Joint Event-Trigger and Control Optimization

Definitions and mathematical statement

An event-triggering policy $\gamma$ specifies when communication occurs, and a control policy $f$ specifies the control actions. The supplied record denotes the control-performance cost by $J_C$, the communication cost by $J_E$, and the average communication budget by $\bar m$. It does not define these costs or the underlying stochastic linear system.

For a fully specified model, let $\mathcal A$ denote its admissible pairs $(f,\gamma)$, including the information available to each policy. Write $J_C(f,\gamma)$ and $J_E(f,\gamma)$ for the resulting random costs, with expectations taken under the probability law induced by the model and the policies. These are placeholders for model data absent from the supplied record. Define, whenever these expectations are meaningful, \[ \mathcal A_{\bar m} =\left\{(f,\gamma)\in\mathcal A: \mathbb E[J_E(f,\gamma)]\leq\bar m\right\}. \] The requested joint optimization is to characterize a pair $(f^*,\gamma^*)\in\mathcal A_{\bar m}$ such that \[ \mathbb E[J_C(f^*,\gamma^*)] \leq \mathbb E[J_C(f,\gamma)] \qquad\text{for every }(f,\gamma)\in\mathcal A_{\bar m}. \] Existence of such a pair is not asserted by the supplied record.

This is only a conditional formulation of the stated target. A precise model-specific problem requires the state dynamics, time domain and horizon, noise and initial-state laws, quadratic cost weights and normalization, communication-cost definition, observation and communication structure, admissibility conditions, and allowed budget values. In particular, the phrase "average communication budget" does not determine how communication is averaged.

Short English statement

Jointly choose when to communicate and how to control a stochastic linear system so as to minimize expected quadratic control cost while keeping expected communication cost at most $\bar m$. The supplied record leaves the model, costs, and policy information unspecified, so it does not determine a complete optimization problem.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001997$, title: "Joint Event-Trigger and Control Optimization"; source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2012; no individual proposer is named.
• Supplied source citation: Control Theory: Mathematical Perspectives on Complex Networked Systems (2012), Oberwolfach Reports, DOI 10.4171/owr/2012/12, pp. 661--732. Supplied source URL and literature-source link labeled "Oberwolfach Report 12 2012": https://doi.org/10.4171/owr/2012/12. The background describes these reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or passage is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports certainty equivalence as an available structural preliminary result, and reports that constrained joint optimization remains difficult and that no general optimal policy characterization was found. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.