We start by splitting the total error and applying triangle inequality as:
$$\begin{aligned} \left\| \varvec{y}-\varvec{y}_h\right\| _{0,\varOmega } \le \left\| \varvec{y}-\varvec{y}_h(\varvec{u})\right\| _{0,\varOmega }+\left\| \varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u})\right\| _{0,\varOmega }+\left\| \varvec{y}_h(\varPi _h \varvec{u})-\varvec{y}_h\right\| _{0,\varOmega }, \end{aligned}$$
(3.44)
where
\( \varPi _h \) represents the
\( \mathbf {L}^2-\)projection operator onto the discrete control space
\(\mathbf {U}_h\). Next, let
\( (\tilde{\varvec{w}}_h,\tilde{r}_h) \in \mathbf {V}_h\times Q_h \) be the unique solution of the auxiliary discrete dual Brinkman problem
$$\begin{aligned} A_h(\tilde{\varvec{w}}_h,\tilde{\varvec{z}}_h)+c_h(\tilde{\varvec{w}}_h,\tilde{\varvec{z}}_h)-B_h(\tilde{\varvec{z}}_h,\tilde{r}_h)&=(\gamma \tilde{\varvec{z}}_h,\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}))_{0,\varOmega }\quad \forall \tilde{\varvec{z}}_h\in \mathbf {V}_h, \end{aligned}$$
(3.45)
$$\begin{aligned} B_h(\tilde{\varvec{w}}_h,\tilde{\psi }_h)&=0 \quad \forall \tilde{\psi }_h \in Q_h. \end{aligned}$$
(3.46)
We then choose
\( \tilde{\varvec{z}}_h=\tilde{\varvec{w}}_h \) and
\( \tilde{\psi }_h=\tilde{r}_h \) in (
3.45) and (
3.46), respectively, next we add the result, and we use the coercivity properties (
2.8) and (
2.12), to derive that
$$\begin{aligned} \left| \left| \left| \tilde{\varvec{w}}_h\right| \right| \right| _{2,h}\le C \left\| \varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u})\right\| _{0,\varOmega }. \end{aligned}$$
(3.47)
After testing (
3.45)–(
3.46) against
\( \tilde{\varvec{z}}_h=\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}) \) and
\( \tilde{\psi }_h=p_h(\varvec{u})-p_h(\varPi _h \varvec{u}) \), respectively, and adding the result, we obtain
$$\begin{aligned}&A_h(\tilde{\varvec{w}}_h,\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}))+c_h(\tilde{\varvec{w}}_h,\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}))-B_h(\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}),\tilde{r}_h)\nonumber \\&\quad -B_h(\tilde{\varvec{w}}_h,p_h(\varvec{u})-p_h(\varPi _h \varvec{u})) =(\gamma (\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u})),\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}))_{0,\varOmega }.\nonumber \\ \end{aligned}$$
(3.48)
In addition, employing the discrete state equation for
\( \varvec{y}_h(\varvec{u})\) and
\( \varvec{y}_h(\varPi _h \varvec{u}) \), we obtain
$$\begin{aligned}&A_h(\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}),\tilde{\varvec{w}}_h)+c_h(\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}),\tilde{\varvec{w}}_h)-B_h(\tilde{\varvec{w}}_h,p_h(\varvec{u})-p_h(\varPi _h \varvec{u}))\nonumber \\&\quad -B_h(\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}),\tilde{r}_h)=(\varvec{u}-\varPi _h \varvec{u},\gamma \tilde{\varvec{w}}_h)_{0,\varOmega }. \end{aligned}$$
(3.49)
We then proceed to subtract (
3.49) from (
