Optimal Time for the Controllability of Linear Hyperbolic Systems in One-Dimensional Space

DOI10.1137/18M1185600zbMATH Open1418.35259arXiv1805.01144MaRDI QIDQ4631455

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Publication date: 29 March 2019

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Abstract: We are concerned about the controllability of a general linear hyperbolic system of the form $partia{l}_{t}w(t,x)=Sigma(x)partia{l}_{x}w(t,x)+gammaC(x)w(t,x)$ ($gammainmR$) 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 $gammainmR$ 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 $Cequiv0$. Our approach is based on the backstepping method paying a special attention on the construction of the kernel and the selection of controls.

Full work available at URL: https://arxiv.org/abs/1805.01144

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