Determine whether selected multipliers degenerate when the joint qualification condition fails

Determine whether the multipliers selected through the first-mesh-cell construction for the free-time controlled sweeping problem can degenerate when the joint positive linear independence condition (JQ_0) fails.

Background

The joint qualification condition (JQ_0) combines linear independence of the active moving-set constraint gradients with positive linear independence of the active moving-set control constraint gradients at the initial time. When it holds, the endpoint atom estimates exclude certain combinations of initial sweeping and control-constraint multiplier atoms in the abnormal case.

The paper constructs multiplier selections at the first mesh cell that yield enhanced nontriviality, but it does not resolve whether those selected multipliers can nevertheless take a degenerate form when (JQ_0) fails. An example shows that a degenerate multiplier collection satisfying the displayed optimality relations need not arise from the specified first-cell construction; thus the general possibility remains unresolved.