Stochastic Control Barrier Certificates

• Stochastic control barrier certificates are functions that convert safety requirements in stochastic systems into conditional expectation inequalities, ensuring probabilistic safety.
• They enforce supermartingale or c-martingale conditions to derive explicit lower bounds on the probability that system trajectories remain safe under various control policies.
• Advanced formulations include multi-function, interpolation-inspired, piecewise, and compositional designs for applications in hybrid, switched, and networked systems.

Stochastic control barrier certificates are certificate functions, or families of functions, for stochastic dynamical systems and stochastic control systems that convert safety requirements into inequalities on conditional expectations or infinitesimal generators. In the discrete-time setting, a barrier certificate typically acts as a nonnegative supermartingale or $c$-martingale and yields an explicit lower bound on the probability that trajectories remain in a safe set; in controlled systems, the same inequalities are enforced under a policy $\pi$ or through a pointwise minimization over admissible controls (Oumer et al., 21 Apr 2025). Across recent literature, the term covers single-function certificates, multi-function interpolation-inspired and $k$-inductive variants, piecewise-constant and compositional constructions, and augmented-space formulations for temporal logic, dynamic obstacles, switched and hybrid systems, wireless communication networks, and data-driven synthesis (Mazouz et al., 23 Jul 2025, Mamduhi et al., 10 May 2026, Mazouz et al., 22 Apr 2026).

1. Formal model and probabilistic safety semantics

A standard discrete-time stochastic dynamical system without control is written as

$S = (X, X_0, w, f),$

where $X \subseteq \mathbb{R}^n$ is a Borel space of states, $X_0 \subseteq X$ is the set of initial states, $w := \{w(t) : \Omega \to W, t \in \mathbb{N}\}$ is a sequence of i.i.d. random variables, and $f : X \times W \to X$ is measurable, with dynamics

$$x(t+1) = f(x(t), w(t)), \quad \forall t \in \mathbb{N}.$$

The process is Markov by construction. With control, the model becomes

$$S_c = (X, X_0, \mathcal{U}, w, f), \qquad x(t+1)=f(x(t),u(t),w(t)),$$

and a stationary policy $\pi$0 induces the closed-loop dynamics $\pi$1 (Oumer et al., 21 Apr 2025).

Safety is specified through a safe set $\pi$2 and an unsafe set $\pi$3. A canonical stopping time is

$\pi$4

and the infinite-horizon safety probability is

$\pi$5

A lower bound $\pi$6 certifies safety if

$\pi$7

(Oumer et al., 21 Apr 2025).

Finite-horizon formulations are equally common. For a discrete-time nonlinear stochastic system with additive noise,

$\pi$8

the probabilistic safety objective over horizon $\pi$9 is

$k$0

(Mazouz et al., 23 Jul 2025). In continuous time, the corresponding controlled Itô model takes the form

$k$1

and the generator-based viewpoint uses

$k$2

(Anand et al., 2021).

This formal setup extends naturally to augmented states. In uncertain temporal specifications, the labeling function depends on a stochastic predicate parameter $k$3, and the augmented state is $k$4 (Mamduhi et al., 10 May 2026). For dynamic obstacles, one augments the state with obstacle configurations, while for switched and hybrid systems one augments with modes, counters, or jump indicators (Mazouz et al., 22 Apr 2026, Lavaei et al., 2022).

2. Core certificate conditions and probability bounds

In the discrete-time single-function case, a barrier certificate is a function $k$5 such that there exists $k$6 with

$k$7

The supermartingale condition ensures nonincreasing expectation along trajectories, and it yields

$k$8

for every $k$9 (Oumer et al., 21 Apr 2025).

For controlled systems, the stochastic control barrier certificate replaces the open-loop expectation by a closed-loop one: $S = (X, X_0, w, f),$0 or, equivalently, uses a robust design condition of the form

$S = (X, X_0, w, f),$1

together with the same initial and unsafe bounds (Oumer et al., 21 Apr 2025). Related discrete-time formulations also use an additive drift term. A function $S = (X, X_0, w, f),$2 is a $S = (X, X_0, w, f),$3-martingale on $S = (X, X_0, w, f),$4 if

$S = (X, X_0, w, f),$5

and this relaxation leads to finite-horizon safety bounds rather than strict supermartingale invariance (Oumer et al., 21 Apr 2025).

