Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions

TL;DR

This paper develops a theory for semiglobal classical solutions to first order quasilinear hyperbolic systems with nonlocal boundary conditions, establishing exact boundary controllability and observability, and demonstrating limitations in control reduction for certain systems.

Contribution

The paper introduces a new theoretical framework for boundary controllability and observability of hyperbolic systems with nonlocal boundary conditions, including constructive methods and limitations analysis.

Findings

Exact boundary controllability and observability are established for the systems.

Control and observation numbers cannot be reduced in the studied examples.

Controllability in networks with loops is generally not achievable.

Abstract

In this paper we establish the theory on semiglobal classical solution to first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions, and based on this, the corresponding exact boundary controllability and observability are obtained by a constructive method. Moreover, with the linearized Saint-Venant system and the 1-D linear wave equation as examples, we show that the number of both boundary controls and boundary observations can not be reduced, and consequently, we conclude that the exact boundary controllability for a hyperbolic system in a network with loop can not be realized generically.