been found that the fractional order model can provide a more realistic description for some kind of complex systems in the fields covering control theory [1], viscoelastic materials [2, 3], anomalous diffusion [46], advection and dispersion of solutes in porous or fractured media [7], etc. [810]. Consequently, an optimal control problem for fractional differential equations initiates a new research direction, and we see a growing interest in this topic from both scientific and engineering communities.

A general formulation and a solution scheme for the fractional optimal control problem (FOCP) were first proposed in [11], where the fractional variational principle and the Lagrange multiplier technique were used. Following this idea, Frederico and Torres [12] formulated a Noether-type theorem in the general context and studied fractional conservation laws. Mophou [13] applied the classical control theory to a fractional diffusion equation, involving a Riemann-Liouville fractional time derivative. Dorville et al. [14] later extended the results of [13] to a boundary fractional optimal control.

Recently, some efforts have been put into developing spectral methods for solving FOCPs. For instance, a numerical direct method based on the Legendre orthonormal basis and operational matrix of Riemann-Liouville fractional integration were introduced in [15] to solve a general class of FOCP, and the convergence of the proposed method was also extensively discussed. In [16], the Legendre spectral-collocation method was applied to obtain approximate solutions for some types of FOCPs. Ye and Xu [17] proposed a Galerkin spectral method to solve a linear quadratic FOCP associated with the time fractional diffusion equation with Caputo fractional derivative, and a detailed error analysis was carried out. However, to the best of our knowledge, much less research is available for the a posteriori error estimation for problems involving fractional derivative, especially the one for FOCP.

The purpose of this paper is to derive a posteriori error estimates for the FOCP governed by the time fractional diffusion equation (TFDE) with Riemann-Liouville fractional derivative. Let \(\Lambda=(-1,1)\), \(I=(0,T)\), \(\Omega=\Lambda\times I\). We consider the following linear-quadratic optimal control problem for the control variable q under constraints:
$$ \min_{q\in K} \biggl\{ \frac{1}{2}\int_{\Omega} \bigl(u(x,t)-\bar{u}(x,t)\bigr)^{2} \,\mathrm{d}x\,\mathrm{d}t+ \frac{\lambda}{2}\int_{\Omega}q^{2}(x,t) \,\mathrm{d}x\, \mathrm{d}t \biggr\} , $$
(1.1)
where λ and \(\bar{u}\) are given, u is governed by the TFDE as follows:
$$\begin{aligned}& {}_{0}^{R}\partial_{t}^{\alpha}u(x,t)- \partial_{x}^{2}u(x,t) =f(x,t)+q(x,t),\quad \forall(x,t)\in \Omega, \\& I_{t}^{1-\alpha}u(x,0) =0, \quad \forall x \in\Lambda, \\& u(-1,t)=u(1,t) =0, \quad \forall t\in I, \end{aligned}$$
(1.2)
with \({}_{0}^{R}\partial_{t}^{\alpha}\) (\(0 <\alpha< 1\)) denoting the left Riemann-Liouville fractional derivative, \(I_{t}^{1-\alpha}\) denoting the Riemann-Liouville fractional integral, and
$$K= \biggl\{ q\in L^{2}(\Omega) : \int_{\Omega}q(x,t)\, \mathrm{d} x\,\mathrm{d}t\geq0 \biggr\} . $$
The main physical purpose for adopting and investigating diffusion equations of fractional order is to describe phenomena of anomalous diffusion usually met in transport processes through complex and/or disordered systems including fractal media [18]. In [19], Nigmatullin used the fractional diffusion equation to describe diffusion in media with fractal geometry. Mainardi [3] pointed out that the propagation of mechanical diffusive wave in viscoelastic media can be modeled by TFDE. An interesting review on the anomalous diffusion by Metzler and Klafter [20] has appeared to which (and references therein) we refer the interested reader. Applying a time fractional integration of order α to both sides of the first equation in (1.2) allows us to eliminate the time fractional derivative on the L.H.S. leading to the integral form
$$ u(x,t)-\frac{1}{\Gamma(\alpha)}\int_{0}^{t} \frac{\partial _{x}^{2}u(x,\tau)}{(t-\tau)^{1-\alpha}}\, \mathrm{d}\tau =\frac{1}{\Gamma(\alpha)}\int_{0}^{t} \frac{f(x,\tau)+q(x,\tau )}{(t-\tau)^{1-\alpha}}\, \mathrm{d}\tau. $$
(1.3)

The outline of the paper is as follows. In the next section we discuss the optimality conditions and spectral discretization of the optimal problem. A posteriori error estimate is derived in Section 3. Finally, in Section 4, we carry out some numerical tests to verify the theoretical results.

For a domain \(\mathcal{O}\), which may be Λ, I or Ω, we use \(L^{2}(\mathcal{O})\), \(H^{s}(\mathcal{O})\), and \(H_{0}^{s}(\mathcal{O})\) to denote the usual Sobolev spaces, equipped with the norms \(\Vert \cdot \Vert _{0,\mathcal{O}}\) and \(\Vert \cdot \Vert _{s,\mathcal{O}}\) respectively. For the Sobolev space X with the norm \(\Vert \cdot \Vert _{X}\), we define the space \(H^{s}(I;X):=\{v|\Vert v(\cdot,t)\Vert _{X} \in H^{s}(I)\} \) endowed with the norm \(\Vert v\Vert _{H^{s} (I;X)}:=\Vert \Vert v({\cdot,t})\Vert _{X}\Vert _{s,I}\). Particularly, when X stands for \(H^{\mu}(\Lambda)\) or \(H^{\mu }_{0}(\Lambda)\), the norm of the space \(H^{s}(I;X)\) will be denoted by \(\Vert \cdot \Vert _{\mu, s,\Omega}\). Hereafter, in cases where no confusion would arise, the domain symbols I, Λ, Ω may be dropped from the notations.

