--- title: Barrier Lyapunov Functions in Control url: https://www.emergentmind.com/topics/barrier-lyapunov-function-blf type: topic --- # Barrier Lyapunov Functions in Control Barrier Lyapunov Functions (BLFs) are a control-design tool for enforcing hard state constraints in nonlinear systems. The basic idea is to build a Lyapunov function that becomes unbounded as the constrained state approaches the boundary of the admissible set; if one can design the closed-loop system so that this BLF is nonincreasing, then the state is prevented from reaching the boundary [2411.06288]. In contemporary safety-critical control, the term also sits within a broader family of one-function certificates, including Lyapunov-like barrier functions and control Lyapunov-barrier functions (CLBFs), that combine convergence and constraint satisfaction in a single scalar object [1802.09921, 2509.12182]. ## 1. Classical BLF mechanism and constrained invariance In the classical formulation for nonlinear systems in strict-feedback form, \[ \dot{x}_i = f_i(x_1,\dots,x_i) + g_i(x_1,\dots,x_i)x_{i+1},\quad i=1,\dots,n-1, \] \[ \dot{x}_n = f_n(x_1,\dots,x_n) + g_n(x_1,\dots,x_n)u, \] one seeks tracking while guaranteeing constraints such as \[ |x_i(t)| < k_{x_i}. \] A canonical example is the logarithmic BLF \[ V(x)=\frac{1}{2}\ln\!\left(\frac{k^2}{k^2-x^2}\right), \] which grows slowly near the origin but sharply near the boundary [2411.06288]. The defining feature is not merely positivity, but singular growth at the admissible boundary. A related formulation appears in constrained learning control, where a BLF is treated as a positive function \(V\mapsto B(V)\) defined on a domain such as \(0\le V0\), forward invariance of \(\Omega_c\) implies that the safety constraints remain satisfied for all time while the coordination error decreases [1802.09921]. This establishes the standard BLF logic: a single scalar function simultaneously blocks boundary crossing and certifies convergence. ## 2. From barrier blow-up to unified Lyapunov-barrier certificates Although the classical BLF relies on singular behavior near a constraint boundary, several converse results show that joint safety and stability can also be encoded by a single smooth Lyapunov-type function. A central statement is that, under appropriate robustness assumptions, stability with safety guarantees admit a smooth converse Lyapunov-barrier characterization on the relevant domain of initial conditions [2009.04432]. In that formulation, there exists an open set \(D\), a smooth \(V:D\to\mathbb{R}_{\ge 0}\), and class-\(\mathcal K_\infty\) functions \(\alpha_1,\alpha_2\) such that \[ \alpha_1(\omega(x)) \le V(x) \le \alpha_2(\omega(x)), \qquad \nabla V(x)\cdot (f(x)+d)\le -V(x), \] for all \(x\in D\) and all \(d\in\delta B\), where \(\omega\) is a proper indicator for the target set. The same work shows that a split Lyapunov-barrier representation can be recovered by setting \(B(x)=c-V(x)\) on a suitable domain [2009.04432]. A sharper control-theoretic equivalence is given by the result that the existence of a strictly compatible pair of control Lyapunov and control barrier functions is equivalent to the existence of a single smooth Lyapunov function that certifies both asymptotic stability and safety [2509.12182]. For a safe set \[ \mathcal{C} = \{x \in \mathbb{R}^n \mid h(x) \ge 0\}, \qquad \partial \mathcal{C} = \{x \in \mathbb{R}^n \mid h(x)=0\}, \] strict compatibility requires that on \(\partial\mathcal C\) the same control satisfy \[ L_f V(x) + L_g V(x)u < 0, \qquad L_f h(x) + L_g h(x)u > 0. \] Under that condition, one obtains a smooth function \(W\) such that \[ \nabla W(x)\cdot F(x) < 0,\qquad x\neq 0, \] \[ \{x\in \mathcal{D} : W(x)\le 1\} = \mathcal{C}, \qquad \{x\in \mathcal{D} : W(x)=1\} = \partial\mathcal{C}. \] This is a one-function certification of both asymptotic stability and exact safe-set invariance [2509.12182]. The same paper makes the conceptual distinction explicit: a BLF is typically designed so that the Lyapunov function blows up or becomes singular near a constraint boundary, whereas the constructed CLBF uses a smooth function whose \(1\)-level set exactly equals the safe boundary and whose sublevel set exactly equals the safe region [2509.12182]. This does not eliminate the BLF viewpoint; rather, it shows that, under strict compatibility, a BLF-like safe stabilizer can be represented by a smooth Lyapunov function without singular barrier behavior. A related converse theorem shows that robustly safe sets admit barrier