3.48) and to rearrange terms, to arrive at
$$\begin{aligned}&(\gamma (\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u})),\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}))_{0,\varOmega } \\&\quad = A_h(\tilde{\varvec{w}}_h,\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}))-A_h(\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}),\tilde{\varvec{w}}_h)\\&\qquad +c_h(\tilde{\varvec{w}}_h,\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}))\\&\qquad -c_h(\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}),\tilde{\varvec{w}}_h) +(\varvec{u}-\varPi _h \varvec{u},\gamma \tilde{\varvec{w}}_h)_{0,\varOmega }. \end{aligned}$$
Using the definition of the norm
\(\left| \left| \left| \cdot \right| \right| \right| _{0,h}\) and its equivalence with the norm
\(\Vert \cdot \Vert _{0,\varOmega }\) we find that
$$\begin{aligned}&\left\| \varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u})\right\| _{0,\varOmega }^2 \\&\quad \le (\varvec{u}-\varPi _h \varvec{u},\gamma \tilde{\varvec{w}}_h)_{0,\varOmega }+|A_h(\tilde{\varvec{w}}_h,\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}))-A_h(\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}),\tilde{\varvec{w}}_h)|\\&\qquad +|c_h(\tilde{\varvec{w}}_h,\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}))-c_h(\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}),\tilde{\varvec{w}}_h)|. \end{aligned}$$
By virtue of the properties of
\( \varPi _h \) applied in the above inequality, we can assert that
$$\begin{aligned} \left\| \varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u})\right\| _{0,\varOmega }^2&\le (\varvec{u}-\varPi _h \varvec{u},\gamma \tilde{\varvec{w}}_h-\tilde{\varvec{w}}_h)_{0,\varOmega }+ (\varvec{u}-\varPi _h \varvec{u}, \tilde{\varvec{w}}_h-\varPi _h \tilde{\varvec{w}}_h)_{0,\varOmega }\nonumber \\&\qquad +|A_h(\tilde{\varvec{w}}_h,\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}))-A_h(\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}),\tilde{\varvec{w}}_h)|\nonumber \\&\qquad +|c_h(\tilde{\varvec{w}}_h,\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}))-c_h(\varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u}),\tilde{\varvec{w}}_h)|\nonumber \\&=: S_1+S_2+S_3+S_4. \end{aligned}$$
(3.50)
Approximation properties of
\(\gamma \) and the
\(\mathbf {L}^2-\)projection readily yield appropriate bounds for
\(S_1\) and
\(S_2\), respectively:
$$\begin{aligned} S_1&\le Ch\left\| \varvec{u}-\varPi _h \varvec{u}\right\| _{0,\varOmega }\left| \left| \left| \tilde{\varvec{w}}_h\right| \right| \right| _{2,h},\quad \text {and} \quad S_2 \le Ch\left\| \varvec{u}-\varPi _h \varvec{u}\right\| _{0,\varOmega }\left| \left| \left| \tilde{\varvec{w}}_h\right| \right| \right| _{2,h}. \end{aligned}$$
Then, a direct application of (
3.47) yields
$$\begin{aligned} S_1+S_2 \le Ch\left\| \varvec{u}-\varPi _h \varvec{u}\right\| _{0,\varOmega }\left\| \varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u})\right\| _{0,\varOmega }. \end{aligned}$$
