Analytic Safety Constraints in Control Systems

• Analytic safety constraints are explicit mathematical conditions, derived from system physics or formal logic, that define invariant safe regions in state or action spaces.
• They are enforced using techniques like control barrier functions, quadratic programming safety filters, and formal runtime monitors in domains such as autonomous driving and robotics.
• Recent approaches integrate learning and synthesis methods to adapt these constraints, balancing rigorous safety guarantees with performance in complex, real-world environments.

Analytic safety constraints are mathematically explicit conditions, often grounded in system physics or formal logics, that guarantee safety properties at all times in the operation of autonomous and cyber-physical systems. Such constraints play a central role in fields including reinforcement learning (RL)-based autonomous driving, model-based and model-free control, data-driven optimization, and interpretable AI system design. Unlike empirical or purely statistical constraints, analytic constraints are synthesized via first-principles derivation, kinematic or dynamic models, or machine-checked logic specifications; they are enforced “hard,” forming invariant sets or control-admissible regions that provably exclude catastrophic or unsafe behavior.

1. Mathematical Foundations of Analytic Safety Constraints

Analytic safety constraints typically define admissible regions in state or state-action space (for control tasks), or in trajectory/behavior space (in formal methods). In control, they are often employed as barrier functions, invariance conditions, or hard constraints:

• Kinematic and dynamic invariance: For example, the longitudinal safe-gap for autonomous driving is formalized by a kinematic inequality guaranteeing that the ego vehicle can always maintain a sufficient stopping distance under worst-case leader braking. Explicitly,

$$g(t) \geq \frac{1}{2}[v_E(t) + v_E(t+1)]\,r + \frac{v_E(t+1)^2}{2d_E} - \frac{v_L(t)^2}{2d_L} + \epsilon$$

where all terms are analytic, kinematically-derived quantities (Shi et al., 2024).

• Barrier functions and quadratic programs (QPs): Control barrier functions (CBFs) encode forward invariance of a safe set $\mathcal{C}=\{x:h(x)\ge 0\}$ as constraints on permissible inputs:

$$L_f h(x) + L_g h(x)\,u + \alpha(h(x)) \geq 0,\ \forall i$$

with admissible control $u\in K_h(x) = \bigcap_{i=1}^N K_{h_i}(x)$ where $K_{h_i}(x)$ is the safe set for each barrier (Reis et al., 20 Mar 2025, Fisher et al., 2024, Tscholl et al., 17 Sep 2025).

• Formal logic invariants: In temporal logic (LTL/STL), analytic constraints are encoded as formulas $\varphi$ over atomic predicates (e.g., positions, states, actions), with satisfaction defined over traces and enforced via runtime automaton monitoring (Yang et al., 2023, Yifru et al., 2024).
• Coalgebraic specification: Safety constraints are equivalently subcoalgebras (invariants) of detector coalgebras that recognize unsafe prefixes, yielding a unifying algebraic semantics for analytic behavioral constraints (Zholtkevych et al., 2020).

The analytic nature of these constraints arises from their explicit logical or algebraic construction: they are typically derived from system models, kinetic theory, invariance principles, or interpreted logic programs, and can be verified or enforced via exact computation.

2. Model Structures and Enactment Mechanisms

The enforcement of analytic safety constraints depends on the underlying system and task setting:

• Statewise-constrained MDPs / RL with hard constraints: Reinforcement learning agents can be trained with action-space projection, where only actions satisfying analytic cost or invariance constraints are allowed. For instance, in SCMDP/DDPG, the policy output is projected onto the safe set

$$A_{\text{safe}}(s) = \{a \mid C_i(s,a,s') \leq 0\ \forall i\}$$

with hard analytic bounds from physics (e.g., maximal safe speed or headway) (Shi et al., 2024).

• Real-time quadratic programming (QP) safety filters: Safety critical controllers for nonlinear or linear systems commonly use a feedback QP that minimizes tracking error subject to one or more analytic input constraints derived from CBFs:

$$u^*(x) = \arg\min_u \|u-u_\textrm{nom}(x)\|^2\ \textrm{s.t.}\ \text{CBF constraints}$$

with analytic feasibility and stability characterized for arbitrary numbers of overlapping constraints (Reis et al., 20 Mar 2025, Fisher et al., 2024).

• Constraint logic automata and formal runtime monitors: For LTL/STL constraints, action sequences are checked in real time against a compiled automaton representing the conjunction of user-supplied or learned analytic formulas,

$\varphi_\text{total} = \bigwedge_{i} \varphi_i$

and any unsafe (dead-end) transitions are pruned or trigger replanning (Yang et al., 2023, Yifru et al., 2024).

• Geometric/hyperplane polytopes in representation spaces: In LLM safety, the safe set in internal representation space is a convex polytope

$$\tilde S = \{x\in\mathbb{R}^d: W^\top x \le \tilde\xi\}$$

providing analytic detection and steerable correction of potentially unsafe generations (Chen et al., 30 May 2025).

• Supervisor "safety filter" integration: In model-free or direct data-driven control, analytic surrogates such as state-action control barrier functions (SACBFs) are constructed and learned to act as QP constraints at policy evaluation time, guaranteeing safety even without explicit physical models (He et al., 21 May 2025).

