equivalently:

$$
\begin{align}
\langle (\bar{P} - hM_{11})\dot{\eta}_k^1, \dot{\eta}_k^1 \rangle &\le \langle hM_{22}\dot{\eta}_k^2, \dot{\eta}_k^2 \rangle + \langle \dot{\eta}_k^1, h(M_{12}^\top + M_{21})\dot{\eta}_k^2 \rangle + \langle \bar{P}\dot{\eta}_{k-1}^1, \dot{\eta}_k^1 \rangle - \langle U^\top R \Delta E_k, \dot{\eta}_k \rangle \nonumber \\
&\le \langle hM_{22}\dot{\eta}_k^2, \dot{\eta}_k^2 \rangle + \frac{1}{2}h||\dot{\eta}_k^1||^2 + \frac{1}{2}h||(M_{12}^\top + M_{21})||^2||\dot{\eta}_k^2||^2 \nonumber \\
&\phantom{\le} + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_{k-1}^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_k^1||^2 + |\langle (U^\top R \Delta E_k)^{1,\top} \dot{\eta}_k^1 \rangle| + |\langle (U^\top R \Delta E_k)^{2,\top} \dot{\eta}_k^2 \rangle| \nonumber \\
&\le \langle h(M_{22} + \frac{1}{2}||(M_{12}^\top + M_{21})||^2)\dot{\eta}_k^2, \dot{\eta}_k^2 \rangle + \frac{1}{2}h||\dot{\eta}_k^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_{k-1}^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_k^1||^2 \nonumber \\
&\phantom{\le} + ||(U^\top R \Delta E_k)^{1}|| ||\dot{\eta}_k^1|| + ||(U^\top R \Delta E_k)^{2}|| ||\dot{\eta}_k^2|| \nonumber \\
&\le \langle h(M_{22} + \frac{1}{2}||(M_{12}^\top + M_{21})||^2)\dot{\eta}_k^2, \dot{\eta}_k^2 \rangle + \frac{1}{2}h||\dot{\eta}_k^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_{k-1}^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_k^1||^2 \nonumber \\
&\phantom{\le} + \alpha_1 h||\dot{\eta}_k^1|| + \alpha_2 h||\dot{\eta}_k^2|| \nonumber \\
&\le \langle h(M_{22} + \frac{1}{2}||(M_{12}^\top + M_{21})||^2)\dot{\eta}_k^2, \dot{\eta}_k^2 \rangle + \frac{1}{2}h||\dot{\eta}_k^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_{k-1}^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_k^1||^2 \nonumber \\
&\phantom{\le} + \frac{1}{2}\alpha_1^2 h + \frac{1}{2}h||\dot{\eta}_k^1||^2 + \frac{1}{2}\alpha_2^2 h + \frac{1}{2}h||\dot{\eta}_k^2||^2, \tag{79}
\end{align}
$$

where the inequalities $|x^\top y| \le ||x|| ||y|| \le \frac{1}{2}||x||^2 + \frac{1}{2}||y||^2$ for any real vectors $x$ and $y$ are used, and item 7 guarantees the existence of bounded $\alpha_1 > 0$ and $\alpha_2 > 0$ (remind that $t_{k+1} - t_k = h$). Hence:

$$
\begin{align}
\langle (\frac{1}{2}\bar{P} - hM_{11} - hI_p)\dot{\eta}_k^1, \dot{\eta}_k^1 \rangle &\le \langle h(M_{22} + \frac{1}{2}||(M_{12}^\top + M_{21})||^2 I_{n-p} + \frac{1}{2}I_{n-p})\dot{\eta}_k^2, \dot{\eta}_k^2 \rangle + \frac{1}{2}||\bar{P}^{\frac{1}{2}}\dot{\eta}_{k-1}^1||^2 \nonumber \\
&\le \frac{1}{2}||\bar{P}^{\frac{1}{2}}\dot{\eta}_{k-1}^1||^2 + (\alpha_1^2 + \alpha_2^2)h = \langle \frac{1}{2}\bar{P}\dot{\eta}_{k-1}^1, \dot{\eta}_{k-1}^1 \rangle + (\alpha_1^2 + \alpha_2^2)h \nonumber \\
&\le \langle (\frac{1}{2}\bar{P} - hM_{11} - hI_p)\dot{\eta}_{k-1}^1, \dot{\eta}_{k-1}^1 \rangle + (\alpha_1^2 + \alpha_2^2)h, \tag{80}
\end{align}
$$

