for the proof of Lemma \[lemma:monotonically\_increasing\].

\[lemma:hilfslemma\] For ${\boldsymbol{H}}^{(0)} = {\boldsymbol{0}}$, ${\boldsymbol{R}}^\intercal = {\boldsymbol{R}} \succ {\boldsymbol{0}}$ and ${\boldsymbol{Q}}^\intercal = {\boldsymbol{Q}} \succeq {\boldsymbol{0}}$, $\forall i > 0$: ${\boldsymbol{L}}^{(i)}{\begin{bmatrix*}[r]{\boldsymbol{x}}_{k+\kappa}^\intercal&{\boldsymbol{r}}_{k+\kappa+1}^\intercal & \dots & {\boldsymbol{r}}_{k+\kappa+N}^\intercal
\end{bmatrix*}}$ is the unique minimizer of $$\begin{aligned}
    \hat{Q}_{\kappa}^{(i)}=\frac{1}{2}{\boldsymbol{z}}_{k+\kappa}^\intercal{\boldsymbol{H}}^{(i)}{\boldsymbol{z}}_{k+\kappa}.
    \end{aligned}$$

Due to ${\boldsymbol{H}}^{(0)}={\boldsymbol{0}}\succeq{\boldsymbol{0}}$, ${\boldsymbol{R}} \succ {\boldsymbol{0}}$ and ${\boldsymbol{Q}} \succeq {\boldsymbol{0}}$, yields ${\boldsymbol{H}}^{(i)}\succeq{\boldsymbol{0}}$. Furthermore, due to ${\boldsymbol{R}} \succ {\boldsymbol{0}}$ it is obvious that ${\boldsymbol{h}}_{uu}^{(i)}\succ{\boldsymbol{0}}$, $\forall i>0$. Therefore, $\frac{\partial \hat{Q}_{\kappa}^{(i)}}{\partial {\boldsymbol{u}}_{k+\kappa}}={\boldsymbol{0}}$ yields ${\boldsymbol{L}}^{(i)}$ and $\frac{\partial ^2 \hat{Q}_{\kappa}^{(i)}}{\partial {\boldsymbol{u}}_{k+\kappa}^2}\succ{\boldsymbol{0}}$ follows due to ${\boldsymbol{h}}_{uu}^{(i)}\succ{\boldsymbol{0}}$ and completes the proof.

Define the operator $$\begin{aligned}
\label{eq:operator}
F\left({\boldsymbol{\Omega}}^{(i)}, {\boldsymbol{\Gamma}}^{(i)}\right)&={\boldsymbol{G}}+\gamma{\boldsymbol{M}}\!\left({\boldsymbol{\Gamma}}^{(i)}\right)^\intercal {\boldsymbol{\Omega}}^{(i)}{\boldsymbol{M}}\!\left({\boldsymbol{\Gamma}}^{(i)}\right),\end{aligned}$$ i.e. $F\left({\boldsymbol{H}}^{(i)}, {\boldsymbol{L}}^{(i)}\right)={\boldsymbol{H}}^{(i+1)}$ according to .

In order to prove that ${\boldsymbol{H}}^{(i)}$ given according to Lemma \[lemma:VI\_matrix\_iteration\] is upper bounded, the following technical Lemma \[lemma:monotonically\_increasing\] is required first, which generalizes [@Landelius.1997 Lemma B.1.1] to cope with the reference-dependent Q-function. Note that knowledge of the exact structure of ${\boldsymbol{H}}$ and therefore the analytical solution by means of Theorem \[Theo:Q\] plays a crucial role for the extension to the tracking case.

\[lemma:monotonically\_increasing\] Let ${\boldsymbol{W}}^{(i)}$ be an arbitrary matrix sequence and ${\boldsymbol{0}}\preceq{\boldsymbol{H}}^{(0)}\preceq{\boldsymbol{Z}}^{(0)}$. Then, given the sequences ${\boldsymbol{Z}}^{(i+1)}=F\left({\boldsymbol{Z}}^{(i)},{\boldsymbol{W}}^{(i)}\right)$ and ${\boldsymbol{H}}^{(i+1)}=F\left({\boldsymbol{H}}^{(i)},{\boldsymbol{L}}\left({\boldsymbol{H}}^{(i)}\right)\right)$ with it follows that ${\boldsymbol{0}}\preceq{\boldsymbol{H}}^{(i+1)}\preceq{\boldsymbol{Z}}^{(i+1)}$.

See \[append:monotonically\_increasing\].

In the next step, upper boundedness of ${\boldsymbol{H}}^{(i)}$ in the sense that ${\boldsymbol{0}}\preceq{\boldsymbol{H}}^{(i)}\preceq {\boldsymbol{Y}}$ is 