2 Optimization and spectral approximation of the problem

For a weak formula of the state equation (1.2), we introduce the control space \(L^{2}(\Omega)\) and the state space [21]
$$B^{s}(\Omega)=H^{s}\bigl(I,L^{2}(\Lambda)\bigr) \cap L^{2}\bigl(I,H_{0}^{1}(\Lambda)\bigr),\quad \forall s>0, $$
equipped with the norm
$$\Vert v\Vert _{B^{s}(\Omega)} = \bigl(\Vert v\Vert _{H^{s}(I,L^{2}(\Lambda))}^{2}+ \Vert v\Vert _{L^{2}(I,H_{0}^{1}(\Lambda))}^{2} \bigr)^{\frac{1}{2}}. $$
Then a weak formulation for the state equation (1.2) reads as follows: given \(q, f \in L^{2}(\Omega)\), find \(u\in B^{\frac{\alpha}{2}}(\Omega)\) such that
$$ \mathcal {A}(u,v)=(f+q,v)_{\Omega},\quad \forall v \in B^{\frac{\alpha}{2}}( \Omega), $$
(2.1)
where the bilinear form \(\mathcal{A}(\cdot,\cdot)\) is defined by
$$ \mathcal{A}(u,v):= \bigl( {}_{0}^{R}\partial_{t}^{\frac{\alpha}{2}} u, {}^{R}_{t}\partial_{T}^{\frac{\alpha}{2}} v \bigr)_{\Omega}+(\partial_{x} u,\partial_{x} v)_{\Omega}. $$
Here, \({}_{0}^{R}\partial_{t}^{\frac{\alpha}{2}}\) and \({}^{R}_{t}\partial _{T}^{\frac{\alpha}{2}}\) respectively denote the left and right Riemann-Liouville fractional derivatives of order \(\frac{\alpha}{2}\).
By defining the cost functional
$$ \mathcal{J}(q,u):=\frac{1}{2}\Vert u-\bar{u}\Vert _{0,\Omega}^{2}+ \frac {\lambda}{2}\Vert q\Vert _{0,\Omega}^{2},\quad (q,u)\in K \times B^{\frac{\alpha}{2}}(\Omega), $$
(2.2)
with the given desired state \(\bar{u} \in L^{2}(\Omega)\), the optimal control problem reads as follows: find \((q^{\ast },u(q^{\ast})) \in K \times B^{\frac{\alpha}{2}}(\Omega)\) such that
$$ \mathcal{J}\bigl(q^{\ast},u\bigl(q^{\ast}\bigr)\bigr) = \min _{(q,u)\in K \times B^{\frac{\alpha}{2}}(\Omega)} \mathcal{J}(q,u)\quad \text{subject to (2.1)}. $$
(2.3)
The well-posedness of the state problem ensures the existence of a control-to-state mapping \(q\mapsto u=u(q)\) defined through (2.1). By means of this mapping we introduce the reduced cost functional \(J : L^{2}(\Omega) \rightarrow \mathbb{R}\) as follows:
$$J(q):=\mathcal{J}\bigl(q,u(q)\bigr), \quad q\in L^{2}(\Omega). $$
Then the optimal control problem (2.3) is equivalent to finding \(q^{\ast}\in K\) such that
$$ J\bigl(q^{\ast}\bigr) = \min_{q\in K}J(q). $$
(2.4)
The first order necessary optimality condition for (2.4) reads
$$ J'\bigl(q^{\ast}\bigr) \bigl(\delta q -q^{\ast} \bigr) \geq0,\quad \forall \delta q \in K, $$
(2.5)
where \(J'(q^{\ast})(\cdot)\) is called the gradient of \(J(q)\), defined through the Gâteaux derivative.
It has been proved [17] that
$$ J'(q) (\delta q)=\bigl(\lambda q +z(q),\delta q \bigr)_{\Omega},\quad \forall\delta q\in L^{2}(\Omega), $$
(2.6)
where \(z(q)=z\in B^{\frac{\alpha}{2}}(\Omega)\) is the solution of the following adjoint state equation:
$$ \mathcal{A}(\varphi,z)=(u-\bar{u},\varphi)_{\Omega},\quad \forall \varphi \in B^{\frac{\alpha}{2}}(\Omega). $$
(2.7)
Now, we consider the spectral approximation of the optimal control problem. We define the polynomial space
$$ P_{M}^{0}(\Lambda)=P_{M}(\Lambda)\cap H_{0}^{1} (\Lambda),\qquad S_{L}=P_{M}^{0}( \Lambda) \otimes P_{N}(I)\subset B^{\frac{\alpha}{2}}(\Omega), $$
where \(P_{M}\) denotes the space of all polynomials of degree less than or equal to M, L stands for the parameter pair \((M,N)\).
Then we consider the spectral approximation to the state equation (2.1) as follows: find \(u_{L}(q)\in S_{L}\) such that
$$ \mathcal{A} \bigl(u_{L}(q),v_{L}\bigr)=(f+q,v_{L})_{\Omega}, \quad \forall v_{L} \in S_{L} . $$
(2.8)
Similar to the continuous case, we introduce the semidiscrete reduced cost functional \(J_{L}:L^{2}(\Omega)\rightarrow\mathbb{R}\) as follows:
$$ J_{L}(q):=\mathcal{J}\bigl(q,u_{L}(q)\bigr), \quad q\in L^{2}(\Omega), $$
(2.9)
where \(u_{L}(q)\) is given by (2.8). Then we consider the following auxiliary optimal problem: find \(q^{\ast}\in K\) such that
$$ J_{L}\bigl(q^{\ast}\bigr) = \min _{q \in K} J_{L}(q). $$
(2.10)
The solution \(q^{\ast}\) of the above problem fulfills the first order optimality condition
$$ J'_{L} \bigl(q^{\ast}\bigr) \bigl(\delta q -q^{\ast}\bigr)\geq0, \quad \forall \delta q \in K , $$
(2.11)
where
$$ J'_{L}(q) (\phi)=\bigl(\lambda q+z_{L}(q), \phi\bigr)_{\Omega}, \quad \forall q, \phi\in K, $$
(2.12)
with \(z_{L}(q)\in S_{L}\) being the solution of the semidiscrete adjoint problem
$$ \mathcal{A}\bigl(\varphi_{L},z_{L}(q)\bigr)= \bigl(u_{L}(q) - \bar{u},\varphi_{L}\bigr)_{\Omega }, \quad \forall\varphi_{L} \in S_{L}. $$
(2.13)
Now we consider the approximation of the control space to obtain the full discrete optimal control problem. To this end, we introduce the finite dimensional subspace for the control variable as follows:
$$K_{L}=K \cap\bigl(P_{M} (\Lambda) \otimes P_{N}(I)\bigr). $$
Then the full discrete optimal control problem reads as follows: find \(q_{L}^{\ast}\in K_{L}\) such that
$$ J_{L}\bigl(q_{L}^{\ast}\bigr) = \min _{q_{L} \in K_{L}} J_{L}(q_{L}), $$
(2.14)
where \(J_{L}(\cdot)\) is defined in (2.9). The unique solution of (2.14), \(q_{L}^{\ast}\), satisfies the following optimality condition:
$$ J_{L}'\bigl(q_{L}^{\ast}\bigr) \bigl( \delta q - q_{L}^{\ast}\bigr)\geq0,\quad \forall\delta q \in K_{L}. $$
(2.15)