where item 5 has been used to obtain the second inequality and item 6 has been used for the third inequality to secure that $\frac{1}{2}\bar{P} \le \frac{1}{2}\bar{P} - hM_{11} - hI_p$. Denote $V(\dot{\eta}_k^1) \triangleq \langle (\frac{1}{2}\bar{P} - hM_{11} - hI_n)\dot{\eta}_k^1, \dot{\eta}_k^1 \rangle$, we obtain from (80)

$$V(\dot{\eta}_k^1) \le V(\dot{\eta}_0^1) + (\alpha_1^2 + \alpha_2^2) \frac{(k+1)T}{n}. \quad (81)$$

Therefore $V(\dot{\eta}_k^1)$ and consequently $\dot{\eta}_k^1$ is uniformly bounded for all $k \in \{0, n\}$, independently of $n$. Thus $\dot{\eta}_h^1(t) = U_1^\top \dot{\xi}^h(t)$ is bounded in $t \in [0, T]$ for any bounded initial data. The sequence $\{\eta_h^1\}$ is thus uniformly bounded (because $\{\xi^h(\cdot)\}$ is), continuous with uniformly bounded derivatives almost everywhere, and thus equicontinuous. From Arzelà-Ascoli Theorem, $\{\eta_h^1\}$ stays in a compact subset of $C^0([0, T]; \mathbb{R}^p)$, and there exists subsequences (which we still denote $\eta_h^1(\cdot)$) that converge uniformly in $C^0([0, T]; \mathbb{R}^p)$ to a limit $\eta^1(\cdot)$.

# F Proof of Proposition 6

Let us start with

$$RPR^{-1}(\xi_{k+1} - \xi_0) - hRAR^{-1}\xi_{k+1} - RPR^{-1}(\xi_k - \xi_0) - hRE_k \in -h \partial f(\xi_{k+1}), \quad (82)$$

equivalently

$$
\begin{align}
& U^\top RPR^{-1} UU^\top (\xi_{k+1} - \xi_0) - hU^\top RAR^{-1} UU^\top \xi_{k+1} - U^\top RPR^{-1} UU^\top (\xi_k - \xi_0) \nonumber \\
&\quad - hU^\top RE_k \in -hU^\top \partial f(\xi_{k+1}), \tag{83}
\end{align}
$$
equivalently:

