Smooth hybrid ODE modes, guards, resets, and deterministic event priority

A smooth hybrid ODE system on a finite horizon $[0,T]$, with $T\ge 0$ and positive integer state dimension $d\ge 1$, consists of a finite mode set $M$, an open state-time domain $D_m\subseteq \mathbb{R}\times {\mathbb{R}}^d$ and a $C^1$ vector field $f_m:D_m\to {\mathbb{R}}^d$ for each $m\in M$, a finite set of directed transitions $(m,n)$, and for each transition a $C^1$ scalar guard $g_{mn}:D_m\to \mathbb{R}$, a declared crossing orientation ${\sigma }_{mn}\in \{-1,+1\}$, and a $C^1$ reset map $R_{mn}:U_{mn}\to {\mathbb{R}}^d$ on an open neighborhood $U_{mn}\subseteq D_m$ of the guard zero set, satisfying $(t,R_{mn}(t,x))\in D_n$ for every $(t,x)\in U_{mn}$. Here $C^1$ has the meaning in Continuously differentiable maps, local inverses, and local diffeomorphisms. In mode $m$, a trajectory is a solution of the first-order initial-value problem $y^{\mathrm{\prime }}=f_m(t,y)$ in the sense of First-order systems, initial value problems, and solutions on intervals. It follows that mode only until the selected event time.

An event time for transition $(m,n)$ is a time $\tau$ at which $g_{mn}(\tau ,y(\tau ))=0$ and the guard crosses in its declared direction, meaning ${\sigma }_{mn}\frac{d}{dt}{g}_{mn}(t,y(t)){\mathrm{\mid }}_{t=\tau }>0$. It is transverse with margin $\nu >0$ on a bracket when ${\sigma }_{mn}\frac{d}{dt}{g}_{mn}(t,y(t))\ge \nu$ throughout that bracket. At an event, the state changes from $y({\tau }^{-})$ to $R_{mn}(\tau ,y({\tau }^{-}))$ and the mode changes from $m$ to $n$. A deterministic event-priority rule is a fixed total order on the finite outgoing transition set of each mode; if several enabled guards vanish at exactly the same first time, the least transition in that order is selected.

A hybrid trajectory has a finite separated event list on $[0,T]$ when it has finitely many event times $0<{\tau }_1<\cdots <{\tau }_N<T$ and some $\delta >0$ with ${\tau }_{j+1}-{\tau }_j\ge \delta$ for each adjacent pair. The definition does not assert that a solution exists, that an event is detected, or that the event list is finite. Those are separate hypotheses or conclusions.