On a stability of time-optimal version of the Boundary Control method

Abstract: Let $Ω$ be a Riemannian manifold with boundary. The time-optimal version of the BC-method determines the parameters in the $T$-neigh-bor-hood $Ω^T$ of $\partialΩ$ from the boundary observations (response operator) $R^{2T}$ on the time segment $[0,2T]$. It visualizes the invisible waves supported in $Ω^T$, by reconstructing the operator $W^T$ that creates these waves. The visualization is based on the triangular factorization of the operator $C^T:=W^{T\,*}W^T$ in the form $C^T:=F^{T\,*}F^T$ with a factor $F^T=U^{T}W^T$, where $U^T$ is a unitary operator. The factorization $C^T\mapsto F^T$ has certain continuity properties, due to which the time-optimal reconstruction $R^{2T}\mapsto C^T\mapsto F^T\mapsto W^T$ turns out to be continuous (stable) in the sense of relevant operator topologies (convergences). As an example, determination of the potential $q$ in the wave equation $u_{tt}-Δu+qu=0$ from $R^{2T}$ is considered. We show that $R^{2T}_j\to R^{2T}$ implies $q_j\to q$ in $H^{-2}(Ω^T)$. However, the question of quantitative estimates of stability (the rate of convergence) remains open.

• The paper establishes that convergence of boundary response data ensures qualitative stability for reconstructing internal potentials in wave equations.
• It employs operator-theoretic techniques and triangular factorization to analyze convergence properties of essential control operators.
• The findings validate the method through rigorous PDE and functional analysis, paving the way for future quantitative stability estimates.

Stability of the Time-Optimal Boundary Control Method: Formal Summary

Overview of the Boundary Control Method and Time-Optimality

The Boundary Control Method (BCm) is a framework for solving inverse problems related to PDEs, particularly the wave equation on a compact Riemannian manifold with boundary. A distinguishing feature of BCm is its time-optimality: for a $T$-neighborhood near the boundary, boundary observations on $[0,2T]$ are necessary and sufficient for determining the internal manifold and parameters, specifically the potential $q$ in the wave equation $u_{tt} - \Delta u + qu = 0$. Observations on smaller intervals yield insufficient information, while longer intervals provide redundant data. This time-optimal reconstruction (TOR) property is a direct reflection of finite wave propagation speed, which fundamentally governs the causal structure of hyperbolic inverse problems.

Previous stability analyses and quantitative estimates have largely addressed BCm variants with observation intervals $[0,2T']$ for $T'>T$. For TOR, theoretical and numerical investigations have motivated the search for qualitative stability, although quantitative rates remain elusive.

Operator-Theoretic Foundations and Triangular Factorization

A central technical aspect of the paper is the operator-theoretic treatment of the reconstruction procedure. The method relies on bounded, positive operators in suitable Hilbert spaces, notably the control operator $W^T$ and the connecting operator $C^T = W^{T*} W^T$. Convergence notions—uniform, strong, and weak—are leveraged across sequences of operators to track stability of the procedure.

The triangular factorization of positive operators plays a pivotal role. For a nest $\mathfrak{f}$ of subspaces, a canonical triangular factorization $C = F^*F$ with a triangular factor $[0,2T]$0 is defined. The factor $[0,2T]$1 is constructively derived using the diagonal of $[0,2T]$2 (obtained via operator integrals), which intertwines projections onto the subspaces of the nest. Crucially, a stability result asserts that if $[0,2T]$3 and $[0,2T]$4 converge regularly on the nest, then the triangular factors $[0,2T]$5 converge weakly and regularly to $[0,2T]$6, and hence the control operators themselves inherit this stability property.

Dynamical System with Boundary Control: Structure and Operators

The underlying PDE system is posed on a compact Riemannian manifold $[0,2T]$7 with boundary $[0,2T]$8, with boundary control and wave propagation governed by the wave equation. The spaces of boundary controls and internal states are $[0,2T]$9 and $q$0, respectively. The control operator $q$1 maps boundary controls to resulting internal wave states at time $q$2, and is injective for $q$3 (the filling time). The connecting operator $q$4 ties the boundary and internal metrics.

Importantly, $q$5 can be expressed via the boundary response operator $q$6, using extension and integration operators. Although $q$7 may be unbounded as an operator, the practical numerical procedure utilizes its Gram matrix for controllable admissible boundary controls. When potentials $q$8 are uniformly bounded in $q$9, convergence $u_{tt} - \Delta u + qu = 0$0 implies strong convergence $u_{tt} - \Delta u + qu = 0$1.

Time-Optimal Reconstruction (TOR): Procedure and Visualization

The TOR procedure proceeds sequentially:

1. Recover the connecting operator $u_{tt} - \Delta u + qu = 0$2 from boundary observations $u_{tt} - \Delta u + qu = 0$3.
2. Obtain $u_{tt} - \Delta u + qu = 0$4.
3. Triangularly factorize $u_{tt} - \Delta u + qu = 0$5 to obtain $u_{tt} - \Delta u + qu = 0$6 via its diagonal.
4. Visualize internal waves with $u_{tt} - \Delta u + qu = 0$7, where $u_{tt} - \Delta u + qu = 0$8 is a time-reversal operator.
5. If the manifold is unknown, reconstruct metric and potential on $u_{tt} - \Delta u + qu = 0$9 (the screen) from the graph of an operator conjugated by the image operator. If known, recover $[0,2T']$0 directly.

For $[0,2T']$1 (regular semi-geodesic coordinates), technical simplifications permit unitarity of certain operators, strengthening the results.

Stability Analysis of TOR

The paper formally analyzes stability by linking the operator convergence properties to convergence of reconstructed quantities. Regular (stable) convergence of $[0,2T']$2 on the relevant nest ensures weak regular convergence of visualization and control operators. If this holds, it further implies convergence of potentials $[0,2T']$3 in $[0,2T']$4 via duality arguments and PDE regularity properties.

The main result is qualitative stability: $[0,2T']$5 leads to $[0,2T']$6 in $[0,2T']$7, assuming regularity and uniform boundedness. However, explicit quantitative estimates (rates of convergence) for TOR remain unestablished, although recent work has begun to address these challenges [Filippas, Oksanen, Journal of Differential Equations, 453, 2026].

Practical and Theoretical Implications

The analysis provides a rigorous foundation for the qualitative stability of the time-optimal BCm, affirming the reliability of the TOR procedure in practical inverse problem settings. The characterization of regularity and diagonal properties in operator sequences ensures robustness of the method in numerically realistic scenarios and guides future algorithmic development.

On the theoretical side, the results delineate the boundaries of current knowledge: uniqueness, constructive procedures, and qualitative stability are established, but quantitative estimates and complete inverse data characterizations remain open. The framework is extendable to general manifolds, as well as PDEs with variable coefficients, but requires further refinement for broader applicability and sharper stability results.

Conclusion

This work establishes the qualitative stability of the time-optimal version of the Boundary Control method for inverse problems associated with the wave equation on Riemannian manifolds. Through a rigorous operator-theoretic framework, the paper demonstrates that convergence of boundary response data implies convergence of reconstructed internal parameters in a weak topology. The results lay the groundwork for future investigations into quantitative stability, sharper estimates, and broader classes of PDEs and inverse problems. Further theoretical exploration and numerical testing are likely to deepen understanding and extend the applicability of the time-optimal BCm.