functions constructed from Lyapunov functions for the closure of the reachable set. If \(W\) is \(\delta\)-robustly safe with respect to an unsafe set \(U\), then the closure \[ \Omega := \overline{R_\delta(W)} \] is robustly asymptotically stable under mild assumptions, and a Lyapunov function for \(\Omega\) can be converted into a robust barrier function, for example through \[ B_c(x)=c-V(x) \] or \(B(x)=-V(x)\) [2007.11086]. In this sense, robust safety implies the existence of a barrier certificate via converse Lyapunov theory. ## 3. PDE characterizations and exact safe-set encoding A distinctive development in recent work is the use of PDEs with prescribed boundary conditions to characterize Lyapunov-barrier functions. In the strict-compatibility theorem, the unified function \(W\) is constructed using a backward hitting time \(T(x)\) to the safe-set boundary and a normalization of a smooth Lyapunov function \(V\), \[ W(x)=\frac{V(x)}{V(\phi(T(x),x))}. \] The integral representation \[ W(x) = \int_0^\infty \omega_1(\phi(t,x))\,dt, \qquad \omega_1(x)=\frac{\omega(x)}{V(\phi(T(x),x))}, \] yields the transport PDE \[ \nabla W(x)\cdot F(x) = -\omega_1(x), \] with boundary information inherited from the hitting-time normalization [2509.12182]. Because \(T(x)=0\) on \(\partial\mathcal C\), one gets \(W=1\) there; because \(T(x)\) is positive outside and negative inside, one gets \(W>1\) outside and \(W<1\) inside. The safe set is therefore encoded exactly as a sublevel set, not merely under-approximated. A complementary PDE-based program arises from a Zubov formulation with Dirichlet boundary conditions for autonomous nonlinear systems. For an unsafe set \(U\) and safe domain \(D=U^c\), a maximal Lyapunov-type function is defined by \[ \tau(x) = \inf\{t\ge 0:\phi(t;x)\in U\}, \] \[ V(x) = \begin{cases} \displaystyle \int_0^{\tau(x)} \omega(\phi(t;x))\,dt + q(\phi(\tau(x);x)), & \tau(x) < \infty, \\[1.2ex] \displaystyle \int_0^{\infty} \omega(\phi(t;x))\,dt, & \tau(x)=\infty, \end{cases} \] and satisfies \[ -\nabla V(x)\cdot f(x) - \omega(x)=0, \qquad V(0)=0. \] After a Zubov transform, \[ W(x) = \begin{cases} \beta(V(x)), & x\in D, \\ 1, & \text{otherwise}, \end{cases} \] one obtains a bounded Lyapunov-barrier function satisfying a Zubov PDE with Dirichlet boundary data \(W=1\) on \(\partial D\) [2511.09523]. The same work states that a physics-informed neural network solution, once formally verified, can serve as a Lyapunov-barrier function that jointly certifies stability and safety and yields a near-optimal certified under-approximation of the true safe domain of attraction [2511.09523]. A distinct PDE construction appears in harmonic CLBFs, where one solves Laplace’s equation \[ \nabla^2 V(s)=0 \quad \forall s\in \mathcal{S}_{safe}, \] with boundary conditions \[ V(s)=0 \quad \forall s\in \overline{\mathcal{S}_{goal}}, \qquad V(s)=c \quad \forall s\in \partial \mathcal{S}\cup \mathcal{S}_{unsafe}. \] By the maximum principle, the resulting harmonic function satisfies the CLBF separation properties, and the control is chosen by \[ \pi^*(x)=\arg\inf_{u\in U}\langle f(x,u), \nabla V(x)\rangle \] to align the dynamics with the steepest descent direction [2310.02869]. This is not a singular BLF construction, but it is a one-function barrier-Lyapunov encoding of reach-avoid structure. ## 4. Variants: fractional, progressive, nonsmooth, and stochastic formulations Several variants modify the classical BLF to address computational, geometric, or robustness issues. Fractional BLFs replace logarithmic expressions by rational forms such as \[ f_{FI}(V)=\frac{V}{b_v-V}, \qquad f_{FII}(V)=\frac{b_vV}{b_v-V}, \] with the claim that the fractional form is algebraically simpler and can provide stronger barrier action than the conventional logarithmic form in finite-duration learning control [2306.06646]. The same paper also introduces variants with the infinite barrier property and shows that fully-saturated learning algorithms are important for assuring boundedness of the estimates and achieving the error-constraint objective [2306.06646]. Progressive BLFs modify the control effort profile inside the safe set. The progressive Barrier Lyapunov Function, or p-BLF, is introduced for output- and full-state-constrained nonlinear control systems with the defining feature that control effort is deliberately small in unconstrained or interior regions and increases smoothly and progressively as the state nears the constraint boundary [2411.06288]. Two forms are given: \[ V(x)=\frac{1}{2\beta}\ln\!