Minimal null control time for arbitrary internal and boundary couplings

Determine the minimal null control time of one-dimensional hyperbolic balance-law systems with constant coefficients for arbitrary internal coupling matrices M and boundary coupling matrices Q.

Background

The paper studies the minimal time required for null controllability of one-dimensional hyperbolic systems with constant characteristic speeds, internal coupling matrix M, and boundary coupling matrix Q. Prior results characterize this time in several special cases, including systems without internal coupling, full-row-rank boundary coupling matrices, and systems with two equations.

A general characterization for arbitrary choices of M and Q is identified as an unresolved problem. The present work addresses an important special case, namely systems with exactly one negative characteristic speed, but does not solve the problem for arbitrary numbers of negative and positive speeds and arbitrary couplings.

References

Finding the minimal null control time for arbitrary M and Q is still an open challenging problem.

— Minimal null control time of some 1D linear hyperbolic balance laws with constant coefficients and properties of related kernel equations  (2609.11118 - Hu et al., 10 Sep 2026) in Section 1.2, Literature

The second problem could not be solved though because, even if the conditions for $f$ are explicit, the map'' $M f$ (defined'' by kern equ-def G, with $m=1$) is quite complicated. It was left as an open problem in the same paper.

kern equ:

{ΛK(x,ξ)+K(x,ξ)Λ+K(x,ξ)M=0,ΛK(x,x)−K(x,x)Λ=M,\begin{dcases} \Lambda{K}(x,\xi) +{K}(x,\xi)\Lambda +K(x,\xi)M=0, \\ \Lambda K(x,x)-K(x,x)\Lambda =M, \end{dcases}

def G:

F(x)=−K(x,0)Λ(IdmQ),F(x) = -K(x,0)\Lambda\begin{pmatrix} Id_m \\ Q \end{pmatrix},

— Minimal null control time of some 1D linear hyperbolic balance laws with constant coefficients and properties of related kernel equations  (2609.11118 - Hu et al., 10 Sep 2026) in Section 2, immediately before Theorem 2.1