Equivalent one-dimensional first-order linear hyperbolic systems and range of the minimal null control time with respect to the internal coupling matrix

TL;DR

This paper characterizes the range of minimal null control times for one-dimensional first-order linear hyperbolic systems with boundary controls, based on the internal coupling matrix, using backstepping and matrix decomposition techniques.

Contribution

It provides an explicit description of how the minimal null control time varies with the internal coupling matrix and identifies conditions for invariance of this time.

Findings

Explicit characterization of minimal null control time range.

Identification of conditions for invariance of control time.

Application of backstepping and LU-decomposition methods.

Abstract

In this paper, we are interested in the minimal null control time of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls. Our main result is an explicit characterization of the smallest and largest values that this minimal null control time can take with respect to the internal coupling matrix. In particular, we obtain a complete description of the situations where the minimal null control time is invariant with respect to all the possible choices of internal coupling matrices. The proof relies on the notion of equivalent systems, in particular the backstepping method, a canonical $LU$-decomposition for boundary coupling matrices and a compactness-uniqueness method adapted to the null controllability property.