On Spectral Petrov–Galerkin Method for Solving Optimal Control Problem Governed by Fractional Diffusion Equations with Fractional Noise

Abstract

In this paper, a spectral Petrov–Galerkin method is developed to solve an optimal control problem governed by a two-sided space-fractional diffusion-reaction equation with additive fractional noise. In order to compensate weak singularities of the solution near boundaries, regularities of both the fractional noise and the optimal control problem are analyzed in weighted Sobolev space. The spectral Petrov–Galerkin method is presented by employing truncated spectral expansion of the fractional Brownian motion (fBm) type noise, and error estimates are given based on the obtained regularity of the optimal control problem. Numerical experiments are carried out to verify the theoretical findings.

A Error Estimate in \(H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }\)-norm

In this section we present error estimates in $H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }$-norm of the spectral Petrov–Galerkin approximation for two-sided fractional diffusion reaction equations. It is motivated by the observation that the convergence order of $\Vert q-q_N\Vert$ tested in Example 5.2 is higher than the overall convergence order given by Theorem 4.4.

We consider the following two-sided fractional diffusion reaction equation, i.e. the model equation of (3.2),

$$\begin{aligned} \begin{aligned} \mathcal {L}_\theta ^\alpha p + \lambda p&= f(x),\ {}&x \in I:=(0,1), \\ p(x)&= 0,\ {}&x \in \partial I.\quad \quad \quad \ \end{aligned}\end{aligned}$$

(A.1)

The Petrov–Galerkin ultra-weak formulation is: Given $f\in H^{-\alpha /2}(I)\cap H_{\omega ^{\sigma ^*,\sigma }}^{r}(I)$, $r\ge -\alpha$, to find $p\in L_{\omega ^{-\sigma ,-\sigma ^*}}^2(I)$ such that

$$\begin{aligned} (p,\mathcal {L}_{1-\theta }^\alpha (\omega ^{\sigma ^*,\sigma } v))+\lambda (p,v)_{\omega ^{\sigma ^*,\sigma }}=(f,v)_{\omega ^{\sigma ^*,\sigma }},\ \forall v\in H_{\omega ^{\sigma ^*,\sigma }}^{\alpha }(I). \end{aligned}$$

(A.2)

The well-posedness of the problem has been constructed as follows.

Lemma A.1

([24, Theorem 3.1]) For the problem (A.2) with $\lambda \ge 0$, if $f\in H^{-\alpha /2}(I)\cap H_{\omega ^{\sigma ^*,\sigma }}^{-\alpha }(I)$, then there exists an unique solution $p\in L^2_{\omega ^{-\sigma ,-\sigma ^*}}(I)$ such that

$$\begin{aligned} \Vert p\Vert _{\omega ^{-\sigma ,-\sigma ^*}}\le C\Vert f\Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{-\alpha }}. \end{aligned}$$

We recall the finite dimensional spaces $U_N$ and $V_N$ defined as (4.3) and (4.4), respectively. Then the spectral Petrov–Galerkin approximation is: Given $f\in H^{-\alpha /2}(I)\cap H_{\omega ^{\sigma ^*,\sigma }}^{r}(I)$, $r\ge -\alpha$, to find $p_N \in U_N$ such that

$$\begin{aligned} (p_N,\mathcal {L}_{1-\theta }^\alpha (\omega ^{\sigma ^*,\sigma } v_N))+\lambda (p_N,v_N)_{\omega ^{\sigma ^*,\sigma }}=(f,v_N)_{\omega ^{\sigma ^*,\sigma }},\ \forall v_N\in V_N. \end{aligned}$$

(A.3)

The error estimate has been obtained in [24].

