Sensor-Actuator Co-Design and Distributed Controllability Algorithms

Sensor-Actuator Co-Design and Distributed Controllability Algorithms

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A controllability matrix for $\dot x=Ax+Bu$ is $[B,AB,\ldots,A^{n-1}B]$; full row rank is the linear-system controllability criterion. A controllability Gramian measures reachable-state energy. ADMM alternates minimization and dual updates for a split constrained optimization problem.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The budget, cost, system and PDE classes, local-information model, and ADMM formulation are not given. The entire bundle is preserved, without claiming one subquestion covers the others.

Source-record limitation: The sensor/control cost models, distributed task, dynamics, and PDE-ADMM formulation are absent; the bundle cannot be recovered as a formal target.

Statement recorded in the supplied source.

Given a sensing or actuation budget and a control-cost functional, how should sensors or actuators be selected to minimize cost? Is there a local distributed algorithm for the rank or controllability-volume problem? Characterize when adding or removing links from the system dynamics or communication network improves sensing or controllability rather than amplifying disturbances. Prove convergence of ADMM to the optimal solution in the PDE setting. [S2]

Source status.

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Short English statement

Optimize sensing and actuation, understand link changes, find distributed controllability algorithms, and prove the stated PDE-optimization convergence.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006089 (OWR-14298800-001). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mini-Workshop: High-Dimensional Control Problems and Mean-Field Equations with Applications in Machine Learning $2025$, Oberwolfach Reports, DOI 10.4171/owr/2024/56, pp. 3211--3254 https://doi.org/10.4171/owr/2024/56.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Preparation and review.

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Research attempt

Literature, source context and scope

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The source combines four questions: cost-based sensor/actuator selection, distributed controllability tests, changes to dynamics or communication links, and PDE-ADMM convergence. They are not one specified optimization problem. This attempt isolates the link-addition question in a finite-dimensional discrete-time model and tests an appealing monotonicity inference.

Adding a dynamics link can destroy controllability

Consider $x_{t+1}=Ax_t+Bu_t$ with \[ A_0=\begin{pmatrix}1/4&0\\0&1/2\end{pmatrix},\quad A_1=\begin{pmatrix}1/4&1/4\\0&1/2\end{pmatrix},\quad B=\begin{pmatrix}1\\1\end{pmatrix}. \] Only one previously zero dynamics entry is added; both matrices have eigenvalues $1/4,1/2$ and are stable.

Where a monotonicity proof actually works

Keep $A$ fixed and add independent actuator columns, replacing $B$ by $[B,D]$. For any integer horizon $T\ge1$, \[ W_T([B,D])-W_T(B)=\sum_{t=0}^{T-1}A^tDD^T(A^T)^t\succeq0, \quad W_T(B)=\sum_{t=0}^{T-1}A^tBB^T(A^T)^t. \] Indeed each summand has quadratic form $\|D^T(A^T)^tv\|^2\ge0$. If $W_T(B)$ is positive definite, writing the new Gramian as $W_T(B)^{1/2}(I+K)W_T(B)^{1/2}$ with $K\succeq0$ proves that its determinant is at least the old determinant: every eigenvalue of $I+K$ is at least one. If the old Gramian is singular, its determinant is zero and the new one is nonnegative. This conclusion applies to genuinely additional unconstrained inputs; it does not justify changing $A$ or relaxing one constraint while changing disturbances.

Verification and remaining gaps

Exact rational checks reproduce both controllability matrices, determinants and the added-actuator PSD difference. The failed inference is transferring actuator monotonicity to new dynamics links. Distributed communication, co-design budgets and costs, disturbance propagation and PDE-ADMM convergence remain untreated.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2024/56. Relevant passages read: OWR 2024/56, printed pages 3239--3240, Conger and four related open questions.

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Completion Estimate

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Final status and attribution

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