$$
\begin{align}
\langle (\bar{P} - hM_{11})\dot{\eta}_k^1, \dot{\eta}_k^1 \rangle &\le \langle hM_{22}\dot{\eta}_k^2, \dot{\eta}_k^2 \rangle + \langle \dot{\eta}_k^1, h(M_{12}^\top + M_{21})\dot{\eta}_k^2 \rangle + \langle \bar{P}\dot{\eta}_{k-1}^1, \dot{\eta}_k^1 \rangle - \langle U^\top R \Delta E_k, \dot{\eta}_k \rangle \nonumber \\
&\le \langle hM_{22}\dot{\eta}_k^2, \dot{\eta}_k^2 \rangle + \frac{1}{2}h||\dot{\eta}_k^1||^2 + \frac{1}{2}h||(M_{12}^\top + M_{21})||^2||\dot{\eta}_k^2||^2 \nonumber \\
&\phantom{\le} + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_{k-1}^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_k^1||^2 + |\langle (U^\top R \Delta E_k)^{1,\top} \dot{\eta}_k^1 \rangle| + |\langle (U^\top R \Delta E_k)^{2,\top} \dot{\eta}_k^2 \rangle| \nonumber \\
&\le \langle h(M_{22} + \frac{1}{2}||(M_{12}^\top + M_{21})||^2)\dot{\eta}_k^2, \dot{\eta}_k^2 \rangle + \frac{1}{2}h||\dot{\eta}_k^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_{k-1}^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_k^1||^2 \nonumber \\
&\phantom{\le} + ||(U^\top R \Delta E_k)^1|| ||\dot{\eta}_k^1|| + ||(U^\top R \Delta E_k)^2|| ||\dot{\eta}_k^2|| \nonumber \\
&\le \langle h(M_{22} + \frac{1}{2}||(M_{12}^\top + M_{21})||^2)\dot{\eta}_k^2, \dot{\eta}_k^2 \rangle + \frac{1}{2}h||\dot{\eta}_k^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_{k-1}^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_k^1||^2 \nonumber \\
&\phantom{\le} + \alpha_1 h||\dot{\eta}_k^1|| + \alpha_2 h||\dot{\eta}_k^2|| \nonumber \\
&\le \langle h(M_{22} + \frac{1}{2}||(M_{12}^\top + M_{21})||^2)\dot{\eta}_k^2, \dot{\eta}_k^2 \rangle + \frac{1}{2}h||\dot{\eta}_k^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_{k-1}^1||^2 + \frac{1}{2}||\bar{P}^\frac{1}{2}\dot{\eta}_k^1||^2 \nonumber \\
&\phantom{\le} + \frac{1}{2}\alpha_1^2 h + \frac{1}{2}h||\dot{\eta}_k^1||^2 + \frac{1}{2}\alpha_2^2 h + \frac{1}{2}h||\dot{\eta}_k^2||^2, \tag{79}
\end{align}
$$

where the inequalities $|x^\top y| \le ||x|| ||y|| \le \frac{1}{2}||x||^2 + \frac{1}{2}||y||^2$ for any real vectors $x$ and $y$ are used, and item 7 guarantees the existence of bounded $\alpha_1 > 0$ and $\alpha_2 > 0$ (remind that $t_{k+1} - t_k = h$). Hence:

$$
\begin{align}
\langle (\frac{1}{2}\bar{P} - hM_{11} - hI_p)\dot{\eta}_k^1, \dot{\eta}_k^1 \rangle &\le \langle h(M_{22} + \frac{1}{2}||(M_{12}^\top + M_{21})||^2 I_{n-p} + \frac{1}{2}I_{n-p})\dot{\eta}_k^2, \dot{\eta}_k^2 \rangle + \frac{1}{2}||\bar{P}^{\frac{1}{2}}\dot{\eta}_{k-1}^1||^2 \nonumber \\
&\le \frac{1}{2}||\bar{P}^{\frac{1}{2}}\dot{\eta}_{k-1}^1||^2 + (\alpha_1^2 + \alpha_2^2)h = \langle \frac{1}{2}\bar{P}\dot{\eta}_{k-1}^1, \dot{\eta}_{k-1}^1 \rangle + (\alpha_1^2 + \alpha_2^2)h \nonumber \\
&\le \langle (\frac{1}{2}\bar{P} - hM_{11} - hI_p)\dot{\eta}_{k-1}^1, \dot{\eta}_{k-1}^1 \rangle + (\alpha_1^2 + \alpha_2^2)h, \tag{80}
\end{align}
$$

where item 5 has been used to obtain the second inequality and item 6 has been used for the third inequality to secure that $\frac{1}{2}\bar{P} \preceq \frac{1}{2}\bar{P} - hM_{11} - hI_p$. Denote $V(\dot{\eta}_k^1) \triangleq \langle (\frac{1}{2}\bar{P} - hM_{11} - hI_n)\dot{\eta}_k^1, \dot{\eta}_k^1 \rangle$, we obtain from (80)

$$ V(\dot{\eta}_k^1) \le V(\dot{\eta}_0^1) + (\alpha_1^2 + \alpha_2^2) \frac{(k+1)T}{n}. \quad (81) $$