Lemma A.2

([24, Theorem 4.1]) Assume that p and $p_N$ are solutions of the formulation (A.2) and its spectral Petrov–Galerkin discrete counterpart (A.3), respectively. If $f\in H^{-\alpha /2}(I)\cap H_{\omega ^{\sigma ^*,\sigma }}^r(I)$, $r\ge -\alpha$, then there exists a positive integer $N_0$ such that when $N>N_0$, we have

$$\begin{aligned} \Vert p_N\Vert _{\omega ^{-\sigma ,-\sigma ^*}}\le C\Vert f\Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{-\alpha }}, \end{aligned}$$

(A.4)

and

$$\begin{aligned} \Vert p-p_N\Vert _{\omega ^{-\sigma ,-\sigma ^*}}\le CN^{-m}\Vert f\Vert _{H_{\omega ^{\sigma ^*,\sigma }}^r}, \end{aligned}$$

where $m=\min \{r+\alpha ,2\alpha +\min (\sigma ,\sigma ^*)+1-\epsilon \},$ is the regularity index of $\omega ^{-\sigma ,-\sigma ^*}p$ in $H_{\omega ^{\sigma ,\sigma ^*}}^m(I)$.

To establish the error estimate in $H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }$-norm of the spectral Petrov–Galerkin approximation, we consider the following weak formulation established in [38] for the problem (A.1): Given $f\in H^{-\alpha /2}(I)\cap H_{\omega ^{\sigma ^*,\sigma }}^{r}(I)$, $r\ge -\alpha /2$, to find $\phi \in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha /2}(I)$ such that $p(x)=\omega ^{\sigma ,\sigma ^*}(x)\phi (x)$ satisfies

$$\begin{aligned} B(\phi ,\psi ):=(\mathcal {L}_\theta ^\alpha p + \lambda p, \psi )_{\omega ^{\sigma ^*,\sigma }}=(f,\psi )_{\omega ^{\sigma ^*,\sigma }},\ \forall \psi \in H_{\omega ^{\sigma ^*,\sigma }}^{\alpha /2}(I). \end{aligned}$$

(A.5)

Correspondingly, the approximation scheme is: Given $f\in H^{-\alpha /2}(I)\cap H_{\omega ^{\sigma ^*,\sigma }}^{r}(I)$, $r\ge -\alpha /2$, to find $\phi _N\in X_N:=\textrm{span}\{\textrm{Q}_\textrm{n}^{\sigma ,\sigma ^{*}}\}_{n=0}^N$ such that $p_N(x)=\omega ^{\sigma ,\sigma ^*}(x)\phi _N(x)$ satisfies

$$\begin{aligned} (\mathcal {L}_\theta ^\alpha p_N + \lambda p_N, \psi _N)_{\omega ^{\sigma ^*,\sigma }}=(f,\psi _N)_{\omega ^{\sigma ^*,\sigma }},\ \forall \psi _N \in Y_N:=\textrm{span}\{\textrm{Q}_\textrm{n}^{\sigma ,\sigma ^*}\}_{n=0}^N. \end{aligned}$$

(A.6)

It is noted that if $f\in H^{-\alpha /2}(I)\cap H_{\omega ^{\sigma ^*,\sigma }}^{r}(I)$, $r\ge -\alpha /2$, the solution $p:=\omega ^{\sigma ,\sigma ^*}(x)\phi (x)$ to the weak formulation (A.5) is also the solution to the ultra-weak formulation (A.2), and this relation still holds for their corresponding approximation scheme (A.6) and (A.3). Thus the following estimates are valid for the ultra-weak formulation (A.2) and its spectral Petrov–Galerkin scheme (A.3) with $f\in H^{-\alpha /2}(I)\cap H_{\omega ^{\sigma ^*,\sigma }}^{r}(I)$, $r\ge -\alpha /2$.

For the above approximation (A.6) we have the following convergence results.

