Replace the discrete BV-penalty inactivity assumption with a data-based condition

Establish a condition on the data of the free-time controlled sweeping problem that replaces the assumption that the bounded-variation penalty is inactive at the discrete optimal solutions, thereby removing the need to impose condition (BV-pen-zero) directly on those solutions.

Background

The discrete approximation problems impose bounded-variation constraints on the discrete moving-set control velocities and perturbation controls. The necessary optimality conditions are stated under an additional assumption that the relevant BV penalty is inactive at the discrete optimal solutions. This assumption is technical and concerns the solutions of the approximating problems rather than the original problem data.

If the penalty remains positive, its subgradient contributes to the discrete adjoint relation with a factor proportional to the inverse mesh size. Although the convergence theorem implies that the penalty excess tends to zero, it provides no rate; a rate of order O(h_k) would suffice to control the additional term. The unresolved problem is therefore to replace the solution-dependent inactivity assumption with a verifiable condition formulated directly in terms of the data of the original sweeping control problem.

References

Replacing the condition on $u$ in `e:BV-pen-zero` by a condition on the data of $(P)$ remains an open question. The difficulty is a mismatch of scales.

e:BV-pen-zero:

$\sum\limits_{j=1}^{k-1}\bigl\|V^u_j-V^u_{j-1}\bigr\|\le\mu,\quad\sum\limits_{j=1}^{k-1}\bigl\|a^k_j-a^k_{j-1}\bigr\|<\mu+1\quad\mbox{with}\quad V^u_j=\frac{u^k_{j+1}-u^k_j}{h_k}. $

— Optimal Control of Prox-Regular Sweeping Processes with Variable Time  (2610.13327 - Cao et al., 9 Oct 2026) in Remark on the BV penalty following Theorem 3.2; Conclusions and Future Research, Section 8