3 A posteriori error estimates

We aim in this section at deriving estimates of the error between a continuous solution and its spectral approximation in terms of known and computable quantities, i.e., a posteriori error estimates. We will confine ourselves to the so-called residual-based estimates [22]. To simplify the notations, we let c be a generic positive constant independent of any functions and of any discretization parameters. We use the expression \(A\lesssim B\) to mean that \(A\leq cB\).

The error analysis will make use of some projection operators. The orthogonal projector \(\Pi _{M}^{1,0}:H_{0}^{1}(\Lambda)\rightarrow P_{M}^{0}(\Lambda)\) is defined by \(\forall v \in H_{0}^{1}(\Lambda)\), \(\Pi_{M}^{1,0} v\in P_{M}^{0}(\Lambda)\) such that
$$\bigl(\bigl(\Pi_{M}^{1,0} v-v\bigr)', \phi_{M} '\bigr)_{\Lambda}=0,\quad \forall \phi_{M} \in P_{M}^{0}(\Lambda). $$
The following estimates hold [22]: \(\forall v \in H^{m}(\Lambda)\cap H_{0}^{1}(\Lambda)\), \(m \geq1\),
$$\begin{aligned}& \bigl\vert \Pi_{M}^{1,0} v -v\bigr\vert _{1,\Lambda} \lesssim M^{1-m} \Vert v\Vert _{m,\Lambda} , \\& \bigl\Vert \Pi_{M}^{1,0} v -v\bigr\Vert _{0,\Lambda} \lesssim M^{-m} \Vert v\Vert _{m,\Lambda}. \end{aligned}$$
For the \(L^{2}\)-orthogonal projector \(\Pi_{N}\), defined by \(\Pi_{N}v \in P_{N}(I)\), such that \((\Pi_{N} v-v,w_{N})_{I}=0\), \(\forall w_{N}\in P_{N}(I)\), we have
$$ \Vert \Pi_{N} v -v\Vert _{0,I}\lesssim N^{-m} \Vert v\Vert _{m,I}, \quad \forall v \in H^{m}(I), m \geq0. $$
The \(L^{2}\)-orthogonal projector \(\Pi_{M}\) in Λ is defined similarly.

The first step is to derive a posteriori error estimates for the approximation to the control variable.

Lemma 3.1

Suppose \(q^{\ast}\) and \(q^{\ast}_{L}\) are the solutions of (2.4) and (2.14) respectively, then the following estimate holds:
$$ \bigl\Vert q^{\ast}-q_{L}^{\ast}\bigr\Vert _{0,\Omega} \lesssim \bigl\Vert z_{L}\bigl(q_{L}^{\ast} \bigr)-z\bigl(q_{L}^{\ast}\bigr)\bigr\Vert _{0,\Omega}, $$
(3.1)
where \(z_{L}(q_{L}^{\ast})\) and \(z(q_{L}^{\ast})\) are respectively the solutions of (2.13) and (2.7) associated to \(q_{L}^{\ast}\).

Proof

Similar to Lemma 4.3 in [23], it follows from (2.6), (2.1) and (2.7) that for all \(p,q \in L^{2}(\Omega)\),
$$ J'(p) (p-q)-J'(q) (p-q)\geq\lambda \Vert p-q \Vert _{0,\Omega}^{2}. $$
Then in virtue of (2.5) and (2.15) we get, for arbitrary \(p_{L} \in K_{L}\),
$$\begin{aligned}& \lambda\bigl\Vert q^{\ast}-q_{L}^{\ast}\bigr\Vert _{0,\Omega}^{2} \\& \quad \leq J'\bigl(q^{\ast}\bigr) \bigl(q^{\ast}-q_{L}^{\ast}\bigr)-J' \bigl(q^{\ast}_{L}\bigr) \bigl(q^{\ast }-q_{L}^{\ast} \bigr) \\& \quad \leq -J'\bigl(q^{\ast}_{L}\bigr) \bigl(q^{\ast}-q_{L}^{\ast}\bigr) \\& \quad = J'_{L}\bigl(q_{L}^{\ast}\bigr) \bigl(q_{L}^{\ast}-q^{\ast}\bigr) -J' \bigl(q^{\ast}_{L}\bigr) \bigl(q^{\ast}-q_{L}^{\ast} \bigr) +J'_{L}\bigl(q_{L}^{\ast}\bigr) \bigl(q^{\ast}-q_{L}^{\ast}\bigr) \\& \quad = J'_{L}\bigl(q_{L}^{\ast}\bigr) \bigl(q_{L}^{\ast}-p_{L}\bigr) +J'_{L} \bigl(q_{L}^{\ast}\bigr) \bigl(p_{L}-q^{\ast} \bigr) \\& \qquad {}-\bigl(\lambda q_{L}^{\ast}+z\bigl(q_{L}^{\ast} \bigr),q^{\ast}-q_{L}^{\ast}\bigr)_{\Omega} + \bigl(\lambda q_{L}^{\ast}+z_{L}\bigl(q_{L}^{\ast} \bigr),q^{\ast}-q_{L}^{\ast}\bigr)_{\Omega} \\& \quad \leq J'_{L}\bigl(q_{L}^{\ast}\bigr) \bigl(p_{L}-q^{\ast}\bigr)+ \bigl(z_{L} \bigl(q_{L}^{\ast}\bigr)-z\bigl(q_{L}^{\ast} \bigr),q^{\ast}-q_{L}^{\ast}\bigr)_{\Omega} \\& \quad = \bigl(z_{L}\bigl(q_{L}^{\ast}\bigr)+\lambda q_{L}^{\ast},p_{L}-q^{\ast} \bigr)_{\Omega }+\bigl(z_{L}\bigl(q_{L}^{\ast} \bigr)-z\bigl(q_{L}^{\ast}\bigr),q^{\ast}-q_{L}^{\ast} \bigr)_{\Omega}. \end{aligned}$$
(3.2)
Furthermore, as shown in Lemma 5 in [17], it holds
$$ \bigl(z_{L}\bigl(q_{L}^{\ast}\bigr)+\lambda q_{L}^{\ast},\Pi_{M}\Pi_{N} q^{\ast }-q^{\ast} \bigr)_{\Omega}=0. $$
(3.3)
Therefore, by taking \(p_{L}=\Pi_{N}\Pi_{M} q^{\ast} \) in (3.2), then using (3.3) and the Cauchy-Schwarz inequality, we obtain (3.1). □

