On a linear differential game in the Hilbert space $\ell^2$

TL;DR

This paper studies a pursuit-evasion differential game and a zero control problem in the infinite-dimensional Hilbert space rak{l}^2, deriving optimal controls and strategies for the pursuer to guarantee success.

Contribution

It introduces a novel approach to solving pursuit-evasion games and zero control problems specifically within the rak{l}^2 space, providing explicit strategies.

Findings

Optimal control for the zero control problem is derived.

A pursuer strategy guaranteeing pursuit success is constructed.

The framework applies to infinite-dimensional Hilbert spaces.

Abstract

Two-player pursuit-evasion differential game and time optimal zero control problem in $\ell^2$ are considered. Optimal control for the corresponding zero control problem is found. A strategy for the pursuer that guarantees the solution for the pursuit problem is constructed.