\left(\frac{k^2}{k^2-x^2}\right), \qquad V(x)=\frac{x^2}{2(k^2-x^2)(1+\beta x^2)}. \] For backstepping designs, the resulting closed-loop derivative takes the form \[ \dot V=-\sum_{j=1}^{n}\kappa_j z_j^2, \] which yields constraint satisfaction, boundedness of all closed-loop signals, and asymptotic tracking [2411.06288]. Nonsmooth formulations address cases in which smooth CLBFs fail. The nonsmooth Control Lyapunov Barrier Function paper introduces a generalized Lyapunov barrier function defined by \[ V(x)=\max(L(x),B(x)), \qquad L(x)=\|x\|^2, \qquad B(x)=\eta_2-\eta_1\|x-x_c\|^2, \] together with an upper generalized derivative condition \[ \overline M_{\mathcal F}V(x)\le -\rho(\|x\|). \] This max structure is used to avoid the gradient cancellation problem endemic to smooth CLBFs and to handle bounded unsafe sets and multiple disjoint unsafe regions directly [2409.13624]. The same work contrasts the construction with smooth CLBFs that can fail when unsafe sets are bounded and can induce undesirable local equilibria. Stochastic extensions replace deterministic Lie-derivative inequalities with generator inequalities. For stochastic nonlinear systems, \[ dx = \big(f(x)+G(x)u\big)\,dt+\Sigma(x)\,dw, \] the infinitesimal generator is \[ \mathcal L V = \frac{\partial V}{\partial x}^T(f+Gu)+\frac12\operatorname{tr}\!\left(\frac{\partial^2 V}{\partial x^2}\Sigma\Sigma^T\right). \] High-relative-degree stochastic control barrier functions are built recursively from \[ \psi_0(x)=h(x), \qquad B_0(x)=\frac{\gamma_0}{\psi_0(x)}, \] and a sequence of \(\psi_i,B_i\) yielding nested safe sets \(\mathcal C_i=\{x:\psi_i(x)\ge0\}\). If \(x(t_0)\in \mathcal C_{r_b}\) and \(\psi_{r_b}(x)\ge0\), then \[ \Pr\{x(t)\in\mathcal C_{\text{safe}}\}=1,\qquad \forall t \] holds almost surely [2004.03856]. A separate stochastic MPC framework defines a stochastic CLBF \(W_c\) by positivity on unsafe states and negativity of \(\mathcal L W_c\) outside the unsafe set and the origin, thereby combining the stability role of a stochastic CLF with the safety role of a stochastic CBF [2211.06175]. ## 5. Synthesis methodologies in control design BLFs and closely related Lyapunov-barrier constructions appear in several synthesis pipelines. In multi-agent coordination, the safe region is approximated from the inside by the sublevel set of an optimal Lyapunov-like barrier function, and the search for the certificate is formulated as an optimization problem over a parameterized family of candidates, often as a sum-of-squares or polynomial optimization problem [1802.09921]. The objective is to maximize the size of the certified under-approximation of the Safety Guaranteed Region of Multi-Task Coordination while preserving collision avoidance, connectivity maintenance, and convergence to the desired coordinated configuration. Predictive control is a major integration point. For an aerial manipulator, BLF-based constraints are embedded in the outer-loop MPC, while the inner loop remains conventional PID control [2212.04625]. The safe set is defined by a differentiable function \(h(\mathbf x)\), and the paper relaxes the invariance condition to \[ \dot h(\mathbf{x}) \ge -\gamma h^z(\mathbf{x}), \] or, for a control-affine system, \[ \frac{\partial h(\mathbf{x})}{\partial x}\big(f(\mathbf{x}) + g(\mathbf{x})\mathbf{u}\big) + \gamma h^z(\mathbf{x}) \ge 0. \] To handle bounded disturbances, the safety condition is strengthened to \[ \dot h_i(\mathbf{x}) + \gamma\big(h_i^z(\mathbf{x}) - \lambda\big) \ge 0. \] The resulting MPC-BLF formulation is used both for wall avoidance and for maintaining the manipulator end effector within the desired workspace [2212.04625]. A related but distinct mechanism appears in Lyapunov redesign for perturbed integrator chains. After a predefined-time reaching phase into a prescribed neighborhood \[ \mathcal S_\varepsilon=\{x\in\mathbb R^n:\ V(x)<\varepsilon\}, \] the controller switches to a barrier-function-based gain \[ \Lambda(t,x)=\frac{V(x)}{\varepsilon - V(x)}, \] which grows unbounded as \(V\to\varepsilon^-\) and keeps trajectories inside \(\mathcal S_\varepsilon\) despite uncertainties and perturbations [2405.09438]. This is BLF-like in the sense that the barrier is placed on a Lyapunov level-set boundary rather than directly on a coordinate constraint. Quadratic-program formulations are often barrier-based but not BLF-based in the narrow sense. Adaptive robust QPs using CLFs and CBFs enforce safety through a robustified CBF inequality inside an online optimization