A finite-horizon stochastic control barrier function formulation used in piecewise synthesis requires

$S = (X, X_0, w, f),$6

which implies

$S = (X, X_0, w, f),$7

(Mazouz et al., 23 Jul 2025).

In continuous time, the same mechanism appears through the generator. A standard stochastic barrier condition is

$S = (X, X_0, w, f),$8

with analogous initial and boundary conditions, yielding lower bounds through supermartingale and optional-stopping arguments (Oumer et al., 21 Apr 2025). A finite-time verification variant for switched stochastic systems uses a common barrier $S = (X, X_0, w, f),$9 satisfying

$X \subseteq \mathbb{R}^n$0

which gives

$X \subseteq \mathbb{R}^n$1

(Anand et al., 2021).

The relation to deterministic CBFs is explicit in several formulations: deterministic CBFs enforce forward invariance through pointwise drift constraints, whereas stochastic control barrier certificates replace those constraints by expectation or generator inequalities and yield probabilistic guarantees rather than deterministic invariance (Oumer et al., 21 Apr 2025, Mazouz et al., 23 Jul 2025).

3. Relaxed, multi-function, and structured certificate families

A major recent development is the replacement of a single barrier by multiple coupled functions. An interpolation-inspired barrier certificate consists of functions $X \subseteq \mathbb{R}^n$2, constants $X \subseteq \mathbb{R}^n$3, $X \subseteq \mathbb{R}^n$4, and conditions

$X \subseteq \mathbb{R}^n$5

Only $X \subseteq \mathbb{R}^n$6 must be a nonnegative supermartingale; $X \subseteq \mathbb{R}^n$7 bridge the initial slice and the tail. The resulting lower bound is

$X \subseteq \mathbb{R}^n$8

(Oumer et al., 21 Apr 2025).

The same paper introduces $X \subseteq \mathbb{R}^n$9-inductive formulations. In a relaxed multi-function form, $X_0 \subseteq X$0 satisfies one-step bounds from the initial set and a supermartingale condition every $X_0 \subseteq X$1 steps: $X_0 \subseteq X$2 leading to

$X_0 \subseteq X$3

Two representative $X_0 \subseteq X$4-inductive interpolation-inspired barrier certificates combine interpolation with $X_0 \subseteq X$5-martingale and $X_0 \subseteq X$6-step supermartingale conditions and preserve explicit safety bounds (Oumer et al., 21 Apr 2025).

Another structured family is the piecewise stochastic control barrier function. On a partition $X_0 \subseteq X$7, the paper on piecewise control barrier functions restricts to

$X_0 \subseteq X$8

with regional controls $X_0 \subseteq X$9, so that the expectation constraint becomes

$w := \{w(t) : \Omega \to W, t \in \mathbb{N}\}$0

This reduces the joint controller-certificate design to a minimax problem and then to a dual linear program with zero duality gap (Mazouz et al., 23 Jul 2025).

Large-scale interconnected and switched systems motivate compositional structures. Control sub-barrier certificates for subsystems are combined into a network-level barrier through dissipativity or max-type small-gain conditions. In one construction,

$w := \{w(t) : \Omega \to W, t \in \mathbb{N}\}$1

while in another,

$w := \{w(t) : \Omega \to W, t \in \mathbb{N}\}$2

These constructions provide finite-horizon or infinite-horizon probabilistic guarantees for interconnected stochastic systems and switched systems with dwell-time (Anand et al., 2021, Anand et al., 2021, Nejati et al., 2020).

4. Synthesis methodologies

When the dynamics, sets, and templates are polynomial or semi-algebraic, sum-of-squares programming is the dominant synthesis mechanism. For interpolation-inspired certificates, one introduces SOS multipliers to enforce nonnegativity and set implications, for example

$w := \{w(t) : \Omega \to W, t \in \mathbb{N}\}$3

$w := \{w(t) : \Omega \to W, t \in \mathbb{N}\}$4

$w := \{w(t) : \Omega \to W, t \in \mathbb{N}\}$5

with analogous constraints for unsafe-set lower bounds and supermartingale tails. The generic workflow is: specify polynomial templates, encode conditions as SOS constraints, compute expectation terms using noise moments, solve the resulting semidefinite program, and report the probability bound from the appropriate theorem (Oumer et al., 21 Apr 2025).

