We stress here that $V_{{r}}^{\textnormal{pr}}$ and $V_{{r}}^{\textnormal{du}}$ might not coincide, this will imply further discussions of the RB approximation of the optimality system .

ROM for the optimality system – Standard approach {#sec:standard_approach}
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Given a RB space $V_{{r}}^{\textnormal{pr}}\subset V_h$ of low dimension $n := \dim V_{{r}}^{\textnormal{pr}}$ and dual RB space $V_{{r}}^{\textnormal{du}}\subset V_h$ of low dimension $m := \dim V_{{r}}^{\textnormal{du}}$, we obtain the RB approximation of state and adjoint equations as follows:

\[eq:optimality\_conditionsRB\]

-   RB approximation for : For each $\mu \in {\mathcal{P}}$ the primal variable $u_{{{r}}, \mu} \in V_{{r}}^{\textnormal{pr}}$ of the *RB approximate primal equation* is defined through $$\begin{aligned}
            {a_{\mu}}(u_{{{r}}, \mu}, v_{{r}}) = {l_{\mu}}(v_{{r}}) &\qquad \text{for all } v_{{r}}\in V_{{r}}^{\textnormal{pr}}.
            \label{eq:state_red}
            \end{aligned}$$

-   RB approximation for : For each $\mu \in {\mathcal{P}}$, $u_{{{r}}, \mu} \in V_{{r}}^{\textnormal{pr}}$ the dual/adjoint variable $p_{{{r}}, \mu} \in V_{{r}}^{\textnormal{du}}$ satisfies the *RB approximate dual equation* through $$\begin{aligned}
            {a_{\mu}}(q_{{r}}, p_{{{r}}, \mu}) = \partial_u {\mathcal{J}}(u_{{{r}}, \mu}, \mu)[q_{{r}}] = {j_{\mu}}(q_{{r}}) + 2 {k_{\mu}}(q_{{r}}, u_{{{r}}, \mu}) &&\text{for all } q_{{r}}\in V_{{r}}^{\textnormal{du}}.
            \label{eq:dual_solution_red}
            \end{aligned}$$

Analogously to Propostion \[prop:solution\_dmu\_eta\], we define the *RB solution map* $\mathcal{S}_{{r}}:{\mathcal{P}}\to V_{{r}}^{\textnormal{pr}}$ by $\mu \mapsto u_{{{r}}, \mu}$ and analogously to Propostion \[prop:dual\_solution\_dmu\_eta\] the *RB dual solution map* $\mathcal{A}_{{r}}:{\mathcal{P}}\to V_{{r}}^{\textnormal{du}}$ by $\mu \mapsto p_{{{r}}, \mu}$, where $u_{{{r}}, 