Theorem 3.1

Let \(q^{\ast}\) be the solution of (2.4), \(u(q^{\ast})\) and \(z(q^{\ast})\) be the corresponding state and the adjoint state respectively. Let \(q^{\ast}_{L}\) be the solution of (2.14) with the corresponding discrete state \(u_{L}(q^{\ast}_{L})\) and the adjoint state \(z_{L}(q^{\ast}_{L})\). Then the following estimate holds:
$$ \bigl\Vert q^{\ast}-q_{L}^{\ast}\bigr\Vert _{0,\Omega} +\bigl\Vert u\bigl(q^{\ast}\bigr)-u_{L} \bigl(q^{\ast}_{L}\bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega)} +\bigl\Vert z\bigl(q^{\ast}\bigr)-z_{L}\bigl(q^{\ast}_{L} \bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega )}\lesssim\eta_{1}+\eta_{2}, $$
where
$$\eta_{1}=\bigl(N^{-\frac{\alpha}{2}} +M^{-1}\bigr) \xi_{1}, \qquad \eta_{2}= \bigl(N^{-\frac{\alpha}{2}} +M^{-1} \bigr)\xi_{2}, $$
with
$$\begin{aligned}& \xi_{1}=\bigl\Vert {}_{t}^{R} \partial_{T}^{\alpha}z_{L}\bigl(q^{\ast }_{L} \bigr)-\partial _{x}^{2}z_{L}\bigl(q^{\ast}_{L} \bigr)-u_{L}\bigl(q^{\ast}_{L}\bigr)+\bar{u} \bigr\Vert _{0,\Omega}, \\& \xi_{2}=\bigl\Vert {}_{0}^{R} \partial_{t}^{\alpha}u_{L}\bigl(q^{\ast }_{L} \bigr)-\partial _{x}^{2} u_{L}\bigl(q^{\ast}_{L} \bigr) -f-q_{L}^{\ast}\bigr\Vert _{0,\Omega}. \end{aligned}$$