Therefore $V(\dot{\eta}_k^1)$ and consequently $\dot{\eta}_k^1$ is uniformly bounded for all $k \in \{0, n\}$, independently of $n$. Thus $\dot{\eta}_h^1(t) = U_1^\top \dot{\xi}^h(t)$ is bounded in $t \in [0, T]$ for any bounded initial data. The sequence $\{\eta_h^1\}$ is thus uniformly bounded (because $\{\xi^h(\cdot)\}$ is), continuous with uniformly bounded derivatives almost everywhere, and thus equicontinuous. From Arzelà-Ascoli Theorem, $\{\eta_h^1\}$ stays in a compact subset of $C^0([0, T]; \mathbb{R}^p)$, and there exists subsequences (which we still denote $\eta_h^1(\cdot)$) that converge uniformly in $C^0([0, T]; \mathbb{R}^p)$ to a limit $\eta^1(\cdot)$.

# F Proof of Proposition 6

Let us start with

$$ RPR^{-1}(\xi_{k+1} - \xi_0) - hRAR^{-1}\xi_{k+1} - RPR^{-1}(\xi_k - \xi_0) - hRE_k \in -h \partial f(\xi_{k+1}), \quad (82) $$

equivalently

$$
\begin{align}
& U^\top RPR^{-1} UU^\top (\xi_{k+1} - \xi_0) - hU^\top RAR^{-1} UU^\top \xi_{k+1} - U^\top RPR^{-1} UU^\top (\xi_k - \xi_0) \nonumber \\
&\quad - hU^\top RE_k \in -hU^\top \partial f(\xi_{k+1}), \tag{83}
\end{align}
$$
that is

$$ \begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_{k+1}^1 - \eta_0^1 \\ \eta_{k+1}^2 - \eta_0^2 \end{pmatrix} - h \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix} \begin{pmatrix} \eta_{k+1}^1 \\ \eta_{k+1}^2 \end{pmatrix} - \begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_k^1 - \eta_0^1 \\ \eta_k^2 - \eta_0^2 \end{pmatrix} \tag{84} \\ -hU^\top RE_k \in -hU^\top \partial f(\xi_{k+1}). $$

The condition $U^\top R \operatorname{Im}(P) \supseteq U^\top R \operatorname{Im}(B)$ implies that $U_2^\top RB = 0 \Rightarrow U_2^\top \zeta_k = 0$ for any $\zeta_k \in \partial f(\xi_{k+1})$. Indeed $U^\top R \operatorname{Im}(P) = \operatorname{Im}(U^\top RPR^{-1}U) = \operatorname{Im}(\begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix})$ and $U^\top R \operatorname{Im}(B) = \operatorname{Im}(U^\top RB) = \operatorname{Im}(\begin{pmatrix} U_1^\top RB \\ U_2^\top RB \end{pmatrix})$. Thus using the second line in the left-hand side of (84) and item 5, we obtain $\eta_{k+1}^2 = M_{22}^{-1}(-M_{21}\eta_{k+1}^1 + U_2^\top RE_k)$. Inserting this in the first line of the second term in the left-hand side of (84) gives $-hM_{11}\eta_{k+1}^1 - hM_{12}\eta_{k+1}^2 = -hM_{\operatorname{sch}}\eta_{k+1}^1 - hM_{12}M_{22}^{-1}U_2^\top RE_k$, where $M_{\operatorname{sch}}$ is in item 4 of the proposition. Now using (84) we obtain:

$$ \begin{aligned} & \langle \begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_{k+1}^1 - \eta_0^1 \\ \eta_{k+1}^2 - \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle - \langle h \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix} \begin{pmatrix} \eta_{k+1}^1 - \eta_0^1 \\ \eta_{k+1}^2 - \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle \\ & \quad - \langle \begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_k^1 - \eta_0^1 \\ \eta_k^2 - \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle - \langle hU^\top RE_k, U^\top (\xi_{k+1} - \xi_0) \rangle \\ & \quad - \langle h \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix} \begin{pmatrix} \eta_0^1 \\ \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle \\ & = \langle h(U^\top \zeta_{k+1} - U^\top \zeta_0 + U^\top \zeta_0), U^\top (\xi_{k+1} - \xi_0) \rangle \le \langle U^\top \zeta_0, U^\top (\xi_{k+1} - \xi_0) \rangle. \end{aligned} \tag{85} $$

