Formal Languages for Hybrid-System Control

Definitions and mathematical statement

For comparison, let $\Sigma$ be a finite alphabet and let $\Sigma^*$ denote the set of finite words over $\Sigma$, including the empty word $\varepsilon$. A deterministic finite automaton consists of a finite state set $Q$, an initial state $q_0\in Q$, a transition map $\delta:Q\times\Sigma\to Q$, and a set $F\subseteq Q$ of accepting states. Define the extension to words by \[ \delta^*(q,\varepsilon)=q,\qquad \delta^*(q,wa)=\delta(\delta^*(q,w),a). \] Its accepted language is \[ L=\{w\in\Sigma^*: \delta^*(q_0,w)\in F\}. \] This supplies the accepting-state analogy invoked by the source; it does not prescribe the hybrid-system model.

Requested specification. Specify suitable formal languages for controlling hybrid systems whose inputs combine a continuous control function $u$, defined on a finite interval, with a symbol string $v$. Develop the corresponding notion of an accepted hybrid input, analogous to acceptance by a finite automaton through its accepting states, and explain how the proposed language describes control of the hybrid system.

Unresolved qualifications. The supplied record does not specify a class of hybrid systems, the control-value space, the endpoints or boundary conventions of the finite interval, the alphabet or permitted lengths of $v$, or how the symbols are synchronized with the continuous input. It also leaves unspecified the dynamics, execution semantics, acceptance criterion, control objective, and conditions that make a language suitable. Consequently, this is a framework-design problem: the supplied evidence does not determine a single quantified existence statement or a uniquely defined class of languages. The finite-word convention in the comparison above is not an additional assumption about $v$.

Short English statement

The problem asks for formal languages that describe acceptable inputs for controlling hybrid systems, combining a continuous control function on a finite interval with a symbol string. Acceptance should have an analogue of the accepting-state interpretation for finite automata. The system model, interaction between the two input components, and criteria for suitability remain unspecified.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30002810$, source problem code catalogue entry, title Formal Languages for Hybrid-System Control; collection Oberwolfach Reports Open Problems. Local registry locations designated by the project are data/unsolvedmath/problems.json and the supplementary catalog/problems.json. This draft uses the supplied complete record; these files were not independently inspected.
• Supplied source citation: Control Theory: A Mathematical Perspective on Cyber-Physical Systems (2015), Oberwolfach Reports, DOI 10.4171/owr/2015/12, pp. 597--659. https://doi.org/10.4171/owr/2015/12. The record attributes the problem to workshop open-problem discussions, gives proposed year 2015, and names no individual proposer or precise problem page.
• Supplied literature source: A computable and compositional semantics for hybrid systems. https://doi.org/10.1016/j.ic.2024.105189. No authors or further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that many hybrid-system languages and compositional semantics exist, while no generally accepted framework fully resolves this specification problem. This is supplied dated curation, not an independently verified literature assessment. The source explicitly requests independent verification of time-sensitive status and cited claims.