\subsection{Affine Uncertainty}

\label{sec:affine}

Many dynamical systems exhibit an affine structure in terms of the disturbance $d \in \Delta$:
\begin{equation}
\label{eq:affine_base}
    \dot{x} = f(t,x,d) = f_0(t,x,d) + \sum_{k=1}^{N_d} d_k(t) f_k(t,x,d)
\end{equation}
% Peak estimation of input-affine systems with time-varying uncertainty may proceed through the methods of Problem \eqref{eq:peak_meas_w} and \eqref{eq:peak_cont_w}. 
In some specific cases the affine structure may be used to simplify computation
\cite{majumdar2014convex, korda2014controller}.
\subsubsection{Polytopic Uncertainty}
\label{sec:polytopic}

Polytopic uncertainty is an instance of time-varying uncertainty where all $d_i(t) \geq 0$ and $\sum_i d_i(t) = 1$. The dynamics from \eqref{eq:affine_base} may be divided into:
\begin{equation}
\label{sec:poly_split}
    \sum_{k=1}^{N_d} d_k(t) (f_0(t,x,d) + f_k(t,x,d) ) = \sum_{w=1}^{N_w} d_k(t) \tilde{f}_k(t,x,d)
\end{equation}
for merged dynamics $\tilde{f}_k(t,x,w)$ on each parameter $w_k$. Dynamics in $f$ are convex combinations of $\tilde{f}_k(t,x,d)$. Mixed dynamics may be realized as a system switching arbitrarily quickly between base dynamics $\tilde{f}_k(t,x,d)$. The occupation measure $\mu \in \Mp{[0, T]\times X}$ and adjoint Lie derivative of constraint \eqref{eq:peak_meas_flow} can be decomposed \cite{henrion2012measures},
\begin{equation}
\label{eq:occ_poly}
    \mu = \textstyle\sum_k \mu_k \qquad \Lie^\dagger \mu = \textstyle\sum_k \Lie^\dagger_{\tilde{f}_k} \mu_k
\end{equation}
A valid auxiliary function $v(t,x)$ from \eqref{eq:peak_cont_f} must decrease along all subsystems \cite{parrilo2008approximation, vlassis2014polytopic}.
\begin{equation}
    \Lie_{\tilde{f}_k} v(t, x) \leq 0 \quad \forall (t, x) \in [0, T] \times X, \ k=1\ldots N_d \label{eq:peak_cont_f_poly}
\end{equation}
The moments $\inp{1}{\mu_k}$ are the expected amount of time that trajectories spend following dynamics $\tilde{f}_k$. 


% The switching sequence $w^*(t)$ connecting the optimal $x_0^* \rightarrow x_p^*$ is not immediately available from the truncated moments of all $\mu_k$, and requires postprocessing by the methods \cite{marx2019tractable} to recover.
% Using a time-dependent uncertainty approach in results in one disturbance-occupation measure $\mu$ with $n + N_d$ variables. With polytopic uncertainty, the $N_d$ occupation measures $\mu_k \in \Mp{[0, T] \times X}$ each have $n$ variables and do not involve $d$. The complexity of the moment-SOS hierarchy scales polynomially on the number of states and linearly on the number of measures, so depending on the relaxation degree it may be favorable to solve peak estimation in switched or polytopic form.

\subsubsection{Switching Uncertainty}
\label{sec:switching}
Switching uncertainty is a generalization of polytopic uncertainty.
Let $\{X^k\}_{k=1}^{N_s}$ be a closed cover of $X$ for $k=1\ldots N_s$, which is a set of not necessarily disjoint closed subsets $X^k \subseteq X$ such that $\cup_k X^k = X$. Further, let $\dot{x} = \tilde{f}_k(t,x)$ be dynamics that evolve on each $X_k$. 
The closed cover formalism generalizes partitions of $X$ where $X^k \cap X^{k'} = \varnothing \ \forall k \neq k'$ and arbitrary switching where $X^k = X \ \forall k$. The switched measure program introduces an occupation measure $\mu_k \in \Mp{[0, T] \times X^k}$ for each subsystem $\tilde{f}_k$ subject to \eqref{eq:occ_poly}. The nonnegativity condition \eqref{eq:peak_cont_f_poly} in $v(t,x)$ is restricted to each region $[0, T] \times X_k$
\begin{equation}
    \Lie_{\tilde{f}_k} v(t, x) \leq 0 \quad \forall (t, x) \in [0, T] \times X_k, \ k=1\ldots N_d \label{eq:peak_cont_f_switch}
\end{equation}

\subsubsection{Box Uncertainty}
\label{sec:box}
% All parameters are in a box

Box uncertainty is a case where the region $\Delta$ is a box, and there are no other constraints on $d(t)$. With a proper scaling of dynamics, all $d$ can be constrained to the region $d_k(t) \in [0,1]$. The methods in \cite{korda2014controller, majumdar2014convex} introduce box-measures $\{\sigma_k \in \Mp{[0,T] \times X} \}_{k=1}^{N_d}$ such that for all subsets $A \times B \in [0, T] \times X$.
\begin{equation}
    \int_{A\times B} d_k(t) d \mu = \int_{A \times B} d \sigma
\end{equation}

Each measure $\sigma_k$ is absolutely continuous with respect to the occupation measure $\mu$ $(\sigma(A \times B) \leq \mu(A \times B), \ \forall A \in [0, T], \ \forall B \in X$. It therefore holds that there exists a nonnegative measure $\hat{\sigma}_k \in \Mp{[0, T] \times X}$ such that $\sigma_k + \hat{\sigma}_k = \mu$ for each $k$.

% By the Jordan decomposition there exists nonnegative complement measures $\hat{\sigma}_k \in \Mp{[0,T] \times X}$ such that $\sigma_k + \hat{\sigma}_k = \mu$ for each $k = 1, \ldots, N_d$. 

The Liouville equation \eqref{eq:liou_d} may be expressed in terms of $\sigma$ for all test functions $v (t, x) \in C^1([0, T] \times X)$ as
\begin{equation}
    \inp{v(t, x)} {\mu_p} = \inp{v(0, x)}{\mu_0} + \inp{\Lie_{f_0}  v(t, x)}{\mu} + + \textstyle\sum_{k = 1}^{N_w} \inp{f_k \cdot \nabla_x v (t,x)}{ \sigma_k}
\end{equation}
\begin{equation}
\label{eq:liou_control}
   \mu_p = \delta_0 \otimes\mu_0 + \Lie_{f_0}^\dagger \mu + \textstyle\sum_{k = 1}^{N_w} (f_k \cdot \nabla_x)^\dagger \sigma_k
\end{equation}

% A system with $N_d$ parameters with box uncertainty may be expressed as an instance of polytopic uncertainty with $2^{N_d}$ subsystems. Each subsystem is comprised of dynamics $\dot{x} = f_d(t,x)= f(t, x, d)$ where $d$ is fixed to a vertex of the box $\Delta = [0, 1]^{N_d}$. Since the number of subsystems (and measures) scale exponentially in $N_w$,  posing boxed uncertainty is generally more tractable than solving as a per-vertex switched system.
