Optimal time for the controllability of linear hyperbolic systems in one dimensional space

TL;DR

This paper determines the minimal time needed for controllability of linear hyperbolic systems in one dimension, analyzing the effects of system parameters and providing explicit control strategies.

Contribution

It establishes the optimal controllability time for linear hyperbolic systems with boundary controls, including generic conditions and feedback law constructions.

Findings

Optimal time for null and exact controllability is characterized.

Null-controllability is achieved for any time greater than the optimal time.

Feedback control can attain null-controllability at the optimal time.

Abstract

We are concerned about the controllability of a general linear hyperbolic system of the form $\partial_t w (t, x) = \Sigma(x) \partial_x w (t, x) + \gamma C(x) w(t, x) $ ($\gamma \in \mR$) in one space dimension using boundary controls on one side. More precisely, we establish the optimal time for the null and exact controllability of the hyperbolic system for generic $\gamma$. We also present examples which yield that the generic requirement is necessary. In the case of constant $\Sigma$ and of two positive directions, we prove that the null-controllability is attained for any time greater than the optimal time for all $\gamma \in \mR$ and for all $C$ which is analytic if the slowest negative direction can be alerted by {\it both} positive directions. We also show that the null-controllability is attained at the optimal time by a feedback law when $C \equiv 0$. Our approach is based on the backstepping method paying a special attention on the construction of the kernel and the selection of controls.