Lemma A.3

([38, Lemma 4.2]) There exists $C>0$ such that for $\phi$ satisfying (A.5) and $\phi _N$ satisfying (A.6),

$$\begin{aligned} \Vert \phi -\phi _N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha /2}}\le C\inf \limits _{\xi _N\in X_N}\Vert \phi -\xi _N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha /2}}. \end{aligned}$$

Lemma A.4

([38, Corollary 4.1]) Let $\phi$ be the solution of (A.5) and $\phi _N$ be the solution of (A.6), respectively. If $f\in H^{-\alpha /2}(I)\cap H_{\omega ^{\sigma ^*,\sigma }}^{r}(I)$, $r\ge -\alpha /2$, then there exists $C>0$ such that

$$\begin{aligned} \Vert \phi -\phi _N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha /2}}\le CN^{-m+\alpha /2}\Vert f\Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{r}}, \end{aligned}$$

where $m=\min \{r+\alpha ,2\alpha +\min (\sigma ,\sigma ^*)+1-\epsilon \}$, is the regularity index of $\phi$ in $H_{\omega ^{\sigma ,\sigma ^*}}^m(I)$.

Let $\pi _N^{\gamma ,\beta }: L^2_{\omega ^{\gamma ,\beta }}(I)\rightarrow P_N(I)$, $\gamma ,\ \beta >-1$, is a $L^2_{\omega ^{\gamma ,\beta }}(I)$-orthogonal projection such that

$$\begin{aligned} (\pi _N^{\gamma ,\beta }u-u,v)_{\omega ^{\gamma ,\beta }}=0, \ \forall v\in P_N(I), \end{aligned}$$

which can be also expressed by

$$\begin{aligned} \pi _N^{\gamma ,\beta }u(x)=\sum \limits _{n=0}^{N}\hat{u}_nQ_n^{\gamma ,\beta },\ \hat{u}_n=\frac{(u,Q_n^{\gamma ,\beta })_{\omega ^{\gamma ,\beta }}}{\Vert Q_n^{\gamma ,\beta }\Vert ^2_{{\omega ^{\gamma ,\beta }}}}. \end{aligned}$$

And we have the following projection error estimate.

Lemma A.5

([19, Theorem 2.1]) For $u\in H_{\omega ^{\gamma ,\beta }}^r(I)$ and for all $0\le r_1\le r$,

$$\begin{aligned} \Vert \pi _N^{\gamma ,\beta }u-u\Vert _{H_{\omega ^{\gamma ,\beta }}^{r_1}}\le C(N(N+\gamma +\beta ))^{\frac{r_1-r}{2}}|u|_{H_{\omega ^{\gamma ,\beta }}^{r}}, \end{aligned}$$

(A.7)

where C is a generic positive constant independent of u, N, $\gamma$ and $\beta$.

Now, we are ready to prove the error estimate in $H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }$-norm of the spectral Petrov–Galerkin approximation.

Theorem A.6

Let p and $p_N$ be determined by (A.5) and (A.6), respectively. If $f\in H^{-\alpha /2}(I)\cap H_{\omega ^{\sigma ^*,\sigma }}^{r}(I)$, $r\ge -\alpha /2$, then there exists a constant $C>0$ such that

$$\begin{aligned} \Vert p-p_N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}&\le C N^{-3\alpha /2}\Vert \phi -\phi _N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha /2}}\nonumber \\&\le CN^{-(m+\alpha )}\Vert f\Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{r}}, \end{aligned}$$

where $m=\min \{r+\alpha ,2\alpha +\min (\sigma ,\sigma ^*)+1-\epsilon \}$, is the regularity index of $\phi$ in $H_{\omega ^{\sigma ,\sigma ^*}}^m(I)$.