Proof

We first estimate \(\Vert q^{\ast}-q_{L}^{\ast} \Vert _{0,\Omega}\). According to (3.1), it suffices to estimate \(\Vert z_{L}(q^{\ast}_{L})-z(q_{L}^{\ast})\Vert _{0,\Omega}\). Let \(e_{z}=z_{L}(q^{\ast}_{L})-z(q_{L}^{\ast})\), \(e_{z}^{L}=\Pi_{N}\Pi_{M}^{1,0} e_{z} \in S_{L}\). It follows from (2.7) and (2.13) that
$$ \mathcal{A}\bigl(e_{z}^{L},e_{z}\bigr)= \bigl(u_{L}\bigl(q^{\ast}_{L}\bigr)-u \bigl(q_{L}^{\ast}\bigr),e_{z}^{L} \bigr)_{\Omega}. $$
It has been proved [21] that for all \(u,v \in B^{\frac{\alpha }{2}}(\Omega)\), the following continuity and coercivity hold:
$$ \mathcal{A}(u,v)\lesssim \Vert u\Vert _{B^{\frac{\alpha }{2}}(\Omega)} \Vert v\Vert _{B^{\frac{\alpha}{2}}(\Omega)},\qquad \mathcal {A}(v,v)\gtrsim \Vert v\Vert ^{2}_{B^{\frac{\alpha }{2}}(\Omega)}. $$
Thus, using (2.1), (2.7), (2.8), and (2.13), we have
$$\begin{aligned}& \bigl\Vert z_{L}\bigl(q^{\ast}_{L}\bigr)-z \bigl(q_{L}^{\ast}\bigr)\bigr\Vert _{B^{\frac {\alpha }{2}}(\Omega)}^{2} \\& \quad \lesssim \mathcal{A}(e_{z},e_{z})=\mathcal {A} \bigl(e_{z}-e_{z}^{L},e_{z}\bigr)+ \mathcal{A}\bigl(e_{z}^{L},e_{z}\bigr) \\& \quad = \mathcal{A}\bigl(e_{z}-e_{z}^{L},z_{L} \bigl(q^{\ast}_{L}\bigr)\bigr)-\mathcal {A}\bigl(e_{z}-e_{z}^{L},z \bigl(q_{L}^{\ast}\bigr)\bigr)+\mathcal{A}\bigl(e_{z}^{L},e_{z} \bigr) \\& \quad = \bigl( {}_{0}^{R}\partial_{t}^{\frac{\alpha}{2}} \bigl(e_{z}-e_{z}^{L}\bigr), {}_{t}^{R} \partial _{T}^{\frac{\alpha}{2}}z_{L}\bigl(q^{\ast}_{L} \bigr) \bigr)_{\Omega} + \bigl(\partial_{x}\bigl(e_{z}-e_{z}^{L} \bigr),\partial_{x} z_{L}\bigl(q^{\ast}_{L} \bigr) \bigr)_{\Omega} \\& \qquad {} - \bigl(u\bigl(q_{L}^{\ast}\bigr)-\bar{u},e_{z}-e_{z}^{L} \bigr)_{\Omega }+\bigl(u_{L}\bigl(q^{\ast}_{L} \bigr)-u\bigl(q_{L}^{\ast}\bigr),e_{z}^{L} \bigr)_{\Omega} \\& \quad = \bigl(e_{z}-e_{z}^{L}, {}_{t}^{R} \partial_{T}^{\alpha}z_{L}\bigl(q^{\ast}_{L} \bigr) \bigr)_{\Omega}-\bigl(e_{z}-e_{z}^{L}, \partial_{x}^{2}z_{L}\bigl(q^{\ast}_{L} \bigr)\bigr)_{\Omega} - \bigl(u\bigl(q_{L}^{\ast}\bigr)- \bar{u},e_{z}-e_{z}^{L} \bigr)_{\Omega} \\& \qquad {}+\bigl(u_{L}\bigl(q^{\ast}_{L}\bigr)-u \bigl(q_{L}^{\ast}\bigr),e_{z}^{L}-e_{z} \bigr)_{\Omega }+\bigl(u_{L}\bigl(q^{\ast}_{L} \bigr)-u\bigl(q_{L}^{\ast}\bigr),e_{z} \bigr)_{\Omega} \\& \quad = \bigl(e_{z}-e_{z}^{L}, {}_{t}^{R} \partial_{T}^{\alpha}z_{L}\bigl(q^{\ast}_{L} \bigr)-\partial _{x}^{2}z_{L}\bigl(q^{\ast}_{L} \bigr) \bigr)_{\Omega} +\bigl(u_{L}\bigl(q^{\ast}_{L} \bigr)-\bar{u},e_{z}^{L}-e_{z} \bigr)_{\Omega} \\& \qquad {}+\bigl(u_{L}\bigl(q^{\ast}_{L}\bigr)-u \bigl(q_{L}^{\ast}\bigr),e_{z}\bigr)_{\Omega} \\& \quad = \bigl(e_{z}-e_{z}^{L}, {}_{t}^{R} \partial_{T}^{\alpha}z_{L}\bigl(q^{\ast}_{L} \bigr)-\partial _{x}^{2}z_{L}\bigl(q^{\ast}_{L} \bigr)-u_{L}\bigl(q^{\ast}_{L}\bigr)+\bar{u} \bigr)_{\Omega} \\& \qquad {} +\bigl(u_{L}\bigl(q^{\ast}_{L}\bigr)-u \bigl(q_{L}^{\ast}\bigr),e_{z}\bigr)_{\Omega} \\& \quad \lesssim \bigl\Vert e_{z}-e_{z}^{L}\bigr\Vert _{0,\Omega}\xi_{1} +\bigl\Vert u_{L} \bigl(q^{\ast}_{L}\bigr)-u\bigl(q_{L}^{\ast} \bigr)\bigr\Vert _{0,\Omega }\Vert e_{z}\Vert _{0,\Omega}. \end{aligned}$$
(3.4)
Furthermore,
$$\begin{aligned}& \bigl\Vert e_{z}-e_{z}^{L}\bigr\Vert _{0,\Omega} \\& \quad \leq \Vert e_{z}-\Pi_{N}e_{z} \Vert _{0,\Omega}+\bigl\Vert \Pi _{N}e_{z}- \Pi_{N}\Pi _{M}^{1,0}e_{z}\bigr\Vert _{0,\Omega} \\& \quad \leq \Vert e_{z}-\Pi_{N}e_{z}\Vert _{0,\Omega} +\bigl\Vert \Pi_{N}\bigl(e_{z}- \Pi_{M}^{1,0}e_{z}\bigr)-\bigl(e_{z}- \Pi_{M}^{1,0}e_{z}\bigr) \bigr\Vert _{0,\Omega} \\& \qquad {}+\bigl\Vert e_{z}-\Pi_{M}^{1,0}e_{z} \bigr\Vert _{0,\Omega} \\& \quad \lesssim \Vert e_{z}-\Pi_{N}e_{z}\Vert _{0,\Omega}+\bigl\Vert e_{z}-\Pi _{M}^{1,0}e_{z} \bigr\Vert _{0,\Omega } \\& \quad \lesssim N^{-\frac{\alpha}{2}}\Vert e_{z}\Vert _{0,\frac{\alpha }{2},\Omega} +M^{-1}\Vert e_{z}\Vert _{1,0,\Omega} \\& \quad \lesssim N^{-\frac{\alpha}{2}}\Vert e_{z}\Vert _{B^{\frac{\alpha }{2}}(\Omega)} +M^{-1}\Vert e_{z}\Vert _{B^{\frac{\alpha}{2}}(\Omega)}. \end{aligned}$$
