Hybrid Barrier Systems

Updated 23 December 2025

Hybrid barrier systems are a synthesis and verification framework that uses barrier certificates to ensure robust safety in dynamical systems with continuous flows and discrete jumps.
Algorithmic synthesis methods, including sum-of-squares and convex relaxations, systematically verify system invariance and reduce conservativeness while accommodating model uncertainties.
Hybrid control strategies that combine control barrier and Lyapunov functions guarantee forward invariance and robust performance even under stochastic and high-dimensional conditions.

A hybrid barrier system is an overview and verification framework for safety, reachability, and performance requirements in dynamical systems that integrate both continuous evolution (flows) and discrete events (jumps, switches). The unifying element is the use of barrier certificates or control barrier functions (CBFs) generalized to hybrid, uncertain, nonlinear, stochastic, and large-scale networks. Hybrid barrier methods systematically encode invariance, safety, or specification satisfaction as inequalities over certain functions (barriers), with conditions designed to guarantee robust forward invariance or set contractivity in the presence of hybrid flows and discrete transitions. This article surveys the technical landscape of hybrid barrier systems, detailing formal conditions, algorithmic synthesis, compositionality, robustness, and high-level specification enforcement across a spectrum of settings.

1. Mathematical Foundations: Hybrid Barrier Certificates

Hybrid System Formalism and Barrier Functions

A hybrid dynamical system is defined by a tuple such as $(C, F, D, G)$ or related variants, where $C \subset \mathbb{R}^n$ is the flow set, $F$ the set-valued flow map, $D$ the jump set, and $G$ the jump map. The system’s execution alternates between flows (where $\dot{x} \in F(x)$ , $x \in C$ ) and jumps (where $x^+ \in G(x)$ , $x \in D$ ) (Meng et al., 2022). In hybrid automata and control applications, discrete states label different continuous dynamics.

A barrier certificate for such a hybrid system is a function (or set of functions) $B(x)$ designed to separate safe and unsafe sets, with conditions to prevent system trajectories from crossing into undesirable regions. The key hybrid barrier inequalities are:

Continuous-flow invariance: $\langle \nabla B(x), f(x) \rangle \geq 0$ for all $x$ in the flow set (ensuring non-decrease along flows towards exiting the safe set) (Yang et al., 26 Jan 2024, Kong et al., 2019).
Jump condition: $B(g(x)) \geq B(x)$ (ensuring non-decrease of the barrier after discrete events).
Separation: $B(x) > 0$ (resp. $< 0$ ) on unsafe (resp. initial) sets.

Multiple formulations exist, such as single Lyapunov-barrier functions, separate Lyapunov and barrier pairs, or mode-indexed families for multi-mode systems (Meng et al., 2022, Kong et al., 2013).

Sufficient Conditions for Forward Invariance and Contractivity

Forward invariance of a closed set $S$ is achieved if, for all trajectories starting in $S$ , solutions remain in $S$ . A function $B(x)$ provides a certificate if it increases (or does not decrease) along flows and across jumps on the boundary of $S$ , with stronger contractivity guarantees (forcing re-entry into the interior) possible if strict inequalities hold and boundary flows always move into $S$ (Meng et al., 2022, Maghenem et al., 2019).

2. Algorithmic Synthesis and Convex Relaxations

Exponential and Generalized Barrier Conditions

Relaxations such as the “exponential condition” allow the barrier function to grow at an exponential rate (not just be non-increasing), thereby reducing conservativeness and enabling lower-degree polynomial parameterizations that are more amenable to semidefinite programming (SDP). The key inequality is: $\frac{\partial B}{\partial x} f(x) - \lambda B(x) \leq 0, \quad \lambda < 0$ which, upon integration, yields $B(x(t)) \leq B(x(0)) e^{\lambda t}$ (Kong et al., 2013). This is amenable to convex optimization since, for fixed $\lambda$ , the constraints remain linear/SOS in the coefficients of $B$ .

Generalizations replace the exponential right-hand side with any function $\psi(B(x))$ such that the scalar ODE $\dot\theta = \psi(\theta)$ , $\theta(0) \leq 0$ , never crosses zero—preserving convexity of the feasible region in the barrier search (Dai et al., 2013).

Piecewise Robust Barrier Tubes

For nonlinear and hybrid systems with uncertainty, piecewise robust barrier tubes (PBT) partition the state space into boxes, constructing local barrier certificates for each box such that the union covers the reachable set. The method avoids interval wrapping by using simulation-based enclosures and LP-relaxed polynomial barriers, and it can accommodate parametric and additive uncertainties by enforcing worst-case derivative conditions on the uncertainty set (Kong et al., 2019).

Data-Driven and Learning-Based Construction

Optimization-based learning of hybrid barrier functions or control barrier functions is enabled by assembling a set of local margin and Lipschitz constraints from demonstration data. The methodology guarantees correctness if the sample data form sufficiently dense $\epsilon$ -nets of safe and unsafe boundary regions, with all hybrid flow and jump conditions enforced at the sampled points (Lindemann et al., 2020, Robey et al., 2021). Neural network or kernel-based parameterizations are used to capture complex, high-dimensional safe sets.