Proof

The adjoint problem to (A.5) is: Given $g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }(I)$, to find $\psi \in H_{\omega ^{\sigma ^*,\sigma }}^{\alpha /2}(I)$ such that $w(x)=\omega ^{\sigma ^*,\sigma }\psi (x)$ satisfies

$$\begin{aligned} (\mathcal {L}_{1-\theta }^\alpha w + \lambda w, \phi )_{\omega ^{\sigma ,\sigma ^*}}=(g,\phi )_{\omega ^{\sigma ,\sigma ^*}},\ \forall \phi \in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha /2}(I). \end{aligned}$$

(A.8)

Correspondingly, the approximation scheme is: Given $g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }(I)$, to find $\psi _N\in Y_N$ such that $w_N(x)=\omega ^{\sigma ^*,\sigma }\psi _N(x)$ satisfies

$$\begin{aligned} (\mathcal {L}_{1-\theta }^\alpha w_N + \lambda w_N, \phi _N)_{\omega ^{\sigma ,\sigma ^*}}=(g,\phi _N)_{\omega ^{\sigma ,\sigma ^*}},\ \forall \phi _N \in X_N. \end{aligned}$$

(A.9)

Note that by Lemma 3.5, for $g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha } (I)$, $\omega ^{\sigma ,\sigma ^*}g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }(I)$, i.e., $\Vert \omega ^{\sigma ,\sigma ^*}g\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}\le c\Vert g\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}$. Thus we have

$$\begin{aligned} \Vert p-p_N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}&=\sup \limits _{g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}\frac{|(p-p_N,g)_{\omega ^{\sigma ,\sigma ^*}}|}{\Vert g\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}} =\sup \limits _{g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}\frac{|(\omega ^{\sigma ,\sigma ^*}\phi -\omega ^{\sigma ,\sigma ^*}\phi _N,g)_{\omega ^{\sigma ,\sigma ^*}}|}{\Vert g\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}}\nonumber \\&=\sup \limits _{g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}\frac{|(\phi -\phi _N,\omega ^{\sigma ,\sigma ^*}g)_{\omega ^{\sigma ,\sigma ^*}}|}{\Vert g\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}}\le C \sup \limits _{g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}\frac{|(\phi -\phi _N,\omega ^{\sigma ,\sigma ^*}g)_{\omega ^{\sigma ,\sigma ^*}}|}{\Vert \omega ^{\sigma ,\sigma ^*}g\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}}\nonumber \\&\le C\Vert \phi -\phi _N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}. \end{aligned}$$

(A.10)

Next we estimate

$$\begin{aligned} \Vert \phi -\phi _N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}=\sup \limits _{g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}\frac{|(\phi -\phi _N,g)_{\omega ^{\sigma ,\sigma ^*}}|}{\Vert g\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}}. \end{aligned}$$

(A.11)

From (A.8), (A.9) and Galerkin orthogonality, we have, for $g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }(I)$,

$$\begin{aligned} (\phi -\phi _N,g)_{\omega ^{\sigma ,\sigma ^*}}&=(\mathcal {L}_{1-\theta }^\alpha w + \lambda w, \phi )_{\omega ^{\sigma ,\sigma ^*}}-(\mathcal {L}_{1-\theta }^\alpha w_N + \lambda w_N, \phi _N)_{\omega ^{\sigma ,\sigma ^*}}\nonumber \\&=(w,\mathcal {L}_{\theta }^\alpha (\omega ^{\sigma ,\sigma ^*}\phi )+\lambda \omega ^{\sigma ,\sigma ^*}\phi )-(w_N,\mathcal {L}_{\theta }^\alpha (\omega ^{\sigma ,\sigma ^*}\phi _N)+\lambda \omega ^{\sigma ,\sigma ^*}\phi _N)\nonumber \\&=B(\phi ,\psi )-B(\phi _N,\psi _N)\nonumber \\&=B(\phi -\phi _N,\psi -\psi _N)+B(\phi _N,\psi -\psi _N)+B(\phi -\phi _N,\psi _N)\nonumber \\&=B(\phi -\phi _N,\psi -\psi _N)\nonumber . \end{aligned}$$

Thus, by the continuity [38, Lemma 3.1] of bilinear form $B(\cdot ,\cdot )$ and Lemma A.3, we obtain

$$\begin{aligned} |(\phi -\phi _N,g)_{\omega ^{\sigma ,\sigma ^*}}|&\le C\Vert \phi -\phi _N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha /2}}\Vert \psi -\psi _N\Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{\alpha /2}}\nonumber \\&\le C\Vert \phi -\phi _N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha /2}}\inf \limits _{\xi _N\in Y_N}\Vert \psi -\xi _N\Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{\alpha /2}}. \end{aligned}$$