(3.5)
Plugging (3.5) into (3.4) yields
$$ \bigl\Vert z_{L}\bigl(q^{\ast}_{L}\bigr)-z \bigl(q_{L}^{\ast}\bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega)} \lesssim \eta_{1} +\bigl\Vert u_{L}\bigl(q^{\ast}_{L} \bigr)-u\bigl(q_{L}^{\ast}\bigr)\bigr\Vert _{0,\Omega}. $$
(3.6)
Similarly, set \(e_{u}=u_{L}(q^{\ast}_{L})-u(q_{L}^{\ast})\), and let \(e_{u}^{L}=\Pi_{N}\Pi_{M}^{1,0} e_{u} \in S_{L}\). Then it follows from (2.1) and (2.8) that \(\mathcal{A}(e_{u},e_{u}^{L})=0\), and thus
$$\begin{aligned}& \bigl\Vert u_{L}\bigl(q^{\ast}_{L}\bigr)-u \bigl(q_{L}^{\ast}\bigr)\bigr\Vert _{B^{\frac {\alpha }{2}}(\Omega)}^{2} \\& \quad \lesssim \mathcal{A}(e_{u},e_{u})=\mathcal{A} \bigl(e_{u},e_{u}-e_{u}^{L}\bigr) \\& \quad = \mathcal{A}\bigl(u_{L}\bigl(q^{\ast}_{L} \bigr),e_{u}-e_{u}^{L}\bigr)-\mathcal{A}\bigl(u \bigl(q_{L}^{\ast }\bigr),e_{u}-e_{u}^{L} \bigr) \\& \quad = \bigl( {}_{0}^{R}\partial_{t}^{\alpha}u_{L} \bigl(q^{\ast}_{L}\bigr)-\partial _{x}^{2}u_{L} \bigl(q^{\ast}_{L}\bigr),e_{u}-e_{u}^{L} \bigr)_{\Omega} - \bigl(f+q_{L}^{\ast},e_{u}-e_{u}^{L} \bigr)_{\Omega} \\& \quad \lesssim \xi_{2}\bigl\Vert e_{u}-e_{u}^{L} \bigr\Vert _{0,\Omega} \\& \quad \lesssim \xi_{2} \bigl(N^{-\frac{\alpha}{2}}\Vert e_{u} \Vert _{0,\frac {\alpha}{2},\Omega }+M^{-1}\Vert e_{u}\Vert _{1,0,\Omega} \bigr) \\& \quad \lesssim \xi_{2} \bigl(N^{-\frac{\alpha}{2}}+M^{-1} \bigr) \Vert e_{u} \Vert _{B^{\frac {\alpha}{2}}(\Omega)}. \end{aligned}$$
This leads to
$$ \bigl\Vert u_{L}\bigl(q^{\ast}_{L}\bigr)-u \bigl(q_{L}^{\ast}\bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega)} \lesssim \eta_{2}. $$
(3.7)
Then combining (3.6) and (3.7) gives
$$\bigl\Vert z_{L}\bigl(q^{\ast}_{L}\bigr)-z \bigl(q_{L}^{\ast}\bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega)} \lesssim \eta_{1} +\bigl\Vert u_{L}\bigl(q^{\ast}_{L} \bigr)-u\bigl(q_{L}^{\ast}\bigr)\bigr\Vert _{0,\Omega} \lesssim\eta_{1}+\eta_{2}. $$
Using the above estimate, the inequality \(\|\cdot\|_{0,\Omega} \lesssim \|\cdot\|_{B^{\frac{\alpha}{2}}(\Omega)}\), and Lemma 3.1, we get
$$ \bigl\Vert q^{\ast}-q_{L}^{\ast}\bigr\Vert _{0,\Omega}\lesssim\eta _{1}+\eta_{2}. $$
(3.8)
Furthermore, using the triangle inequalities
$$\begin{aligned}& \bigl\Vert z_{L}\bigl(q^{\ast}_{L}\bigr)-z \bigl(q^{\ast}\bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega)} \leq \bigl\Vert z_{L}\bigl(q^{\ast}_{L}\bigr)-z\bigl(q_{L}^{\ast} \bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega)} +\bigl\Vert z\bigl(q_{L}^{\ast} \bigr)-z\bigl(q^{\ast}\bigr)\bigr\Vert _{B^{\frac{\alpha }{2}}(\Omega )}, \\& \bigl\Vert u_{L}\bigl(q^{\ast}_{L}\bigr)-u \bigl(q^{\ast}\bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega)} \leq \bigl\Vert u_{L}\bigl(q^{\ast}_{L}\bigr)-u\bigl(q_{L}^{\ast} \bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega)} +\bigl\Vert u\bigl(q_{L}^{\ast} \bigr)-u\bigl(q^{\ast}\bigr)\bigr\Vert _{B^{\frac{\alpha }{2}}(\Omega)}, \end{aligned}$$
and the following obvious estimates
$$ \bigl\Vert z\bigl(q_{L}^{\ast}\bigr)-z\bigl(q^{\ast} \bigr)\bigr\Vert _{B^{\frac{\alpha }{2}}(\Omega)} \lesssim \bigl\Vert u\bigl(q_{L}^{\ast} \bigr)-u\bigl(q^{\ast}\bigr)\bigr\Vert _{0,\Omega} \lesssim \bigl\Vert q_{L}^{\ast}-q^{\ast}\bigr\Vert _{0,\Omega}, $$
we obtain
$$ \bigl\Vert u_{L}\bigl(q^{\ast}_{L}\bigr)-u \bigl(q^{\ast}\bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega )}+\bigl\Vert z_{L}\bigl(q^{\ast}_{L}\bigr)-z\bigl(q^{\ast} \bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega)} \lesssim\eta_{1}+\eta_{2}. $$
This completes the proof. □

