Null controllability and finite-time stabilization in minimal time of one-dimensional first-order $2 \times 2$ linear hyperbolic systems

TL;DR

This paper determines the minimal time required for null controllability and finite-time stabilization of 1D first-order 2x2 linear hyperbolic systems, proving optimality using backstepping and Titchmarsh convolution theorem.

Contribution

It establishes the exact minimal time for control and stabilization of these systems and demonstrates that this time cannot be improved.

Findings

Derived the minimal time for null controllability and stabilization.

Proved the optimality of the minimal time using mathematical theorems.

Combined backstepping method with Titchmarsh convolution theorem for proof.

Abstract

The goal of this article is to present the minimal time needed for the null controllability and finite-time stabilization of one-dimensional first-order $2 \times 2$ linear hyperbolic systems. The main technical point is to show that we cannot obtain a better time. The proof combines the backstepping method with the Titchmarsh convolution theorem.