3. Control Synthesis: Hybrid CBFs and Hybrid CLF–CBF Control

Hybrid Control Barrier Functions (HCBF) and Safety Control

The hybrid CBF framework defines a family $h_q(x)$ for each discrete mode $q$ , with flow constraints

$\forall x \in D, \quad \sup_{u \in U_q} \nabla h_q(x) \cdot f_q(x,u) \geq -\alpha_q(h_q(x)),$

and jump constraints

$h_{q'}(g(x,u,q)) \geq 0.$

A switching CBF-based controller enforces these via quadratic programming to minimally intervene and guarantee forward invariance of the safe set in each mode. For global safety, a refinement step excludes backward-reachable unsafe states near transitions, relying on Hamilton–Jacobi reachability analysis, dynamic programming refinements, or learning of the viability kernel (Yang et al., 2023, Yang et al., 26 Jan 2024).

Hybrid CLF–CBF Approaches and Deadlock Elimination

Combining control Lyapunov functions (CLF) and CBFs through a hybrid automaton resolves deadlocks endemic to continuous CLF/CBF-QP schemes. The system flows under a QP-based subproblem controller (for the current CBF region), with discrete jumps triggered by penetration into adjacent safe regions, ensuring strict progress toward the goal. Recursive hybrid CLF–CBF backstepping extends this to higher-order (strict-feedback) nonlinearities. Global asymptotic stabilization and safety are guaranteed under appropriate feasibility conditions (Matias et al., 13 Apr 2025).

Handling Model Uncertainty and Robustness Guarantees

Robust hybrid CBFs (RHCBF) subtract Lipschitz constants times model error from derivative conditions, ensuring forward invariance under bounded uncertainty in both flows and jumps. Data-driven algorithms enforce margin and covering conditions in the presence of uncertainty, yielding certificates that expand the robust domain of attraction (Robey et al., 2021).

4. Compositionality and Large-Scale Systems

Compositional Hybrid Barrier Synthesis

For networks of interconnected subsystems (including stochastic hybrid systems), local augmented control sub-barrier certificates (A-CSBCs) are computed per subsystem, with small-gain or max-type interconnection inequalities ensuring that the system-level sum (or max) of local barriers yields a valid certificate for the entire network. This reduces complexity from polynomial or exponential in the network size to linear scaling via independent local SOS programs or CEGIS loops. Probabilistic finite-horizon safety can be quantified by supermartingale arguments on the sum barrier (Zaker et al., 16 Sep 2024, Anand et al., 2021, Nejati et al., 2020).

Handling Temporal Logic and Automata Specifications

Temporal logic (e.g., LTL/scLTL) requirements are enforced via product construction of the hybrid system with a finite-state automaton encoding the specification. Hybrid barrier certificates for the product system guarantee that executions satisfy eventuality properties or diagnosability, with SOS programming or counter-example guided inductive synthesis used for implementation (Bisoffi et al., 2020, Zhong et al., 13 Aug 2024).

5. Specialized Barrier Structures and Applications

Min-Quadratic and Synergistic Barriers

The min-quadratic barrier approach constructs a global barrier as a pointwise minimum over ellipsoidal certificates (from LMI optimizations at sampled equilibria), each paired with a safety controller, wrapped in a two-mode hybrid automaton with hysteresis thresholds. This ensures hard state and input constraints are always met, intervening only when boundaries are approached (Thomas et al., 2018).

For nonholonomic agents and cases where standard CBFs fail (e.g., due to loss of control authority at singular configurations), hybrid switching between synergistic CBFs (differently rotated velocity terms) restores feasibility. A hybrid automaton manages discrete switching between CBF "branches" to maintain safety and forward invariance under actuation limits (Haraldsen et al., 7 Apr 2025).

Stochastic Hybrid Barrier Systems

Barrier-based approaches naturally extend to stochastic hybrid systems, including Brownian and Poisson noise, and sampled-data resets. The key is to enforce a generalized supermartingale condition on the barrier under both flow and jump (reset) transitions, which places explicit upper bounds on the probability of reaching unsafe sets within a finite horizon (Lavaei et al., 2022). Compositional approaches allow scaling to thousands of subsystems with certified probabilistic guarantees (Zaker et al., 16 Sep 2024).

6. Computational Tools and Practical Considerations

Sum-of-Squares and Convex Optimization

All principal hybrid barrier constructions (including relaxed and exponential variants, multi-function combinations, and robust extensions) remain within the cone of convex optimization—sum-of-squares programs or linear programming—provided polynomial templates and fixed relaxation parameters are used. Post-processing of numerical SDP output with symbolic nonnegativity checks or rational reconstruction addresses numeric unsoundness (Dai et al., 2013). When polynomial parameterizations are too restrictive, learning-based or neural approaches provide scalable alternatives for high-dimensional systems (Lindemann et al., 2020).

Efficiency, Scalability, and Limitations

Scalability is limited by polynomial degree and state dimension; compositional schemes and neural parameterizations alleviate these for large-scale or high-dimensional systems. For specifications requiring backward reachability or HJ-based value function computation, learning-based or sample-efficient PDE solvers (CBVF neural nets) can replace grid-based algorithms (Yang et al., 26 Jan 2024, Lindemann et al., 2020). Not all dynamics (e.g., highly non-polynomial, non-smooth) are directly amenable to current techniques, but overapproximation schemes or embedding strategies can extend applicability.