Piecewise certificates admit a different route. The piecewise-constant formulation reduces the inner maximization over admissible kernels to a dual LP, with variables $w := \{w(t) : \Omega \to W, t \in \mathbb{N}\}$6 and constraints such as

$w := \{w(t) : \Omega \to W, t \in \mathbb{N}\}$7

This produces a single-level linear program for simultaneous barrier and controller synthesis (Mazouz et al., 23 Jul 2025).

Counterexample-guided inductive synthesis and SMT-based search remain common in polynomial and switched settings. Parametric certificates are synthesized on finite witness sets, then verified over larger domains; counterexamples are added iteratively until no violations remain (Nejati et al., 2020, Anand et al., 2021). Neural parameterizations replace polynomial templates by neural barrier functions and certify the barrier inequalities using linear bound propagation, linear programming, and branch-and-bound. In that framework, the discrete-time condition

$w := \{w(t) : \Omega \to W, t \in \mathbb{N}\}$8

is upper-bounded on partitions of $w := \{w(t) : \Omega \to W, t \in \mathbb{N}\}$9, and the final certificate gives

$f : X \times W \to X$0

(Mathiesen et al., 2022).

Data-driven synthesis replaces known models by finite samples. One line formulates barrier search as a robust convex program and then a scenario convex program, with sample complexity expressed through the regularized incomplete beta function and empirical expectation errors controlled by Chebyshev-type variance bounds (Salamati et al., 2021). Another line learns $f : X \times W \to X$1 with Gaussian processes, constructs piecewise stochastic barrier functions from learned transition kernels, and sequentially prunes unsafe state-control regions to obtain a maximal permissible strategy set (Mazouz et al., 2024).

5. Extensions to temporal logic, uncertain predicates, hybrid and networked systems

Barrier certificates are not limited to basic invariance. For finite-trace temporal logic, the standard construction translates the negation of a safe-LTL or LTL$f : X \times W \to X$2 formula into a DFA, decomposes accepting runs into sequential reachability tasks, and computes upper bounds for those tasks using barrier certificates. If $f : X \times W \to X$3 is the set of elementary triples for a run $f : X \times W \to X$4, then one obtains bounds of the form

$f : X \times W \to X$5

for continuous-time switched systems, or the corresponding discrete-time reachability products in LTL$f : X \times W \to X$6 settings (Anand et al., 2021, Jagtap et al., 2018, Jagtap et al., 2019).

When predicates themselves evolve randomly, the system is augmented with the uncertainty state $f : X \times W \to X$7, producing a deterministic specification on the product space $f : X \times W \to X$8. A $f : X \times W \to X$9-averaged control barrier certificate then imposes expectation bounds over the stochastic predicate state on initial and unsafe sets, and an expectation bound over the process noise in the drift condition: $$x(t+1) = f(x(t), w(t)), \quad \forall t \in \mathbb{N}.$$0 This yields

$$x(t+1) = f(x(t), w(t)), \quad \forall t \in \mathbb{N}.$$1

on the augmented space (Mamduhi et al., 10 May 2026).

Dynamic obstacles induce time-varying unsafe sets $$x(t+1) = f(x(t), w(t)), \quad \forall t \in \mathbb{N}.$$2. A time-varying barrier $$x(t+1) = f(x(t), w(t)), \quad \forall t \in \mathbb{N}.$$3 satisfying

$$x(t+1) = f(x(t), w(t)), \quad \forall t \in \mathbb{N}.$$4

gives

$$x(t+1) = f(x(t), w(t)), \quad \forall t \in \mathbb{N}.$$5

This Bellman-inspired formulation directly captures temporal structure and is reported to be less conservative than time-invariant augmented-state alternatives (Mazouz et al., 22 Apr 2026).

Hybrid and switched architectures use augmented states to encode modes, counters, or jump phases. For stochastic hybrid systems with Brownian and Poisson components and instantaneous stochastic jumps, the augmented control barrier certificate satisfies a one-step inequality

$$x(t+1) = f(x(t), w(t)), \quad \forall t \in \mathbb{N}.$$6

and this yields an explicit finite-time bound

$$x(t+1) = f(x(t), w(t)), \quad \forall t \in \mathbb{N}.$$7

with $$x(t+1) = f(x(t), w(t)), \quad \forall t \in \mathbb{N}.$$8 given piecewise in terms of $$x(t+1) = f(x(t), w(t)), \quad \forall t \in \mathbb{N}.$$9 (Lavaei et al., 2022). In networked control systems with packet losses and delays, the barrier is defined on an augmented state containing true and predicted states and inputs, and the expected drift condition is enforced through an SDP-LMI coupled with SOS constraints (Akbarzadeh et al., 2023).

