Rebecka Aulin: Controllability of the Lasso graph

Abstract: In this article we study the Lasso graph with a boundary control at its leaf, and with scaling invariant or Robin-type vertex conditions at its loop vertex. We solve the wave equation for small times and establish the well-definedness of the control operator. For scaling invariant conditions, we show that the system attains controllability at a minimal time precisely when the scattering matrix fulfills S₂₁ ≠ ± S₃₁. If this condition is not fulfilled then the system does not attain controllability, no matter the control time. A similar condition is derived for the Lasso graph with a magnetic potential and standard conditions at the loop vertex. Notably, none of the scaling invariant vertex conditions at the loop vertex fulfill the non-symmetry condition identified by P. Kurasov as sufficient for solvability of the inverse spectral problem.