Let us now define $w_0 \stackrel{\Delta}{=} U^\top \zeta_0 + \begin{pmatrix} M_{11} & M_{21} \\ M_{21} & M_{22} \end{pmatrix} U^\top \xi_0$, with $\zeta_0 \in \partial f(\xi_0)$. Using the monotonicity of $\partial f(\cdot)$, the definition of $w_0$ and $\zeta_{k+1} \in \partial f(\xi_{k+1})$:

$$ \begin{aligned} & \langle \begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_{k+1}^1 - \eta_0^1 \\ \eta_{k+1}^2 - \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle - \langle h \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix} \begin{pmatrix} \eta_{k+1}^1 - \eta_0^1 \\ \eta_{k+1}^2 - \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle \\ & \quad - \langle \begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_k^1 - \eta_0^1 \\ \eta_k^2 - \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle - \langle h(U^\top RE_k + w_0), U^\top (\xi_{k+1} - \xi_0) \rangle \le 0 \end{aligned} \tag{86} $$

Using the calculations made just above (85) for $M_{11}\eta_{k+1}^1 + M_{12}\eta_{k+1}^2$ with the Schur complement $M_{\operatorname{sch}}$, the fact that $U^\top(\xi_{k+1} - \xi_0) = \begin{pmatrix} \eta_{k+1}^1 - \eta_0^1 \\ \eta_{k+1}^2 - \eta_0^2 \end{pmatrix}$, it follows from (86) that:

$$ \begin{aligned} \langle (\bar{P} - hM_{\operatorname{sch}})(\eta_{k+1}^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle & \le \langle \bar{P}(\eta_k^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle \\ & \quad +h\langle U_1^\top RE_k + w_0^1 - M_{11}\eta_0^1 - M_{12}\eta_0^2, \eta_{k+1}^1 - \eta_0^1 \rangle \\ & \quad +h\langle U_2^\top \zeta_0, M_{12}M_{22}^{-1}U_2^\top RE_k \rangle - h\langle M_{12}M_{22}^{-1}U_2^\top \zeta_0, \eta_{k+1}^1 - \eta_0^1 \rangle \\ & \quad -h\langle M_{12}M_{22}^{-1}U_2^\top \zeta_0, \eta_0^1 \rangle \\ & \le \langle \bar{P}(\eta_k^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle \\ & \quad +\frac{h}{2}\|U_1^\top RE_k + w_0^1 - M_{11}\eta_0^1 - M_{12}\eta_0^2\|^2 + \frac{h}{2}\|\eta_0^1\|^2 \\ & \quad +\frac{h}{2}\|M_{12}M_{22}^{-1}U_2^\top \zeta_0\|^2 + \frac{h}{2}\|U_2^\top \zeta_0\|^2 + \frac{h}{2}\|M_{12}M_{22}^{-1}U_2^\top RE_k\|^2 \\ & \quad +h\|\eta_{k+1}^1 - \eta_0^1\|^2. \end{aligned} \tag{87} $$
that is

$$ \begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_{k+1}^1 - \eta_0^1 \\ \eta_{k+1}^2 - \eta_0^2 \end{pmatrix} - h \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix} \begin{pmatrix} \eta_{k+1}^1 \\ \eta_{k+1}^2 \end{pmatrix} - \begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_k^1 - \eta_0^1 \\ \eta_k^2 - \eta_0^2 \end{pmatrix} \tag{84} \\ -hU^\top RE_k \in -hU^\top \partial f(\xi_{k+1}). $$