We next use relations (
2.11), (
2.13) and (
3.47) to obtain
$$\begin{aligned} S_3+S_4\le & {} Ch\left| \left| \left| \varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u})\right| \right| \right| _{2,h}\left| \left| \left| \tilde{\varvec{w}}_h\right| \right| \right| _{2,h}\\\le & {} Ch\left\| \varvec{u}-\varPi _h \varvec{u}\right\| _{0,\varOmega }\left\| \varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u})\right\| _{0,\varOmega }. \end{aligned}$$
Consequently, substituting the estimates for the terms
\(S_1\),
\(S_2\),
\(S_3\) and
\(S_4\) back into (
3.50), one straightforwardly arrives at
$$\begin{aligned} \left\| \varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u})\right\| _{0,\varOmega }\le Ch\left\| \varvec{u}-\varPi _h \varvec{u}\right\| _{0,\varOmega }. \end{aligned}$$
(3.51)
The third term in (
3.44) is bounded using (
2.7) and proceeding as in the proof of Lemma
4:
$$\begin{aligned} \left\| \varvec{y}_h(\varPi _h \varvec{u})-\varvec{y}_h\right\| _{0,\varOmega } \le \left| \left| \left| \varvec{y}_h(\varPi _h \varvec{u})-\varvec{y}_h\right| \right| \right| _{2,h} \le C\left\| \varPi _h \varvec{u}-\varvec{u}_h\right\| _{0,\varOmega }. \end{aligned}$$
(3.52)
Using the discrete variational inequality along with the projection property of
\( \varPi _h \) and (
3.37), we have the following relation
$$\begin{aligned} \lambda \left\| \varPi _h \varvec{u}-\varvec{u}_h\right\| _{0,\varOmega }^2= & {} \lambda (\varvec{u}-\varvec{u}_h,\varPi _h \varvec{u}-\varvec{u}_h)_{0,\varOmega } \le (\varvec{w}-\varvec{w}_h,\varvec{u}_h-\varPi _h \varvec{u})_{0,\varOmega }\nonumber \\= & {} (\varvec{w}-\varvec{w}_h({\varvec{u}}),\varvec{u}_h-\varPi _h \varvec{u})_{0,\varOmega }+(\varvec{w}_h({\varvec{u}})\nonumber \\&-\,\varvec{w}_h({\varvec{y}}_h(\varPi _h {\varvec{u}})),\varvec{u}_h-\varPi _h \varvec{u})_{0,\varOmega }\nonumber \\&+\,(\varvec{w}_h({\varvec{y}}_h(\varPi _h {\varvec{u}}))-\varvec{w}_h,\varvec{u}_h-\varPi _h \varvec{u})_{0,\varOmega }\nonumber \\= & {} (\varvec{w}-\varvec{w}_h({\varvec{u}}),\varvec{u}_h-\varPi _h \varvec{u})_{0,\varOmega }+(\varvec{w}_h({\varvec{u}})\nonumber \\&-\,\varvec{w}_h({\varvec{y}}_h(\varPi _h {\varvec{u}})),\varvec{u}_h-\varPi _h \varvec{u})_{0,\varOmega }\nonumber \\&+\,(\varvec{w}_h({\varvec{y}}_h(\varPi _h {\varvec{u}}))-\varvec{w}_h-\gamma (\varvec{w}_h({\varvec{y}}_h(\varPi _h {\varvec{u}}))-\varvec{w}_h),\varvec{u}_h-\varPi _h \varvec{u})_{0,\varOmega }\nonumber \\&+\,(\gamma (\varvec{w}_h({\varvec{y}}_h(\varPi _h {\varvec{u}}))-\varvec{w}_h),\varvec{u}_h-\varPi _h \varvec{u})_{0,\varOmega }\nonumber \\= & {} J_1+J_2+J_3+J_4. \end{aligned}$$
(3.53)
Next, we use Cauchy–Schwarz inequality and (
3.23) to bound the first term:
$$\begin{aligned} J_1 \le \left\| \varvec{w}-\varvec{w}_h(u)\right\| _{0,\varOmega }\left\| \varvec{u}_h-\varPi _h \varvec{u}\right\| _{0,\varOmega }\le Ch^2\left\| \varvec{u}_h-\varPi _h \varvec{u}\right\| _{0,\varOmega }. \end{aligned}$$
For
\( J_2 \), an application of Lemma
