\subsection{Uncertain LMI}

\label{sec:lmi_cont}
The compact semialgebraic sets in the uncertain peak estimation setting are
\begin{align}
    \label{eq:peak_sets}
    X &= \{x \mid g_i(x)\geq 0 & &\mid \ i = 1, \ldots, N_c\} \\
    X_0 &= \{x \mid  g_{0i}(x)\geq 0 & &\mid \ i = 1, \ldots, N_c^0\}\\
    W &= \{w \mid g_{wi}(w)\geq 0 & &\mid \ i = 1, \ldots, N_c^w\}\\
    \Delta &= \{d \mid g_{di}(d)\geq 0 & &\mid \ i = 1, \ldots, N_c^d\}.
\end{align}

The degree of $g_i(x)$ is $d_i$, and other degrees $d_{0i}, \ d_{wi}, \ d_{di}$ are defined on corresponding polynomials. Monomials forming moments may be indexed as  $x^\alpha t^\beta w^\gamma d^\delta$ for multi-indices $\alpha \in \N^{N_x}, \ \beta \in \N, \ \gamma \in \N^{N_w}, \ \delta \in \N^{N_d}$. Define $y^0 = \{y^0_{\alpha \gamma} \}, \ y^p = \{y^p_{\alpha \beta \gamma}\}$ as the moment sequences for measures $\mu_0$ and $\mu_p$, just as in Section \ref{sec:peak_lmi}. The moment sequence for the occupation measure $\mu^k$ is $y^k = \{y^k_{\alpha \beta \gamma \delta}\}$ for each switching subsystem $k$. The box-affine disturbance occupation measures $\sigma_{k\ell}, \hat{\sigma}_{k\ell}$ likewise have moment sequences $s^{k\ell} = \{s^{k \ell}_{\alpha \beta \gamma \delta}\}, \ \hat{s}^{k \ell} = \{\hat{s}^{k\ell}_{\alpha \beta \gamma \delta}\}$. The Liouville equation \eqref{eq:peak_meas_un_flow} with test function $v(t, x, w) = x^\alpha t^\beta w^\gamma$ has the form,
\begin{equation}
\label{eq:liou_lmi_unc}
    \inp{x^\alpha t^\beta w^\gamma}{ {\delta_0 \otimes} \mu_0} + \textstyle \sum_k\inp{\Lie_{f_{0k}}( x^\alpha t^\beta w^\gamma)}{\mu_k} + \textstyle\sum_{k \ell}\inp{f_{k\ell} \cdot \nabla_x(x^\alpha t^\beta w^\gamma)}{\sigma_{k\ell}} - \inp{x^\alpha t^\beta w^\gamma}{\mu_p} = 0.
\end{equation}
Define the operator $\textrm{Liou}_{\alpha \beta\gamma }(y^0, y^p, y^k, s^{k \ell})$ as the linear relation between the moment sequences induced by \eqref{eq:liou_lmi_unc}, just as in problem \eqref{eq:peak_lmi}.
The degree-$d$ LMI relaxation of the uncertain peak estimation problem in  \eqref{eq:peak_meas_un} resulting in an upper bound $p^*_d \geq P^*$ is, 
\begin{subequations}
\label{eq:peak_lmi_un}
\begin{align}
    p^*_d = & \textrm{max} \quad \textstyle\sum_{\alpha} p_\alpha y_{\alpha 0 0}^p \label{eq:peak_lmi_un_obj} \\
    & \textrm{Liou}_{\alpha \beta \gamma}(y^0, y^p, y^k, s^{k \ell}) = 0 \quad \textrm{by \eqref{eq:liou_lmi_unc}} & &  \forall \abs{\alpha} + \abs{\beta} + \abs{\gamma} \leq 2d \label{eq:peak_lmi_un_flow}\\
    & y^0_0 = 1 \\
    & \forall k \ell: s^{k \ell}_{\alpha \beta \gamma \delta} + \hat{s}^{k \ell}_{\alpha \beta \gamma \delta}  = y^k_{\alpha \beta \gamma \delta}  & &  \forall \abs{\alpha} + \abs{\beta} + \abs{\gamma} + \abs{\delta} \leq 2d \label{eq:peak_lmi_un_abscont}\\
    & \M_d(y^0), \M_d(y^p), \forall k: \M_d(y^k) \succeq 0 \label{eq:peak_lmi_un_psd} \\
    & \forall k \ell: \M_d(s^{k \ell}), \M_d(s^{k \ell}) \succeq 0 \label{eq:peak_lmi_un_psd_box} \\
    &\M_{d - d_{0i}}(g_{0i} y^0) \succeq 0  \label{eq:peak_lmi_un_init}  & &\forall i = 1, \ldots, N_c^0\\ 
    &\M_{d - d_{wi}}(g_{wi} y^0) \succeq 0  \label{eq:peak_lmi_un_init_w}  & &\forall i = 1, \ldots, N_c^w\\ 
    &\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 \\ 
    &\forall k \ell: \ \M_{d - d_i}(g_{i} s^{k \ell}), \ \M_{d - d_i}(g_{i} \hat{s}^{k \ell}) \succeq 0  & & \forall i = 1, \ldots, N_c  \label{eq:peak_lmi_un_state} \\
    &\M_{d - d_{wi}}(g_{wi} y^p), \ \forall k: \M_{d - d_{wi}}(g_{wi} y^k) \succeq 0 & & \forall i = 1, \ldots, N_c^w \\ 
    &\forall k \ell: \ \M_{d - d_{wi}}(g_{wi} s^{k \ell}), \ \M_{d - d_{wi}}(g_{wi} \hat{s}^{k \ell}) \succeq 0  & & \forall i = 1, \ldots, N_c^w  \label{eq:peak_lmi_un_w} \\
    &\M_{d-2}(t(T-t)y^p), \forall k: \M_{d-2}(t(T-t)y^k) \succeq  0 \label{eq:peak_lmi_un_supp2} \\
    & \forall k \ell: \ \M_{d-2}(t(T-t)s^{k \ell}), \  \M_{d-2}(t(T-t)\hat{s}^{k \ell}) \succeq 0 & &   \\
    &\forall k: \M_{d - d_{di}}(g_{di} y^k) \succeq 0 & & \forall i = 1, \ldots, N_c^d \\ 
    &\forall k \ell: \ \M_{d - d_{di}}(g_{di} s^{k \ell}), \ \M_{d - d_{di}}(g_{di} \hat{s}^{k \ell}) \succeq 0  & & \forall i = 1, \ldots, N_c^d \label{eq:peak_lmi_un_d}
\end{align}
\end{subequations}

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.



