2 Catching-Up Algorithm with Errors

This chapter presents the enhanced catching-up algorithm using approximate projections and establishes its main properties.

2.1 The Sweeping Process

Definition 2.1 Moreau’s sweeping process

Given a Hilbert space \(H\) and a family of closed moving sets \((C(t))_{t\in [0,T]}\), Moreau’s sweeping process is the differential inclusion

\begin{equation} \label{eq:sweeping_process} \begin{cases} \dot{x}(t) \in -N(C(t); x(t)) & \text{a.e. } t \in [0, T], \\ x(0) = x_0 \in C(0). \end{cases} \end{equation}

2.2 Classical Catching-Up Algorithm

The classical catching-up algorithm, developed by J.J. Moreau for convex moving sets, consists of taking a time discretization \(\{ t_k^n\} _{k=0}^n\) of the interval \([0, T]\) and defining a piecewise linear continuous function \(x^n : [0, T] \to H\) with nodes

\[ x_{k+1}^n := \text{proj}_{C(t_{k+1}^n)}(x_k^n) \quad \text{for all } k \in \{ 0, \dots , n-1\} \]

Under suitable assumptions, the sequence \((x^n)\) converges to the unique solution of ??.

2.3 Enhanced Algorithm with Approximate Projections

Definition 2.2 Catching-up algorithm with approximate projections

Given a time discretization \(\{ t_k^n\} _{k=0}^n\) of \([0,T]\) and a sequence \((\varepsilon _k^n)\) of positive numbers, the catching-up algorithm with errors defines a piecewise linear function \(x^n: [0,T] \to H\) with nodes

\[ x_{k+1}^n \in \text{proj}_{C(t_{k+1}^n)}^{\varepsilon _k^n}(x_k^n) \]

for all \(k \in \{ 0, \ldots , n-1\} \), with \(x_0^n = x_0\).

Definition 2.3 Admissible error schedule

A sequence \((\varepsilon _k)\) is admissible if \(\varepsilon _k {\gt} 0\) for every \(k\).

Lemma 2.4 Self-membership under admissible errors

If \(x \in S\) and \((\varepsilon _k)\) is admissible, then for every \(k\):

\[ x \in \text{proj}_{S}^{\varepsilon _k}(x). \]

2.4 Main Properties

Theorem 2.5 Properties of the algorithm

✓

Under suitable assumptions on the moving set \(C\) and the error sequence \((\varepsilon _k^n)\), the sequence \((x^n)\) of functions generated by the catching-up algorithm satisfies certain boundedness and convergence properties.