Stochastic Barrier Certificates

• Stochastic barrier certificates are analytical functions that provide formal, probabilistic safety guarantees for stochastic dynamical systems by bounding the probability of reaching unsafe states.
• They leverage drift conditions and SOS-based optimization to extend deterministic safety methods to systems with probabilistic state transitions and complex dynamics.
• Applications include safety verification, controller synthesis, and enforcing temporal logic specifications in continuous, discrete, switched, and hybrid stochastic systems.

Stochastic barrier certificates are analytical constructs used to provide formal, rigorous bounds on the probability that a stochastic system satisfies finite-horizon safety or temporal logic specifications. These certificates generalize deterministic barrier and Lyapunov functions to systems with probabilistic state transitions, and serve as an indispensable component in safety verification, controller synthesis, and formal specification enforcement for stochastic dynamical and control systems.

1. Foundational Definitions and System Classes

A stochastic barrier certificate is typically a twice-differentiable or polynomial function $B: \mathbb{R}^n \to [0,\infty)$ (or extended to vector or piecewise forms), associated with a set of system regions:

• $X_0$ (“safe”/initial set)
• $X_1$ (“unsafe”/target set)
• $X$ (state space)

For continuous-time switched stochastic systems, the evolution is governed by an Itô SDE in mode $m$:

$d\xi = f_m(\xi) \,dt + g_m(\xi) \,dW_t$

where $f_m$ is drift, $g_m$ is diffusion, and $W_t$ is Brownian motion. The system may switch modes according to a piecewise-constant, càdlàg signal with finitely many jumps on bounded intervals (Anand et al., 2021).

For discrete-time systems, transitions take the form

$x_{k+1} = f(x_k, u_k, w_k)$

with $u_k$ as control and $w_k$ an i.i.d. noise (Jagtap et al., 2019, Mazouz et al., 23 Jul 2025). Generalizations include switched systems, hybrid systems with jumps, and data-driven setups using unknown functions and sample-based learning (Mazouz et al., 2024).

2. Barrier Certificate Conditions and Safety Bounds

The certificate $B$ must meet a sequence of safety and drift inequalities:

• Boundary Conditions/Region Constraints:
  • $B(x) \leq \gamma$ for all $x \in X_0$ (initial region)
  • $B(x) \geq 1$ for all $x \in X_1$ (unsafe region)
• Drift or Supermartingale Condition:
  • In continuous time: For each mode $m$,

  $$\mathcal{D}B(x, m) = \nabla B(x) \cdot f_m(x) + \tfrac{1}{2} \mathrm{Tr}\left(g_m(x)^\top \nabla^2 B(x) g_m(x)\right) \leq c$$

  where $c \geq 0$ (Anand et al., 2021, Santoyo et al., 2019, Santoyo et al., 2019). - In discrete time: For controls $u$ and all $x$,

  $\mathbb{E}[B(f(x, u, w)) | x, u] \leq B(x) + c$

  If $c = 0$, this enforces a supermartingale; for $c > 0$, a relaxed $c$-martingale (Jagtap et al., 2019, Mazouz et al., 23 Jul 2025, Chen et al., 23 Jul 2025).

Finite-Time Safety Probability Bound: If the above conditions are met, for any horizon $T>0$,

$$\mathbb{P}_{x_0}\{\exists t \in [0, T]: \xi^\mu(t) \in X_1\} \leq \gamma + c T$$

with analogous expressions for discrete time and multi-mode systems (including reachability decompositions for temporal logic) (Anand et al., 2021, Jagtap et al., 2019, Jagtap et al., 2018).

3. Temporal Logic and Automata-Based Decomposition

Barrier certificate results enable compositional temporal logic verification, most commonly for fragments of safe-LTL over finite traces. The workflow involves:

• Translating the negation of the temporal logic formula $\phi$ into a DFA $\mathcal{A}_{\neg\phi}$.
• Decomposing all accepting runs into sequential reachability subtasks (triples of automaton states and associated regions).
• For each subtask, synthesizing a barrier certificate and bounding its reach probability $\gamma_{\nu} + c_{\nu} T$.
• The overall probability of violation is bounded by a sum-product over runs, yielding a final lower bound for property satisfaction as $1 - \mathbb{P}\{\neg \phi\}$ (Anand et al., 2021, Jagtap et al., 2018, Jagtap et al., 2019).

4. Computational Synthesis and Data-Driven Methods

Certificate construction is tractable only via numerical optimization, leveraging templates (polynomial, piecewise-constant, neural-network) and convex relaxations:

• Sum-of-Squares (SOS): Converts the barrier and drift inequalities into SOS constraints over semialgebraic sets, optimizable via SDP toolchains (Santoyo et al., 2019, Santoyo et al., 2019, Oumer et al., 21 Apr 2025). For controller synthesis, control-affine systems allow joint SOS programs in $B$ and the feedback law.
• Counterexample-Guided Inductive Synthesis (CEGIS): Parameterizes $B$ and iteratively refines through synthesis and verification over sampled sets, using SMT solvers such as Z3, dReal (Anand et al., 2021, Jagtap et al., 2019).
• Piecewise-Constant and Piecewise-Affine Barriers: For large-scale or data-driven systems, grid partitioning is used, converting the design to linear or minimax programs with feasible occupation measures (Mazouz et al., 2024, Mazouz et al., 23 Jul 2025).
• Data-Driven Convex Programs: For unknown dynamics, robust convex programs are approximated by scenario convex programs, with sample complexity and confidence guarantees determined by scenario theory (Salamati et al., 2021, Salamati et al., 2021). Gaussian process learning facilitates model uncertainty quantification (Mazouz et al., 2024).
• Neural Barrier Functions: Barrier $B$ is parameterized as a neural network, with robust linear-program-based certification via bound propagation and branch-and-bound strategies (Mathiesen et al., 2022).

