\section{Introduction}
\label{sec:introduction}

Peak estimation under uncertainty aims to bound extreme values of a state function subject to an adversarial noise process. Examples include finding the maximum height of an aircraft subject to wind, the maximum voltage in a transmission line subject to thermal noise, and the maximum speed of a motor subject to impedance within a tolerance.
% Introduce the paper. Pose the peak estimation problem under uncertianty. Detail which kinds of uncertainty will be discussed (Time-independent, Time-varying, Affine).
A system with finite-dimensional state $x \in \R^{N_x}$ evolves under Ordinary Differential Equation (ODE) dynamics defined by a locally Lipschitz vector field $f$ perturbed by uncertainty over the time-range $t \in [0, T]$. The time-independent uncertainty $\theta \in \Theta \subset \R^{N_\theta}$ is fixed (such as the unknown mass of a system component within tolerance), while the time-dependent uncertainty $w(t)$ may change arbitrarily in time within the region $W \subset \R^{N_w}$. Let $x(t \mid x_0, \theta, w(t))$ denote a trajectory in time starting from an initial point $x_0$ subject to uncertainties $(\theta, w(t))$. The uncertain peak estimation problem with variables $(t, x_0, \theta, w(t))$ may be posed as,
    \begin{align}
    P^* = & \max_{t \in [0, T],\, x_0 \in X_0,\,  \theta \in \Theta,\,  w(t)} p(x(t \mid x_0, \theta, w(t))) &\label{eq:peak_traj}\\
    & \dot{x}(t) = f(t, x(t), \theta, w(t)), \quad 
    w(t) \in W & \forall t \in [0, T]. \nonumber
    \end{align}
This paper produces an infinite-dimensional linear program (LP) in occupation measures to upper bound the quantity $P^*$ from \eqref{eq:peak_traj}. Measure-based peak estimation was introduced in \cite{helmes2001computing} and \cite{cho2002linear} for a stochastic setting, and was numerically approximated by a  discretizing set of finite LPs. The work in \cite{fantuzzi2020bounding} forms a sum-of-squares program from an LP dual to the measure LP in \cite{cho2002linear}.
% , solving the infinite LP with a hierarchy of Linear Matrix Inequalities (LMI) \cite{lasserre2009moments}. 
Each of these are variations on the optimal control framework in \cite{lewis1980relaxation, lasserre2008nonlinear}, with an optimal stopping cost rather than an average integral (running) cost.
% with a state rather than a running cost. 

Occupation measure-based bounds for uncertain peak estimation may be developed by adapting methods from optimal control. 
% The reachability set is the locus of states that can be accessed by trajectories of a dynamical system given an initial set $X_0$, and may be outer-approximated by maximizing the volume of the set connected to $X_0$ by state dynamics \cite{korda2014controller, magron2017discrete}.
Time-dependent uncertainty is an instance of an adversarial optimal control which aims to maximize the state function.
Time-independent parameter uncertainty may be incorporated by adding states, and switched systems can be analyzed by splitting the occupation measure \cite{henrion2012measures}.
% This paper uses insights from uncertain reachability set approximation to form an uncertain peak estimation framework.
The true peak cost $P^*$ is upper bounded with an infinite dimensional LP in occupation measures. The infinite LP is then truncated into a sequence of LMIs by the moment-SOS hierarchy \cite{lasserre2009moments}. 