6. Case studies, limitations, and current directions

The literature reports that relaxed and structured certificates can succeed where classical single-function barriers fail. In a one-dimensional stochastic system $$S_c = (X, X_0, \mathcal{U}, w, f), \qquad x(t+1)=f(x(t),u(t),w(t)),$$0 with $$S_c = (X, X_0, \mathcal{U}, w, f), \qquad x(t+1)=f(x(t),u(t),w(t)),$$1, $$S_c = (X, X_0, \mathcal{U}, w, f), \qquad x(t+1)=f(x(t),u(t),w(t)),$$2, $$S_c = (X, X_0, \mathcal{U}, w, f), \qquad x(t+1)=f(x(t),u(t),w(t)),$$3, and $$S_c = (X, X_0, \mathcal{U}, w, f), \qquad x(t+1)=f(x(t),u(t),w(t)),$$4, a standard cubic barrier is infeasible, while an interpolation-inspired certificate with degree-3 polynomials and $$S_c = (X, X_0, \mathcal{U}, w, f), \qquad x(t+1)=f(x(t),u(t),w(t)),$$5 yields the lower bound $$S_c = (X, X_0, \mathcal{U}, w, f), \qquad x(t+1)=f(x(t),u(t),w(t)),$$6, and a $$S_c = (X, X_0, \mathcal{U}, w, f), \qquad x(t+1)=f(x(t),u(t),w(t)),$$7-inductive interpolation-inspired variant yields approximately $$S_c = (X, X_0, \mathcal{U}, w, f), \qquad x(t+1)=f(x(t),u(t),w(t)),$$8 (Oumer et al., 21 Apr 2025). In a logistic map with noise, standard, interpolation-inspired, and single-function $$S_c = (X, X_0, \mathcal{U}, w, f), \qquad x(t+1)=f(x(t),u(t),w(t)),$$9-inductive barriers are all infeasible, while a $\pi$00-IBC v1 certificate gives a lower bound of approximately $\pi$01 (Oumer et al., 21 Apr 2025). Piecewise methods report $\pi$02 on a 2D linear system and $\pi$03 on a 4D nonlinear unicycle, with empirical Monte Carlo safety exceeding the certified lower bounds (Mazouz et al., 23 Jul 2025).

Compositional and large-scale constructions have been demonstrated on a room temperature network containing $\pi$04 rooms and on a network of $\pi$05 switched subsystems, providing explicit lower bounds on finite-horizon safety probabilities (Nejati et al., 2020). Wireless-network-aware synthesis has been demonstrated on a permanent magnet synchronous motor, with a reported lower bound of $\pi$06 over $\pi$07 steps under $\pi$08 (Akbarzadeh et al., 2023). Data-driven variants show that increasing the dataset size enlarges the permissible strategy set in both linear and nonlinear systems (Mazouz et al., 2024).

The limitations are equally consistent across papers. SOS relaxations and global polynomial templates are conservative and sensitive to degree, monomial basis, and multiplier choice (Oumer et al., 21 Apr 2025). Scalability remains difficult in high dimensions because SDP size grows quickly, LP size can grow quadratically with the number of cells in partition-based methods, and automata-based decompositions can accumulate conservative sum-product bounds (Mazouz et al., 23 Jul 2025, Anand et al., 2021). Time-invariant certificates on augmented spaces face dimensionality issues in dynamic-obstacle settings (Mazouz et al., 22 Apr 2026). Data-driven guarantees require large sample sizes, Lipschitz constants, and variance bounds, and are conservative on high-dimensional domains (Salamati et al., 2021).

A recent refinement removes a boundedness assumption that earlier finite-time stochastic barrier conditions imposed on auxiliary functions. In discrete time, the refined condition uses

$\pi$09

together with boundary anchoring on $\pi$10 and $\pi$11, and yields an upper bound on finite-time safety probabilities without requiring $\pi$12 to be bounded on unbounded domains. In continuous time, the refined reach-avoid conditions use

$\pi$13

with boundary-time inequalities and provide lower bounds on finite-time reach-avoid probabilities, again without a boundedness requirement (Xue et al., 23 Sep 2025). This suggests that one current direction is the systematic extension of stochastic control barrier certificates from compact semi-algebraic settings toward unbounded domains, richer policy classes, and less conservative finite-time guarantees.