The condition $U^\top R \operatorname{Im}(P) \supseteq U^\top R \operatorname{Im}(B)$ implies that $U_2^\top RB = 0 \Rightarrow U_2^\top \zeta_k = 0$ for any $\zeta_k \in \partial f(\xi_{k+1})$. Indeed $U^\top R \operatorname{Im}(P) = \operatorname{Im}(U^\top RPR^{-1}U) = \operatorname{Im}(\begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix})$ and $U^\top R \operatorname{Im}(B) = \operatorname{Im}(U^\top RB) = \operatorname{Im}(\begin{pmatrix} U_1^\top RB \\ U_2^\top RB \end{pmatrix})$. Thus using the second line in the left-hand side of (84) and item 5, we obtain $\eta_{k+1}^2 = M_{22}^{-1}(-M_{21}\eta_{k+1}^1 + U_2^\top RE_k)$. Inserting this in the first line of the second term in the left-hand side of (84) gives $-hM_{11}\eta_{k+1}^1 - hM_{12}\eta_{k+1}^2 = -hM_{\operatorname{sch}}\eta_{k+1}^1 - hM_{12}M_{22}^{-1}U_2^\top RE_k$, where $M_{\operatorname{sch}}$ is in item 4 of the proposition. Now using (84) we obtain:

$$ \begin{aligned} & \langle \begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_{k+1}^1 - \eta_0^1 \\ \eta_{k+1}^2 - \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle - \langle h \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix} \begin{pmatrix} \eta_{k+1}^1 - \eta_0^1 \\ \eta_{k+1}^2 - \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle \\ & \quad - \langle \begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_k^1 - \eta_0^1 \\ \eta_k^2 - \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle - \langle hU^\top RE_k, U^\top (\xi_{k+1} - \xi_0) \rangle \\ & \quad - \langle h \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix} \begin{pmatrix} \eta_0^1 \\ \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle \\ & = \langle h(U^\top \zeta_{k+1} - U^\top \zeta_0 + U^\top \zeta_0), U^\top (\xi_{k+1} - \xi_0) \rangle \le \langle U^\top \zeta_0, U^\top (\xi_{k+1} - \xi_0) \rangle. \end{aligned} \tag{85} $$

Let us now define $w_0 \stackrel{\Delta}{=} U^\top \zeta_0 + \begin{pmatrix} M_{11} & M_{21} \\ M_{21} & M_{22} \end{pmatrix} U^\top \xi_0$, with $\zeta_0 \in \partial f(\xi_0)$. Using the monotonicity of $\partial f(\cdot)$, the definition of $w_0$ and $\zeta_{k+1} \in \partial f(\xi_{k+1})$:

$$ \begin{aligned} & \langle \begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_{k+1}^1 - \eta_0^1 \\ \eta_{k+1}^2 - \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle - \langle h \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix} \begin{pmatrix} \eta_{k+1}^1 - \eta_0^1 \\ \eta_{k+1}^2 - \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle \\ & \quad - \langle \begin{pmatrix} \bar{P} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \eta_k^1 - \eta_0^1 \\ \eta_k^2 - \eta_0^2 \end{pmatrix}, U^\top (\xi_{k+1} - \xi_0) \rangle - \langle h(U^\top RE_k + w_0), U^\top (\xi_{k+1} - \xi_0) \rangle \le 0 \end{aligned} \tag{86} $$

Using the calculations made just above (85) for $M_{11}\eta_{k+1}^1 + M_{12}\eta_{k+1}^2$ with the Schur complement $M_{\operatorname{sch}}$, the fact that $U^\top(\xi_{k+1} - \xi_0) = \begin{pmatrix} \eta_{k+1}^1 - \eta_0^1 \\ \eta_{k+1}^2 - \eta_0^2 \end{pmatrix}$, it follows from (86) that:

$$ \begin{aligned} \langle (\bar{P} - hM_{\operatorname{sch}})(\eta_{k+1}^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle & \le \langle \bar{P}(\eta_k^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle \\ & \quad +h\langle U_1^\top RE_k + w_0^1 - M_{11}\eta_0^1 - M_{12}\eta_0^2, \eta_{k+1}^1 - \eta_0^1 \rangle \\ & \quad +h\langle U_2^\top \zeta_0, M_{12}M_{22}^{-1}U_2^\top RE_k \rangle - h\langle M_{12}M_{22}^{-1}U_2^\top \zeta_0, \eta_{k+1}^1 - \eta_0^1 \rangle \\ & \quad -h\langle M_{12}M_{22}^{-1}U_2^\top \zeta_0, \eta_0^1 \rangle \\ & \le \langle \bar{P}(\eta_k^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle \\ & \quad +\frac{h}{2}\|U_1^\top RE_k + w_0^1 - M_{11}\eta_0^1 - M_{12}\eta_0^2\|^2 + \frac{h}{2}\|\eta_0^1\|^2 \\ & \quad +\frac{h}{2}\|M_{12}M_{22}^{-1}U_2^\top \zeta_0\|^2 + \frac{h}{2}\|U_2^\top \zeta_0\|^2 + \frac{h}{2}\|M_{12}M_{22}^{-1}U_2^\top RE_k\|^2 \\ & \quad +h\|\eta_{k+1}^1 - \eta_0^1\|^2. \end{aligned} \tag{87} $$
Using items 3 and 4 we have $\tilde{P} \triangleq \bar{P} - hM_{sch} - hI_p > 0$, then:

$$ \langle \tilde{P}(\eta_{k+1}^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle \le \langle \bar{P}(\eta_k^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle + \frac{h}{2}\Upsilon(E_k, \zeta_0, \eta_0) \\ \le \langle \tilde{P}(\eta_k^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle + \frac{h}{2}\Upsilon(E_k, \zeta_0, \eta_0) \quad (88) $$

where $\Upsilon(E_k, \zeta_0, \eta_0)$ is easy to define. From the assumptions of the proposition, $\frac{1}{2} \sum_{j=1}^{n-1} h\Upsilon(E_k, \zeta_0, \eta_0) \le C\frac{h}{2}(n-1) = C\frac{T}{2}\frac{n-1}{n} \le \frac{CT}{2}$ for some constant $C > 0$. Thus $\langle \tilde{P}(\eta_{k+1}^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle \le \frac{CT}{2} + \langle \tilde{P}(\eta_1^1 - \eta_0^1), \eta_1^1 - \eta_0^1 \rangle$ for all $k \in \{1, n-1\}$, while $\langle \tilde{P}(\eta_1^1 - \eta_0^1), \eta_1^1 - \eta_0^1 \rangle \le \frac{CT}{2}$. We conclude that $||\eta_{k+1}^1 - \eta_0^1||$ is bounded uniformly in $k$, thus $\eta_k^1$ and $\eta_k^2$ are uniformly bounded as well. This ends the proof.

**Remark 5** Item 5 in Proposition 6 guarantees a decoupling between the algebraic equation and the multiplier $\lambda$. If this assumption is relaxed, one has to deal with the term $\langle U_2^\top \zeta_0, M_{12}M_{22}^{-1}U_2^\top \zeta_k \rangle$ in the right-hand side of (87). Another way to decouple is thus to assume that $M_{12} = 0$. Item 5 could be replaced by: $U^\top R \text{Im}(P) \subseteq U^\top R \text{Im}(P)$ or $M_{12} = 0$. If none of these conditions is satisfied, then one has to resort to another kind of proof, as outlined in section 7.

## References

[1] V. Acary, O. Bonnefon, and B. Brogliato. *Nonsmooth Modeling and Simulation for Switched Circuits*, volume 69 of *Lecture Notes in Electrical Engineering*. Springer Science+Business Media, 2011. Erratum at https://hal.inria.fr/hal-01311078.

[2] V. Acary and B. Brogliato. *Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics*, volume 35 of *Lecture Notes in Applied and Computational Mechanics*. Springer Verlag, Berlin Heidelberg, 2008.

[3] K. Addi, B. Brogliato, and D. Goeleven. A qualitative mathematical analysis of a class of linear variational inequalitites via semi-complementarity problems: applications in electronics. *Mathematical Programming A*, 126:31–67, 2011.

[4] S. Adly, T. Haddad, and L. Thibault. Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities. *Mathematical Programming A*, 148(1):5–47, 2014.

[5] S. Adly and B.K. Le. On semicoercive sweeping process with velocity constraint. *Optimization Letters*, 12(4):831–843, 2018.

