Projected contact dynamics and explicit projection Euler step

Let $d\ge 1$, let $C\subseteq {\mathbb{R}}^d$ be nonempty, closed, and convex, and let $P_C(y)$ be the unique nearest point in $C$ to $y$, whose existence and uniqueness are Every point has a unique nearest point in a nonempty closed Euclidean convex set. For $x\in C$, set $N_C(x)=\{n\in {\mathbb{R}}^d:⟨n,z-x⟩\le 0\text{ for every }z\in C\};$ for $x\mathrm{\notin }C$, set $N_C(x)=\mathrm{\varnothing }$. On $C$ this is the outward normal cone of Scaled KKT residual and convex normal cone. The projection variational inequality of Metric projection onto a closed convex set satisfies the variational inequality says $P_C(y)=p$ exactly when $p\in C$ and $y-p\in N_C(p)$.

For $T>0$, a solution of the projected contact dynamics with forcing $f:[0,T]\times C\to {\mathbb{R}}^d$ and initial state $x_0\in C$ is an absolutely continuous map $x:[0,T]\to C$ satisfying $f(t,x(t))-\dot{x}(t)\in N_C(x(t))\text{for almost every }t\in [0,T]\mathrm{.}$ Equivalently, $\dot{x}(t)+N_C(x(t))\ni f(t,x(t))$. For an integer $N\ge 1$, put $h=T\mathrm{/}N$, $t_k=kh$, and define the exact explicit projection Euler nodes by $x_{k+1}=P_C(x_k+hf(t_k,x_k)),k=0,\dots ,N-1,$ starting from $x_0$. Its affine interpolant joins consecutive nodes linearly on each interval $[t_k,t_{k+1}]$.

For a map $x:[0,T]\to {\mathbb{R}}^d$, absolutely continuous means that each coordinate is absolutely continuous in the scalar sense of Absolute continuity on a compact interval, and $\dot{x}(t)$ is the vector of coordinate derivatives, from The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set, wherever they all exist. The solution condition requires a Lebesgue-null set $N\subseteq [0,T]$ such that for every $t\mathrm{\notin }N$ all coordinate derivatives exist and the displayed inclusion holds. Here almost every $t$ is relative to Lebesgue measure $\lambda$ from Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$, with the convention of Measure-null sets and almost-everywhere statements relative to a measure.