(A.12)

It follows from Lemma 3.4 that for $g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }(I)$ in (A.8), we have $\psi \in H_{\omega ^{\sigma ^*,\sigma }}^{2\alpha }(I)$, and

$$\begin{aligned} \Vert \psi \Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{2\alpha }}\le C\Vert g\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}. \end{aligned}$$

Then by the estimate (A.7) of the projection error,

$$\begin{aligned} \inf \limits _{\xi _N\in Y_N}\Vert \psi -\xi _N\Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{\alpha /2}}&\le C \Vert \psi -\pi _N^{\sigma ^*,\sigma }\psi \Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{\alpha /2}}\nonumber \\&\le CN^{-3\alpha /2}\Vert \psi \Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{2\alpha }}\le CN^{-3\alpha /2}\Vert g\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}. \end{aligned}$$

(A.13)

With (A.10)-(A.13) and Lemma A.4, we obtain

$$\begin{aligned} \Vert p-p_N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}&\le C\sup \limits _{g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}\frac{|(\phi -\phi _N,g)_{\omega ^{\sigma ,\sigma ^*}}|}{\Vert g\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}}\\&\le C \sup \limits _{g\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}\frac{\Vert \phi -\phi _N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha /2}}\inf \limits _{\xi _N\in Y_N}\Vert \psi -\xi _N\Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{\alpha /2}}}{\Vert g\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}}\\&\le C N^{-3\alpha /2}\Vert \phi -\phi _N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha /2}}\\&\le CN^{-3\alpha /2} N^{-m+\alpha /2} \Vert f\Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{r}}\le CN^{-(m+\alpha )}\Vert f\Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{r}}.\qquad \qquad \square \end{aligned}$$

With the help of error estimate in $H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }$-norm, the convergence result of control $q_N$ in Theorem 4.4 can be improved for the deterministic control problem (3.1)-(3.2) with $f\in H_{\omega ^{\sigma ^*,\sigma }}^{r_1}(I)$, $r_1\ge -\alpha /2$.

Theorem A.7

Assume that (u, z, q) is the solution of the weak formulation of optimality conditions (3.3)-(3.5), and $(u_N,z_N,q_N)$ is the solution of corresponding spectral Petrov–Galerkin scheme. If $f\in H^{r_1}_{\omega ^{\sigma ^*,\sigma }}(I)$, $u_d\in H^{r_2}_{\omega ^{\sigma ,\sigma ^*}}(I)$ with $r_1, r_2\ge -\alpha /2$, then the control $q_N$ has the following error estimate

$$\begin{aligned} \Vert q-q_N\Vert \le C N^{-(\min \{r_1+\alpha ,r_2,s\}+\alpha )}, \end{aligned}$$

where $s=3\min \{\sigma ,\sigma ^*\}+1-\epsilon$.

Proof

Refer to the proof of Theorem 4.5 in [28] or following the proof of Theorem 4.3 for the optimal control problem with fractional noise (see (4.19)), we can derive that the error $\Vert q-q_N\Vert$ in Theorem 4.4 is bounded by $\Vert z-z_N(q)\Vert _{\omega ^{-\sigma ^*,-\sigma }}$, and

$$\begin{aligned} \Vert q-q_N\Vert&\le C\Vert z-z_{N}(u)\Vert _{\omega ^{-\sigma ^*,-\sigma }}+C\Vert z_{N}(u)-z_{N}(q)\Vert _{\omega ^{-\sigma ^*,-\sigma }}, \end{aligned}$$

where $z_N(u)$ and $z_N(q)$ are the intermediate variables defined by

$$\begin{aligned} b(z_N(q),w_N)&=(u_N(q)-u_d,w_N)_{\omega ^{\sigma ,\sigma ^*}},\ \forall w_N \in W_N, \end{aligned}$$