problem, while performance is encoded through a CLF with relaxation [2010.04699]. Event- and self-triggered greedy synthesis with CLF-CBF constraints likewise uses barrier inequalities, not BLFs, to keep the state inside a safe set and to maximize inter-execution times [2302.12435]. These methods belong to the same safety-control family, but their barrier mechanism is an inequality on \(\dot h\) rather than a barrier term embedded in a Lyapunov function. ## 6. Learning-based BLFs and conceptual limitations Learning-based methods have adapted BLF ideas to policy optimization and data-driven robot control. In constrained Markov decision processes, Lyapunov Barrier Policy Optimization introduces a Lyapunov-induced safe policy set and adds a logarithmic barrier \[ \psi\!\left(Q^{C_i}_{\pi_B}(s,\pi_\theta(s))\right) = -\beta \log\!\Big( \hat{\epsilon}(s) - \big(Q^{C_i}_{\pi_B}(s,\pi_\theta(s)) - Q^{C_i}_{\pi_B}(s,\pi_B(s))\big) \Big), \] which diverges as the policy update approaches the safety boundary [2103.09230]. This is BLF-like because the penalty becomes prohibitive near the boundary, but the constrained object is the policy update rather than the physical state. Model-free safe RL also uses unified Lyapunov-barrier certificates learned from data. In Lyapunov barrier actor-critic, a single function \(V(s)\) is required to satisfy a goal condition, a safe/unsafe threshold condition at \(\hat c\), and a decrease condition \[ \mathbb{E}_{s'\sim P(\cdot|s,\pi(s))}\big[V(s')-V(s)+\lambda V(s)\big]\le 0. \] Unsafe states are assigned a large terminal cost \(C\), and the learned critic \(Q_{\text{LB}}\) is used to define the CLBF via \(V(s)=Q_{\text{LB}}(s,\pi_\theta(s))\) [2305.09793]. The method is demonstrated on a 2D quadrotor navigation task and is motivated in part by the observation that separate CLF-CBF-QP control can get stuck due to conflict between attraction to the goal and repulsion from obstacles [2305.09793]. Robust neural Lyapunov-barrier certificates extend this logic to perturbed dynamics. A robust certificate is defined by the conditions \[ V(x) \leq \beta, \quad \forall x \in \mathcal{X}_I, \] \[ V(x) - V(y) \geq \epsilon, \quad \forall x \in \left\{ x \in \mathcal{X} \setminus \mathcal{X}_G \mid V(x) \leq \beta \right\}, \forall y \in \mathcal{B}_{\delta,p}(f(x,\pi(x))), \] \[ V(x) \geq \alpha, \quad \forall x \in \mathcal{X}_U, \] and the effective robustness margin is \[ \epsilon_{\text{rob}} = \epsilon_r - L_p \cdot \delta. \] Training objectives combine adversarial training, Lipschitz neighborhood bounds, and global Lipschitz regularization, and the reported improvements include certified robustness bounds up to \(4.6\) times and empirical success rates under strong perturbations up to \(2.4\) times compared to the baseline [2602.05311]. The literature also clarifies several limitations. First, not every constrained stabilization problem admits a globally well-behaved barrier-Lyapunov representation. If a safety and stability specification cannot be certified by a single smooth Lyapunov function, then any CLF-CBF pair necessarily leads to a conflict and cannot be satisfied simultaneously in a robust sense [2509.12182]. Second, smooth unified certificates can fail in the presence of bounded unsafe regions or gradient cancellation, motivating nonsmooth constructions [2409.13624]. Third, in harmonic formulations, superharmonic alternatives can make the safe/unsafe boundary more visible but introduce more interior local minima, which can hurt goal-reaching [2310.02869]. Fourth, many algorithmic realizations remain conservative because feasibility, disturbance margins, sampling time, or value-function estimation accuracy strongly affect the certified set and the practical controller [2010.04699, 2212.04625, 2103.09230]. Taken together, these developments show that BLF now denotes both a classical singular-barrier methodology and a broader research direction in which safety and stability are encoded by one scalar certificate. The classical blow-up mechanism remains central for direct constrained control design, while converse theorems, PDE formulations, nonsmooth constructions, MPC embeddings, and learning-based certificates demonstrate that the same safety-critical objective can also be represented through exact level-set geometry, transport or Laplace equations, recursive high-order inequalities, and formally verified neural surrogates [2411.06288, 2509.12182, 2511.09523]. Source: https://www.emergentmind.com/topics/barrier-lyapunov-function-blf