[6] H. Attouch and J. Peypouquet. Convergence of inertial dynamics and proximal algorithms governed by maximally monotone operators. *Mathematical Programming*, 174(1-2):391–432, 2019.

[7] J.P. Aubin. *Applied Functional Analysis*. John Wiley & Sons, New York, 1979.

[8] J.P. Aubin and A. Cellina. *Differential Inclusions. Set-Valued Maps and Viability Theory*. Springer-Verlag, Berlin Heidelberg, 1984.

[9] J. Bastien and M. Schatzman. Numerical precision for differential inclusions with uniqueness. *ESAIM Mathematical Modelling and Numerical Analysis*, 36(3):427–460, 2002.
Using items 3 and 4 we have $\tilde{P} \triangleq \bar{P} - hM_{sch} - hI_p > 0$, then:

$$ \begin{aligned} \langle \tilde{P}(\eta_{k+1}^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle &\le \langle \bar{P}(\eta_k^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle + \frac{h}{2}\Upsilon(E_k, \zeta_0, \eta_0) \\ &\le \langle \tilde{P}(\eta_k^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle + \frac{h}{2}\Upsilon(E_k, \zeta_0, \eta_0) \end{aligned} \quad (88) $$

where $\Upsilon(E_k, \zeta_0, \eta_0)$ is easy to define. From the assumptions of the proposition, $\frac{1}{2} \sum_{j=1}^{n-1} h\Upsilon(E_k, \zeta_0, \eta_0) \le C\frac{h}{2}(n-1) = C\frac{T}{2}\frac{n-1}{n} \le \frac{CT}{2}$ for some constant $C > 0$. Thus $\langle \tilde{P}(\eta_{k+1}^1 - \eta_0^1), \eta_{k+1}^1 - \eta_0^1 \rangle \le \frac{CT}{2} + \langle \tilde{P}(\eta_1^1 - \eta_0^1), \eta_1^1 - \eta_0^1 \rangle$ for all $k \in \{1, n-1\}$, while $\langle \tilde{P}(\eta_1^1 - \eta_0^1), \eta_1^1 - \eta_0^1 \rangle \le \frac{CT}{2}$. We conclude that $||\eta_{k+1}^1 - \eta_0^1||$ is bounded uniformly in $k$, thus $\eta_k^1$ and $\eta_k^2$ are uniformly bounded as well. This ends the proof.

**Remark 5** Item 5 in Proposition 6 guarantees a decoupling between the algebraic equation and the multiplier $\lambda$. If this assumption is relaxed, one has to deal with the term $\langle U_2^\top \zeta_0, M_{12}M_{22}^{-1}U_2^\top \zeta_k \rangle$ in the right-hand side of (87). Another way to decouple is thus to assume that $M_{12} = 0$. Item 5 could be replaced by: $U^\top R \mathrm{Im}(P) \subseteq U^\top R \mathrm{Im}(P)$ or $M_{12} = 0$. If none of these conditions is satisfied, then one has to resort to another kind of proof, as outlined in section 7.

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[2] V. Acary and B. Brogliato. *Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics*, volume 35 of Lecture Notes in Applied and Computational Mechanics. Springer Verlag, Berlin Heidelberg, 2008.

[3] K. Addi, B. Brogliato, and D. Goeleven. A qualitative mathematical analysis of a class of linear variational inequabilities via semi-complementarity problems: applications in electronics. *Mathematical Programming A*, 126:31–67, 2011.

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[5] S. Adly and B.K. Le. On semicoercive sweeping process with velocity constraint. *Optimization Letters*, 12(4):831–843, 2018.

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[7] J.P. Aubin. *Applied Functional Analysis*. John Wiley & Sons, New York, 1979.

[8] J.P. Aubin and A. Cellina. *Differential Inclusions. Set-Valued Maps and Viability Theory*. Springer-Verlag, Berlin Heidelberg, 1984.

[9] J. Bastien and M. Schatzman. Numerical precision for differential inclusions with uniqueness. *ESAIM Mathematical Modelling and Numerical Analysis*, 36(3):427–460, 2002.
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