3. Examples of Analytic Safety Constraints Across Domains

Analytic safety framework instantiations are found throughout safety-critical control and learning:

Domain	Constraint Type	Example Equation or Logical Formula
Autonomous driving RL	Kinematic headway, safe gap/speed	Eq (1)-(4) in (Shi et al., 2024)
Robot control	CBF in QP	$L_f h(x) + L_g h(x)\,u + \alpha(h(x)) \geq 0$
Multi-agent/coalgebraic	Prefix-free trace invariants	$L_P = \{\sigma : \text{no prefix in } P\}$
Formal verification	LTL over atomic predicates	$G\,\neg(agent\_at(B)) \lor agent\_at(A)$
LLM representation space	Polytope in hidden feature space	$\{x: W^\top x \le \tilde\xi\}$
Data-driven control	SACBF constraint on learned state/action value	$Q^B_\omega(x,u)\le 0$

These constraints are typically enforced per-step (in feedback) or on each candidate action (via projection or filtering), ensuring that the closed-loop trajectory or generated behavior provably remains inside the analytically-characterized safe region (Shi et al., 2024, Reis et al., 20 Mar 2025, Fisher et al., 2024, Chen et al., 30 May 2025).

4. Learning, Synthesis, and Adaptation of Analytic Constraints

While early analytic safety constraints were typically hand-engineered, recent frameworks offer methods for direct synthesis, adaptation, and learning:

• Demonstration-based learning: One-class decision trees are fit on feature vectors from expert demonstrations to carve out safely occupied regions, which are then converted into DNF logic for RL constraint enforcement (Baert et al., 2023).
• Simultaneous learning of constraints and policies: In bilevel frameworks, parametric logical formulas (e.g., pSTL) are synthesized by analyzing labeled safe/unsafe trajectories, alternating with policy optimization under the current candidate constraint (Yifru et al., 2024).
• Direct data-driven safety certificates: Methods such as SACBF (He et al., 21 May 2025) learn forward-invariant safety certificates directly from data, via regression, robust optimization, or value iteration, without requiring explicit plant models.
• Safe Bayesian optimization: Analytic characterization of the safe set via confidence tubes (Lipschitz bounds (Fiedler et al., 23 Jan 2025) or Bayesian/posterior high-probability upper bounds (Luebsen et al., 11 Mar 2025)) provides provable per-evaluation safety constraints in black-box optimization problems.
• Coalgebraic and logic-based specification: Safety languages are specified as prefix-free sets or as automata, with tools (e.g., LTL-to-automaton compilers, coalgebraic proof assistants) available for construction and verification (Zholtkevych et al., 2020, Yang et al., 2023).

5. Formal Guarantees and Proof Techniques

The hallmark of analytic safety constraints is their amenability to rigorous, mathematical proof of invariance properties:

• Closed-form induction: For kinematic headway, induction over the discrete system (e.g., $v_E(t)\le s(t)$ for all $t$) ensures that safety constraints are maintained stepwise under worst-case leader action (Shi et al., 2024).
• Lyapunov and CBF invariance: Barrier certificates and CBFs provide analytic conditions for forward invariance, usually by showing that the barrier function never decreases past a violation threshold under the constrained input (Reis et al., 20 Mar 2025, Fisher et al., 2024).
• Automaton soundness: Running a deterministic automaton compiled from LTL/STL constraints with any unsafe trace reaching a sink state ensures that unsafe executions are strictly excluded (Yang et al., 2023).
• Sample-wise constraint satisfaction: For data-driven and Bayesian optimization settings, confidence sets and robust analytic bounds offer probabilistic but still analytic guarantees—e.g., no unsafe parameter or action is ever evaluated with prescribed high probability (Fiedler et al., 23 Jan 2025, Luebsen et al., 11 Mar 2025).
• Error-to-state robustness: For function-approximation settings, analytic margins (e.g., slack $\kappa(\varepsilon)$ as a function of the SACBF regression error) quantify the degree of tightening required to recover safety guarantees in learned constraints (He et al., 21 May 2025).

These proofs rely on the explicit construction and algebraic/logic reasoning enabled by the analytic form of the constraints, in contrast to empirical or purely statistical risk controls.

6. Limitations, Trade-offs, and Practical Considerations

While analytic safety constraints provide strong invariance properties, their practical realization involves inherent trade-offs:

• Model accuracy and conservatism: Analytic models may be conservative if derived under worst-case assumptions (e.g., uniform deceleration bounds, input saturation), trading off operational efficiency for provable safety (Shi et al., 2024, Fisher et al., 2024).
• Expressiveness and complexity: High expressiveness (e.g., many intersecting hyperplanes, high-dimensional polytopes, intricate logic formulas) can increase computational or representational burden, especially for online enforcement (Reis et al., 20 Mar 2025, Chen et al., 30 May 2025).
• Scalability and sample efficiency: Data-driven analytic constraint learning minimizes unnecessary conservatism when the critical unsafe set can be empirically carved out, but this approach may struggle in high-dimensional state-action spaces or under significant noise (Massiani et al., 2021, He et al., 21 May 2025).
• Integration with performance objectives: Analytic constraints are often coupled with soft (reward-based) comfort, efficiency, or task performance metrics—multiobjective QPs and Lagrangian RL schemes are common (Shi et al., 2024, Reis et al., 20 Mar 2025, Baert et al., 2023).
• Specification and verification: While formal methods provide machine-checkable correctness, they are limited by the expressiveness of the analytic specification formalism (e.g., pure safety vs. liveness) (Zholtkevych et al., 2020).

Overall, the adoption of analytic safety constraints is motivated by the need for certification, reliability, and verifiability in the operation of learning and control systems deployed in safety-critical or high-assurance domains. They have become a central unifying concept across modern reinforcement learning, optimization, cyber-physical systems control, and AI safety verification.