(A.14)

$$\begin{aligned} b(z_N(u),w_N)&=(u-u_d,w_N)_{\omega ^{\sigma ,\sigma ^*}},\ \forall w_N \in W_N. \end{aligned}$$

(A.15)

Then subtracting (A.14) from (A.15), by (4.23) and Lemma 4.4 in [18] we have

$$\begin{aligned} \Vert z_{N}(u)-z_{N}(q)\Vert _{\omega ^{-\sigma ^*,-\sigma }}&\le C \sup \limits _{0\not =w_N\in W_N}\frac{b(z_{N}(u)-z_{N}(q),w_N)}{\Vert w_N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}}\\&\le C\sup \limits _{0\not =w_N\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}\frac{b(z_{N}(u)-z_{N}(q),w_N)}{\Vert w_N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}}\nonumber \\&=C\sup \limits _{0\not =w_N\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}\frac{(u-u_N(q),w_N)_{\omega ^{\sigma ,\sigma ^*}}}{\Vert w_N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}}.\nonumber \\&\le C\sup \limits _{0\not =w_N\in H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}\frac{\Vert u-u_N(q)\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}\Vert w_N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}}{\Vert w_N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{\alpha }}}\nonumber \\&\le C\Vert u-u_N(q)\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert q-q_N\Vert&\le \Vert z-z_{N}(u)\Vert _{\omega ^{-\sigma ^*,-\sigma }}+C\Vert u-u_N(q)\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}. \end{aligned}$$

(A.16)

Note that in (A.16) $z_N(u)$ and $u_N(q)$ are the spectral Petrov–Galerkin approximation to z and u, respectively. Then for $f\in H^{r_1}_{\omega ^{\sigma ^*,\sigma }}(I)$, $r_1\ge -\alpha /2$, by error estimate of spectral Petrov–Galerkin approximation in $L^2_{\omega ^{-\sigma ^*,-\sigma }}$-norm in Theorem A.2 and $H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }$-norm in Theorem A.6, we obtain

$$\begin{aligned} \Vert q-q_N\Vert&\le \Vert z-z_{N}(u)\Vert _{\omega ^{-\sigma ^*,-\sigma }}+C\Vert u-u_N(q)\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}\\&\le CN^{-(\min \{r_1+\alpha ,r_2,s\}+\alpha )}+CN^{-(\min \{r_1,r_2+\alpha ,s\}+2\alpha )}\\&\le CN^{-(\min \{r_1+\alpha ,r_2,s\}+\alpha )}. \end{aligned}$$

where $s=3\min \{\sigma ,\sigma ^*\}+1-\epsilon$. $\square$

Remark A.8

By Theorem 3.7, we have $\omega ^{-\sigma ^* , -\sigma }q \in H^{\min \{ r_1+\alpha ,\,r_2 ,\, s \}+ \alpha }_{\omega ^{\sigma ^* , \sigma }} (I)$. It indicates that the convergence order of $q_N$ in standard $L^2$-norm can be raised to the same level as its regularity in weighted Sobole space.

Similarly, we also can obtain

$$\begin{aligned} \Vert u-u_N\Vert _{\omega ^{-\sigma ,-\sigma ^*}}&\le \Vert u-u_N(q)\Vert _{\omega ^{-\sigma ,-\sigma ^*}}+C\Vert q-q_N\Vert _{H_{\omega ^{\sigma ^*,\sigma }}^{-\alpha }},\\ \Vert z-z_N\Vert _{\omega ^{-\sigma ,-\sigma ^*}}&\le \Vert z-z_N(u)\Vert _{\omega ^{-\sigma ,-\sigma ^*}}+C\Vert u-u_N\Vert _{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}, \end{aligned}$$

which indicates the convergence of state u and adjoint state z are also likely to be improved. As here we could not presented the negative norm estimation for discrete control $q_N$ and state $u_N$ due to technical difficulties, it is tested by numerical experiments in Example 5.2.

