Efficient representation of the closure \overline{C^T} applicable to all controls

Construct an efficient representation for the closure \overline{C^T} of the connecting operator C^T=W^{T*}W^T that is valid for all controls f\in\mathscr F^T=L_2(\Sigma^T), beyond the restricted identity C^T=\tfrac12 S^{T*}R^{2T}J^{2T}S^T which applies only when S^T f lies in the domain of R^{2T}J^{2T}.

Background

Although CT is bounded on \mathscr FT, its commonly used representation via boundary measurements, CT=\tfrac12 S{T*}R{2T}J{2T}ST, is only immediately applicable to a restricted class of controls. For general f\in\mathscr FT one must interpret CT through closure and continuity arguments.

The authors note that, despite longstanding efforts, no efficient closed‑form representation of \overline{CT} that can be directly applied to arbitrary controls is known, hindering certain analytical and numerical developments.

References

However, no efficient representation of $\overline{CT}$ that allows its application to all $f\in\mathscr FT$ is known. This question has remained open for over 30 years.

— On a stability of time-optimal version of the Boundary Control method  (2604.02957 - Belishev, 3 Apr 2026) in Section 2.2 (System and operators), paragraph following equation (C^T via R^{2T})