4 Optimization algorithm and numerical results

4.1 Projection gradient optimization algorithm

In what follows, we propose a projection gradient optimization algorithm to solve the resulting minimization problems. The key of the algorithm is to determine a suitable projector to guarantee that the imposed constraint on the control variable is satisfied. To this end, to any \(q_{L} \in P_{M} (\Lambda) \otimes P_{N}(I)\) we associate the function \(q_{K} =-\min\{0,\overline{q_{L}}\}+q_{L} \) such that \(q_{K}\in K\).

Then we propose the following projection gradient algorithm for the optimal control problem (2.14):
  • Start with an initial control \(q_{L}^{(0)}\).

  • Repeat for \(k =0, 1,\ldots\) .
    • Determine a descent direction: \({J_{L}' (q_{L}^{(k)})}\).

    • Choose a step size \(\rho_{k}\).

    • Update: \({q_{L}^{(k+\frac{1}{2})}}=q_{L}^{(k)}-\rho_{k} J_{L}' (q_{L}^{(k)}) \).

    • Projection: \(q_{L}^{(k+\frac{1}{2})} \rightarrow q_{L}^{(k+1)}:=q_{K}^{(k+\frac{1}{2})}\).

  • Until stopping criterion is satisfied.

The proposed stopping criterion is
$$ \bigl\Vert {J_{L}'\bigl(q_{L}^{(k)} \bigr)}\bigr\Vert \leq\varepsilon, $$
(4.1)
where ε is a pre-defined tolerance. Whenever (4.1) is satisfied for some k, the optimal control variable \(q_{L}^{\ast}\) is supposed to be obtained, i.e., \(q_{L}^{\ast}=q_{L}^{(k)}\).
The key components of the above algorithm include:
  1. (i)

    Determination of the descent direction.

     
  2. (ii)

    Choice of the step size.

     
The details are described below. Given an initial control \(q_{L}^{(0)}\), the corresponding state \(u_{L} (q_{L}^{(0)} )\) is given by the solution of the state equation in (2.8). To apply the stopping criterion \(\Vert {J_{L}' (q_{L}^{(0)} )}\Vert \leq\varepsilon \), we need information on the adjoint state \(z_{L} (q_{L}^{(0)} )\), which is obtained from the adjoint state equation (2.13) for given \(u_{L} (q_{L}^{(0)} )\) and \(q_{L}^{(0)}\). Then the descent direction, that is, the gradient of the objective functional at \(q_{L}^{(0)}\), is calculated through
$$d_{L}^{(0)}:=J_{L}' \bigl(q_{L}^{(0)} \bigr)=z_{L} \bigl(q_{L}^{(0)} \bigr)+\lambda q_{L}^{(0)}. $$
Then, assuming known \(q_{L}^{(k)}\) and \(d_{L}^{(k)}\) at the current (kth) iteration, we update \(q_{L}^{(k)}\) via
$$q_{L}^{(k+\frac{1}{2})}=q_{L}^{(k)}- \rho_{k} d_{L}^{(k)},\qquad q_{L}^{(k+1)}=- \min \bigl\{ 0,\overline{q_{L}^{(k+\frac{1}{2})}} \bigr\} +q_{L}^{(k+\frac{1}{2})}, $$
where \(\rho_{k}\) is the iteration step size determined in a way such that
$$J_{L} \bigl(q_{L}^{(k)}-\rho_{k} d_{L}^{(k)} \bigr)=\min_{\rho>0} J_{L} \bigl(q_{L}^{(k)}-\rho d_{L}^{(k)} \bigr). $$
Such a \(\rho_{k}\) is characterized by
$$ \bigl(z_{L}^{(k+\frac{1}{2})}+\lambda\bigl(q_{L}^{(k)}- \rho_{k} d_{L}^{(k)}\bigr),d_{L}^{(k)} \bigr)_{\Omega}=0, $$
(4.2)
where \(z_{L}^{(k+\frac{1}{2})}\in S_{L}\) is the solution of
$$ \mathcal{A}\bigl(\varphi_{L},z_{L}^{(k+\frac{1}{2})} \bigr) = \bigl(u_{L}^{(k+\frac{1}{2})}-\bar{u},\varphi_{L} \bigr)_{\Omega},\quad \forall\varphi_{L} \in S_{L} $$
(4.3)
with \(u_{L}^{(k+\frac{1}{2})}\in S_{L}\) given by
$$ \mathcal{A}\bigl(u_{L}^{(k+\frac{1}{2})},v_{L} \bigr) = \bigl(f+q_{L}^{(k)}-\rho_{k} d_{L}^{(k)},v_{L}\bigr)_{\Omega},\quad \forall v_{L} \in S_{L}. $$
(4.4)
The optimal iteration step size \(\rho_{k}\) can be efficiently calculated through solving (4.2). Indeed we first notice that there exists an explicit expression of \(z_{L}^{(k+\frac{1}{2})}\) on \(\rho_{k}\). Let \(\tilde{u}_{L}^{(k)}\) and \(\tilde{z}_{L}^{(k)}\) denote respectively the solutions of
$$\begin{aligned}& \mathcal{A}\bigl(\tilde{u}_{L}^{(k)},v_{L}\bigr) = \bigl(d_{L}^{(k)},v_{L}\bigr)_{\Omega},\quad \forall v_{L} \in S_{L}, \end{aligned}$$
(4.5)
$$\begin{aligned}& \mathcal{A}\bigl(\varphi_{L},\tilde{z}_{L}^{(k)} \bigr) = \bigl(\tilde{u}_{L}^{(k)},\varphi_{L} \bigr)_{\Omega},\quad \forall\varphi_{L} \in S_{L}. \end{aligned}$$
(4.6)
\(u_{L} (q_{L}^{(k)} )\) and \(z_{L} (q_{L}^{(k)} )\) are respectively the solutions of
$$\begin{aligned}& \mathcal{A} \bigl(u_{L} \bigl(q_{L}^{(k)} \bigr),v_{L} \bigr) = \bigl(f+q_{L}^{(k)},v_{L} \bigr)_{\Omega},\quad \forall v_{L} \in S_{L}, \end{aligned}$$
(4.7)
$$\begin{aligned}& \mathcal{A} \bigl(\varphi_{L},z_{L} \bigl(q_{L}^{(k)} \bigr) \bigr) = \bigl(u_{L} \bigl(q_{L}^{(k)} \bigr)-\bar{u},\varphi_{L} \bigr)_{\Omega},\quad \forall \varphi_{L} \in S_{L} . \end{aligned}$$
(4.8)
Then it can be checked that \(z_{L} (q_{L}^{(k)} )-\rho_{k} \tilde {z}_{L}^{(k)}\) solves (4.3) and (4.4), that is,
$$z_{L}^{(k+\frac{1}{2})}=z_{L} \bigl(q_{L}^{(k)} \bigr)-\rho_{k} \tilde{z}_{L}^{(k)}. $$
Bringing this expression into (4.2) gives
$$\bigl(z_{L} \bigl(q_{L}^{(k)} \bigr)- \rho_{k} \tilde{z}_{L}^{(k)}+\lambda \bigl(q_{L}^{(k)}-\rho_{k} d_{L}^{(k)} \bigr),d_{L}^{(k)} \bigr)_{\Omega}=0. $$
Let \(\tilde{d}_{L}^{(k)}=\tilde{z}_{L}^{(k)}+\lambda d_{L}^{(k)}\), then we obtain
$$ \rho_{k}=\frac{(d_{L}^{(k)},d_{L}^{(k)})_{\Omega}}{(\tilde {d}_{L}^{(k)},d_{L}^{(k)})_{\Omega}}. $$
(4.9)

The overall process is summarized below.