5. Compositional and Scalable Construction

For interconnected or switched networks, compositional approaches use small-gain or dissipativity-type conditions to compose local barrier certificates:

• Subsystem Control Sub-Barrier Certificates (CSBCs): Each subsystem admits a CSBC with local safety and drift constraints (Nejati et al., 2020, Anand et al., 2021, Anand et al., 2021).
• Global Aggregation: When max-type small-gain conditions are satisfied, the maximum of the local certificates yields a valid global barrier certificate with explicit probabilistic guarantees (Nejati et al., 2020, Anand et al., 2021).
• Switching and Hybrid Systems: For stochastic hybrid systems with SDE flows and Poisson jumps, augmented control barrier certificates (ACBC) combine flow and jump certificates, yielding finite-horizon safety bounds (Lavaei et al., 2022).

6. Extensions, Limitations, and Recent Refinements

• Interpolation-Inspired and $k$-Induction Certificates: To reduce conservatism, families of barrier functions are allowed, using interpolation and $k$-induction to relax per-step requirements (Oumer et al., 21 Apr 2025).
• Refined Dynamic Programming Conditions: Dynamic programming perspectives show that classical $c$-martingale-based barrier conditions may be overly conservative on unsafe sets. Relaxed conditions allow tighter finite-horizon probability bounds through stage-wise inequalities and tailored terminal constraints, as formalized for both safety and reach–avoid specifications (Chen et al., 23 Jul 2025, Xue et al., 23 Sep 2025).
• SOS Formulation on Unbounded Domains: Recent formulations remove requirements for bounded auxiliary functions, facilitating SOS programming on unbounded state spaces (Xue et al., 23 Sep 2025).

7. Numerical Case Studies and Applications

Stochastic barrier certificates have been instantiated for diverse systems:

Application Domain	Methodology	Verified Probability Bound
Room temperature network (1000 rooms)	Compositional CBC + SOS	$\geq 0.87$ over $T_d=10$
Vehicle lane-keeping	SOS + CEGIS	$\geq 0.8688$ over $N=400$
Nonlinear hybrid system with jumps	ACBC + SOS	$\geq 0.9443$ over $T=100$
Data-driven unknown systems	Scenario Convex Program	$\geq 0.9$ (99% confidence)
Permissible control set (learned GP)	Piecewise LP/GP bound	$\geq 0.991$

Empirical simulation results consistently validate the analytical bounds, confirming that conservative barrier-based guarantees are often matched or exceeded in practice (Anand et al., 2021, Mazouz et al., 2024, Nejati et al., 2020).

• Verification of Switched Stochastic Systems via Barrier Certificates (Anand et al., 2021)
• A Barrier Function Approach to Finite-Time Stochastic System Verification and Control (Santoyo et al., 2019)
• Data-Driven Permissible Safe Control with Barrier Certificates (Mazouz et al., 2024)
• Formal Synthesis of Stochastic Systems via Control Barrier Certificates (Jagtap et al., 2019)
• Verification and Control for Finite-Time Safety of Stochastic Systems via Barrier Functions (Santoyo et al., 2019)
• Piecewise Control Barrier Functions for Stochastic Systems (Mazouz et al., 23 Jul 2025)
• Safety Certification for Stochastic Systems via Neural Barrier Functions (Mathiesen et al., 2022)
• Safety Barrier Certificates for Stochastic Hybrid Systems (Lavaei et al., 2022)
• Data-driven verification and synthesis of stochastic systems via barrier certificates (Salamati et al., 2021)
• Data-driven Safety Verification of Stochastic Systems via Barrier Certificates (Salamati et al., 2021)
• $k$-Inductive and Interpolation-Inspired Barrier Certificates for Stochastic Dynamical Systems (Oumer et al., 21 Apr 2025)
• On the Construction of Barrier Certificate: A Dynamic Programming Perspective (Chen et al., 23 Jul 2025)
• Refined Barrier Conditions for Finite-Time Safety and Reach-Avoid Guarantees in Stochastic Systems (Xue et al., 23 Sep 2025)
• Safety Barrier Certificates for Stochastic Control Systems with Wireless Communication Networks (Akbarzadeh et al., 2023)
• Compositional Synthesis of Control Barrier Certificates for Networks of Stochastic Systems against $ω$-Regular Specifications (Anand et al., 2021)
• Temporal Logic Verification of Stochastic Systems Using Barrier Certificates (Jagtap et al., 2018)
• Compositional Construction of Control Barrier Certificates for Large-Scale Stochastic Switched Systems (Nejati et al., 2020)
• From Small-Gain Theory to Compositional Construction of Barrier Certificates for Large-Scale Stochastic Systems (Anand et al., 2021)