Remark A.9

For the spectral Petrov–Galerkin approximation to the deterministic control problem (3.1)-(3.2), refer to Remark 4.3 in [37] and by using (3.6), we can prove that for $\sigma ,\sigma ^* \in (0,1)$,

$$\begin{aligned} \Vert q-q_N\Vert _{\omega ^{-\sigma ^*,-\sigma }}\le C\Vert z-z_N\Vert _{\omega ^{-\sigma ^*,-\sigma }}. \end{aligned}$$

Then similar to (4.29) and (4.31), we have

$$\begin{aligned} \Vert z-z_N\Vert _{\omega ^{-\sigma ^*,-\sigma }}\le C\Vert q-q_N\Vert _{\omega ^{-\sigma ^*,-\sigma }}+\Vert z-z_{N}(u)\Vert _{\omega ^{-\sigma ^*,-\sigma }}+C\Vert u-u_N(q)\Vert _{\omega ^{-\sigma ,-\sigma ^*}}. \end{aligned}$$

It means that discrete control $q_N$ and adjoint state $z_N$ will stay same convergence order in weighted $L^2_{\omega ^{-\sigma ^*,-\sigma }}$-norm. It is verified in Example 5.2.

Remark A.10

For the optimal control problem with fractional noise, similar to (A.16), by (4.19), (4.20) and (4.27), we have

$$\begin{aligned} \Vert q-q_{MN}\Vert ^2&\le C\mathbb E[ \Vert z-z_{MN}(u)\Vert ^2_{\omega ^{-\sigma ^*,-\sigma }}]+C \mathbb E[ \Vert z_{MN}(u)-z_{MN}(q)\Vert ^2_{\omega ^{-\sigma ^*,-\sigma }}]\nonumber \\&\le CN^{-2m}+C \mathbb E[ \Vert u-u_{MN}(q)\Vert ^2_{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}], \end{aligned}$$

where $m=\min \{H-1+\alpha -\epsilon ,3\min (\sigma ,\sigma ^*)+1-\epsilon ,r\}+\alpha$, and r is the regularity index of $u_d$. Unlike deterministic case, $u_{MN}$ is not the spectral Petrov–Galerkin approximation of u but $u_M$, which defined by

$$\begin{aligned} a(u_M,v)=(\dot{W}^H_M+q,v)_{\omega ^{\sigma ^*,\sigma }},\ \forall v\in \mathcal { H}_{\omega ^{\sigma ^*,\sigma }}^{\alpha }(I). \end{aligned}$$

Thus,

$$\begin{aligned} \Vert q-q_{MN}\Vert ^2&\le CN^{-2m}+ 2C \mathbb E[ \Vert u-u_M\Vert ^2_{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}]+ 2C \mathbb E[ \Vert u_M-u_{MN}(q)\Vert ^2_{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}],\nonumber \\&\le CN^{-2m}+ 2C \mathbb E[ \Vert u-u_M\Vert ^2_{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}]+ CN^{-2(H-1+2\alpha -\epsilon )}, \end{aligned}$$

and similarly we have

$$\begin{aligned} \mathbb E[\Vert z-z_{MN}\Vert ^2_{\omega ^{-\sigma ^*,-\sigma }}] \le C \mathbb E[\Vert z-z_{MN}(u)\Vert ^2_{\omega ^{-\sigma ^*,-\sigma }}]+C\mathbb E[\Vert u-u_{MN}\Vert ^2_{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}], \end{aligned}$$

which indicates that a further analysis of $\mathbb E[\Vert u-u_M\Vert ^2_{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}]$ and $\mathbb E[\Vert u-u_{MN}\Vert ^2_{H_{\omega ^{\sigma ,\sigma ^*}}^{-\alpha }}]$ allows for an increase in the convergence orders of adjoint state $z_{MN}$ and control $q_{MN}$, see numerical results in Example 5.1.