5 Fixed Set Case
This chapter addresses the convergence of the catching-up algorithm for a fixed boundedly compact closed set.
5.1 The Fixed Set Problem
In this chapter, we consider the simplified case where \(C(t) \equiv C\) for all \(t \in [0,T]\), i.e., the set does not move in time. The sweeping process reduces to:
\begin{equation} \label{eq:fixed_sweeping_process} \begin{cases} \dot{x}(t) \in -N(C; x(t)) & \text{a.e. } t \in [0, T] \\ x(0) = x_0 \in C \end{cases} \end{equation}
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5.2 Simplified Algorithm
For a fixed set \(C\), the enhanced catching-up algorithm simplifies to:
\[ x_{k+1}^n \in \text{proj}_C^{\varepsilon _k^n}(x_k^n) \quad \text{for all } k \in \{ 0, \dots , n-1\} \]
Let \(C \subset H\) be a boundedly compact closed set. Under appropriate assumptions on the error sequence, the catching-up algorithm with approximate projections converges to a solution of the sweeping process with fixed set \(C\).
5.3 Remarks
The fixed set case provides simpler conditions for convergence and serves as a foundation for understanding the more general moving set case.