\subsection{Discrete LMI}

The LMI relaxation of \eqref{eq:peak_meas_un_disc} can be developed in the same manner as in Section \ref{sec:lmi_cont}. The sets $(X_0, X, W, D)$ are defined in the same way as in equation \eqref{eq:peak_sets}. As there is no $t$ term in discrete systems, monomials forming moments are indexed as $x^\alpha \theta^\gamma w^\eta$. The moment sequences are $y^0$, $y^p$, and a $y^k$ for each switching subsystem $k = 1, \ldots, N_s$. 
The Liouville equation \eqref{eq:peak_meas_un_disc_flow} with a given test function $v(x, \theta) = x^\alpha \theta^\gamma$  is ,
 \begin{align}
 \label{eq:liou_disc_lmi_unc}
    0&= \inp{x^\alpha \theta^\gamma}{ \delta_0 \otimes \mu_0} - \inp{x^\alpha  \theta^\gamma}{\mu_p} \\
    &+ \textstyle \sum_k\inp{( f_{k}(x, \theta, w)^\alpha \theta^\gamma - x^\alpha \theta^\gamma}{\mu_k}. \nonumber
\end{align}

The operator $\textrm{Liou}_{\alpha \gamma }(y^0, y^p, y^k)$  is defined as the relation induced by the discrete Liouville equation \eqref{eq:liou_lmi_unc_disc}. The discrete degree-$d$ LMI truncation of \eqref{eq:peak_meas_un_disc} is,
\begin{subequations}
\label{eq:peak_lmi_un_disc}
\begin{align}
    p^*_d = & \textrm{max} \quad \textstyle\sum_{\alpha} p_\alpha y_{\alpha 0}^p \label{eq:peak_lmi_un_disc_obj} \\
    & \textrm{Liou}_{\alpha \gamma}(y^0, y^p, y^k) = 0 \qquad \textrm{by \eqref{eq:liou_disc_lmi_unc}} & &  \forall \abs{\alpha} + \abs{\gamma} \leq 2d \label{eq:peak_lmi_un_disc_flow}\\
    & y^0_0 = 1 \\
    & \textstyle\sum_{k} y^k_0 \leq T \label{eq:peak_lmi_un_time_limit} \\
    & \M_d(y^0), \M_d(y^p), \forall k: \M_d(y^k) \succeq 0 \label{eq:peak_lmi_un_disc_psd} \\
    &\M_{d - d_{0i}}(g_{0i} y^0) \succeq 0  \label{eq:peak_lmi_un_disc_init}  & &\forall i = 1, \ldots, N_c^0\\ 
    &\M_{d - d_{\theta i}}(g_{\theta i} y^0) \succeq 0  \label{eq:peak_lmi_un_disc_init_w}  & &\forall i = 1, \ldots, N_c^{\theta}\\ 
    &\M_{d - d_i}(g_{i} y^p), \ \forall k: \M_{d - d_i}(g_{i} y^k) \succeq 0 & & \forall i = 1, \ldots, N_c \label{eq:peak_lmi_un_disc_peak} \\
    &\M_{d - d_{\theta i}}(g_{\theta i} y^p), \ \forall k: \M_{d - d_{\theta i}}(g_{\theta i} y^k) \succeq 0 & & \forall i = 1, \ldots, N_c^{\theta }\\
    &\forall k: \M_{d - d_{wi}}(g_{wi} y^k) \succeq 0 & & \forall i = 1, \ldots, N_c^w. \label{eq:peak_lmi_un_disc_d}
\end{align}
\end{subequations}

Constraint \eqref{eq:peak_lmi_un_time_limit} enforces the time limit constraint on occupation measures \eqref{eq:peak_meas_un_disc_time}. The structure of \eqref{eq:peak_lmi_un_disc} is similar to \eqref{eq:peak_lmi_un} with the affine, moment matrix and localizing matrix constraints.


% Many of the constraints and definitions are similar to the standard peak estimation problem in \eqref{eq:peak_lmi}. Constraint \eqref{eq:peak_lmi_un_abscont} enforces the absolute continuity condition $\sigma^{k \ell} + \hat{\sigma}^{k \ell} = \mu^k$ in \eqref{eq:peak_meas_un_abscont}. Constraints \eqref{eq:peak_lmi_un_psd}- \eqref{eq:peak_lmi_un_d} are moment and localizing matrix PSD constraints ensuring that there exist representing measures to the moment sequences $(y^0, y^p, y^k, s^{k \ell}, \hat{s}^{k \ell})$ supported on the appropriate spaces.



