A method to determine the minimal null control time of 1D linear hyperbolic balance laws
Long Hu, Guillaume Olive
TL;DR
The paper develops a general, constructive method to determine the minimal null control time $T_{\mathrm{inf}}$ for 1D first-order linear hyperbolic systems with boundary control. By combining backstepping, kernel equations, and a systematic row-reduction procedure, it reduces complex boundary and internal couplings to tractable forms $(Q^{[n^*]},G^{[n^*]}_{+-})$ and yields an explicit time formula $T_{\mathrm{inf}}=\max_{1\le k\le n^*}\{T_{m+k}+T_{c_k}, T_m\}$, where $c_k$ are read from the canonical form of the transformed $Q$. The work also provides a precise mechanism to compute derivatives of the kernel at the origin, linking them to data from $M$ and $\Lambda$, and shows how to handle zero rows and derivative conditions to preserve controllability times. Through detailed examples and special cases (including no boundary coupling and $p=1$ systems), the approach unifies and extends existing results, offering a practical pathway to exact minimal-time predictions in a broad class of hyperbolic balance laws.
Abstract
In this paper we introduce a method to find the minimal control time for the null controllability of 1D first-order linear hyperbolic systems by one-sided boundary controls when the coefficients are regular enough.