Projection gradient optimization algorithm

Choose an initial control \(q_{L}^{(0)}\), set \(k=0\).
  1. (a)

    Solve problems (4.7) and (4.8), let \(d_{L}^{(k)}=z_{L}(q_{L}^{(k)})+\lambda q_{L}^{(k)}\).

     
  2. (b)

    Solve problems (4.5) and (4.6), and set \(\tilde{d}_{L}^{(k)}=\tilde{z}_{L}^{(k)}+\lambda d_{L}^{(k)}\), \(\rho_{k}=\frac{(d_{L}^{(k)},d_{L}^{(k)})_{\Omega}}{(\tilde {d}_{L}^{(k)},d_{L}^{(k)})_{\Omega}}\).

     
  3. (c)

    Update: \(q_{L}^{(k+\frac{1}{2})}=q_{L}^{(k)}-\rho_{k} d_{L}^{(k)}\), \(q_{L}^{(k+1)}=-\min \{0,\overline{q_{L}^{(k+\frac{1}{2})}} \} +q_{L}^{(k+\frac{1}{2})}\).

     
  4. (d)

    If \(\Vert d_{L}^{(k)}\Vert \leq \mbox{tolerance}\), then take \(q_{L}^{\ast}=q_{L}^{(k+1)}\) and solve problems (2.8) and (2.13) to get \(u_{L}(q_{L}^{\ast})\) and \(z_{L}(q_{L}^{\ast})\).

    Else, set \(k=k+1\), repeat (a)-(d).

     

4.2 Numerical results

In this subsection we carry out some numerical experiments to validate the a posteriori error estimates for the numerical solutions. In our calculation, we take \(T=1\), \(\lambda=1\).

Example 4.1

We consider problem (2.3) with exact analytical solutions as
$$u\bigl(q^{\ast}\bigr)=\sin\pi x \cos\pi t,\qquad z\bigl(q^{\ast} \bigr)=\sin\pi x \sin\pi (1-t), \qquad q^{\ast}= \max\bigl\{ 0,\overline{z \bigl(q^{\ast}\bigr)}\bigr\} - z\bigl(q^{\ast}\bigr). $$
The right-hand side f and the desired state \(\bar{u}\) here are respectively numerically calculated through (2.1) and (2.7) using \(u(q^{\ast})\), \(z(q^{\ast})\) and \(q^{\ast}\).
In order to validate the a posteriori error estimate, we compare the error indicator \(\eta=\eta_{1}+\eta_{2}\), which is defined in Theorem 3.1 and the real error of the numerical solution measured by
$$ e=\bigl\Vert q^{\ast}-q_{L}^{\ast}\bigr\Vert _{0,\Omega} +\bigl\Vert u\bigl(q^{\ast}\bigr)-u_{L} \bigl(q^{\ast}_{L}\bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega)} +\bigl\Vert z\bigl(q^{\ast}\bigr)-z_{L}\bigl(q^{\ast}_{L} \bigr)\bigr\Vert _{B^{\frac {\alpha}{2}}(\Omega)}. $$
These two errors are compared in Figure 1 as functions of M (=N). We observe that the indicator η has almost the same exponential decay as the error e, whereas it overestimates the error, which is consistent with our theoretical results.
Figure 1

Performance of error indicator η with \(\pmb{M=N}\) , \(\pmb{\alpha=0.6}\) .

Example 4.2

We choose other exact analytical solutions as
$$u\bigl(q^{\ast}\bigr)=\sin\pi xe^{t},\qquad z \bigl(q^{\ast}\bigr)=\sin\pi x (1-t)e^{2t}, \qquad q^{\ast}= \max\bigl\{ 0,\overline{z\bigl(q^{\ast}\bigr)}\bigr\} - z \bigl(q^{\ast}\bigr). $$
The a posteriori error indicator η and the real error e are compared in Figure 2 as functions of M (=N) with \(\alpha=0.3\). It can be also observed from Figure 2 that the indicator η has almost the same exponential decay as the error e, and the reliability of the proposed estimator is confirmed again.
Figure 2

Performance of error indicator η with \(\pmb{M=N}\) , \(\pmb{\alpha=0.3}\) .

5 Concluding remarks

We have obtained a posteriori upper bound of the spectral method for the fractional control problem. This is an important step towards developing an adaptive spectral method for solving FOCPs. In the future, we will consider the efficiency of the a posteriori estimator to obtain an optimal estimate. As for the classical parabolic equation, we guess such an optimal estimate will have to make use of some Jacobi-weighted Sobolev spaces and polynomial inverse inequalities. Furthermore, many computational issues have to be addressed. For example, an adaptive refinement strategy should be investigated for efficiently implementing the adaptive spectral method for FOCPs, and the adaptive spectral method should be also used to solve some real examples from physical and engineering sciences.

Declarations

Acknowledgements

The work of XY Ye is partially supported by the Science Foundation of Jimei University, China (Grant Nos. ZQ2013005 and ZC2013021), the Foundation (Class B) of Fujian Educational Committee (Grant No. FB2013005), the Foundation of Fujian Educational Committee (No. JA14180). The work of CJ Xu was partially supported by the National NSF of China (Grant Nos. 11471274 and 11421110001).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Science, Jimei University
(2)
School of Mathematical Sciences, Xiamen University

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© Ye and Xu; licensee Springer. 2015