A priori bound on the number of orthogonality conditions

Determine an a priori bound, in terms of the system dimension n, on the smallest integer N for which the orthogonality conditions defining the sets C_k are equivalent to checking only the first N terms of the nonlinear recurrence sequence, for arbitrary n.

Background

The minimal null control time is characterized through infinitely many orthogonality conditions involving a sequence defined by a nonlinear recurrence. Although finite-dimensionality guarantees that some finite number N of conditions suffices, the relevant N depends on the particular recurrence sequence.

The paper establishes the bounds N ≤ 3 when n = 3 and N ≤ 6 when n = 4. It does not provide a general dimension-dependent bound for arbitrary n, leaving unresolved how many initial orthogonality conditions must be checked in higher dimensions.

References

However, such a $N$ depends on the sequence and it is a priori unknown, so that, in practice, we do not know when we have to stop checking the orthogonality conditions.

— 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 1.3, immediately before Theorem 2.2