4 and (
3.51) suffices to get
$$\begin{aligned} J_2 \le \left\| \varvec{y}_h(\varvec{u})-\varvec{y}_h(\varPi _h \varvec{u})\right\| _{0,\varOmega }\left\| \varvec{u}_h-\varPi _h \varvec{u}\right\| _{0,\varOmega }\le Ch\left\| \varvec{u}-\varPi _h \varvec{u}\right\| _{0,\varOmega }\left\| \varvec{u}_h-\varPi _h \varvec{u}\right\| _{0,\varOmega }. \end{aligned}$$
To bound the third term we use the approximation property of
\( \gamma \) and Lemma
4$$\begin{aligned} J_3\le & {} Ch\left| \left| \left| \varvec{w}_h(\varvec{y}_h(\varPi _h \varvec{u}))-\varvec{w}_h\right| \right| \right| _{2,h}\left\| \varvec{u}_h-\varPi _h \varvec{u}\right\| _{0,\varOmega }\\\le & {} Ch\left\| \varvec{y}_h(\varPi _h \varvec{u})-\varvec{y}_h\right\| _{0,\varOmega }\left\| \varvec{u}_h-\varPi _h \varvec{u}\right\| _{0,\varOmega } \le Ch\left\| \varvec{u}_h-\varPi _h \varvec{u}\right\| _{0,\varOmega }^2. \end{aligned}$$
Proceeding analogously to the proof of Lemma
6, using (
2.11) and (
2.13), the last term of the expression (
3.53) can be estimated as
$$\begin{aligned} J_4\le & {} A_h(\varvec{y}_h-\varvec{y}_h(\varPi _h \varvec{u}), \varvec{w}_h(\varPi _h \varvec{u})-{\varvec{w}}_h)-A_h(\varvec{w}_h(\varPi _h \varvec{u})-{\varvec{w}}_h,\varvec{y}_h-\varvec{y}_h(\varPi _h \varvec{u}))\\&+\,c_h(\varvec{y}_h-\varvec{y}_h(\varPi _h \varvec{u}), \varvec{w}_h(\varPi _h \varvec{u})-{\varvec{w}}_h)-c_h(\varvec{w}_h(\varPi _h \varvec{u})-{\varvec{w}}_h,\varvec{y}_h-\varvec{y}_h(\varPi _h \varvec{u}))\\\le & {} Ch\left| \left| \left| \varvec{y}_h-\varvec{y}_h(\varPi _h \varvec{u})\right| \right| \right| _{2,h}\left| \left| \left| \varvec{w}_h(\varPi _h \varvec{u})-{\varvec{w}}_h\right| \right| \right| _{2,h} \le Ch \left\| \varvec{u}_h-\varPi _h \varvec{u}\right\| _{0,\varOmega }^2. \end{aligned}$$
Plugging the bounds for
\( J_1, J_2, J_3 \) and
\( J_4 \) in (
3.53), putting (
3.51) and (
3.52) into (
3.44), and using interpolation estimate
\( \left\| \varvec{u}-\varPi _h \varvec{u}\right\| _{0,\varOmega } \le Ch\left\| \varvec{u}\right\| _{1,\varOmega } \) along with Lemma
5; we can assert that
$$\begin{aligned} \left\| \varvec{y}-\varvec{y}_h\right\| _{0,\varOmega }\le Ch^2 \left[ \left\| \varvec{y}\right\| _{2,\varOmega }+\left\| p\right\| _{1,\varOmega } +\left\| \varvec{u}\right\| _{1,\varOmega } +\left\| \varvec{f}\right\| _{1,\varOmega } \right] . \end{aligned}$$
(3.54)
Finally, splitting the co-state velocity error as
\( \varvec{w}-\varvec{w}_h=\varvec{w}-\varvec{w}_h(\varvec{y})+\varvec{w}_h(y)-\varvec{w}_h \), using triangle inequality and Lemmas
4,
5, and relation (
3.54), we get the second desired estimate
$$\begin{aligned} \left\| \varvec{w}-\varvec{w}_h\right\| _{0,\varOmega }\le \left\| \varvec{w}-\varvec{w}_h(\varvec{y})\right\| _{0,\varOmega }+\left\| \varvec{w}_h(\varvec{y})-\varvec{w}_h\right\| _{0,\varOmega } \le \left\| \varvec{w}-\varvec{w}_h(\varvec{y})\right\| _{0,\varOmega }+\left\| \varvec{y}-\varvec{y}_h\right\| _{0,\varOmega }. \end{aligned}$$
\(\square \)