Strict Control Lyapunov Functions

• Strict CLFs are Lyapunov certificates that impose a strictly negative decrease condition away from equilibrium, guaranteeing global asymptotic or exponential stability in control-affine systems.
• They enable feedback synthesis via methods like CLF-QP and Sontag-type controllers, which parameterize admissible inputs to enforce strict stability margins.
• Strict CLFs integrate with control barrier functions to jointly certify safety and stability, supporting computational synthesis in high-dimensional, switched, and data-driven regimes.

A strict Control Lyapunov Function (CLF) is a Lyapunov certificate for controlled dynamics that requires a strictly negative decrease condition away from the equilibrium, rather than mere nonincrease. For control-affine systems of the form $\dot{x} = f(x) + g(x)u$, one standard formulation takes a continuously differentiable function $V : X \to \mathbb{R}_+$ and class $\mathcal{K}_\infty$ functions $\alpha_1,\alpha_2,\alpha_3$ such that

$$\alpha_1(\|x\|) \leq V(x) \leq \alpha_2(\|x\|), \qquad \inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|), \quad \forall x \in X.$$

In the literature summarized here, this strictness is used to obtain global asymptotic or exponential stability, to parameterize stabilizing feedback laws, to certify joint safety and stability when combined with barrier functions, and to support computational synthesis in settings ranging from nonlinear MPC and switched systems to high-dimensional decomposition, reinforcement learning, and data-driven bilinear models (Grandia et al., 2020, Quartz et al., 15 Sep 2025).

1. Formal definition and the meaning of strictness

The defining feature of a strict CLF is that the Lyapunov decrease condition is negative definite away from the target. In the control-affine setting, the strict condition is commonly written as $\inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|)$, and the admissible stabilizing inputs at a state can be collected in a set such as

$$U_{\mathrm{clf}}(x) = \{u \in U : \dot{V}(x,u) \leq -\alpha_3(\|x\|)\}.$$

The same idea appears in exponential forms used for synthesis under input constraints, where a polynomial CLF is required to satisfy

$L_f V(x) + L_g V(x)u < -\kappa_V V(x)$

for all $x \neq 0$ inside a certified sublevel region. In that form, strictness is encoded by the positive margin $-\kappa_V V(x)$ and yields strict exponential decrease (Grandia et al., 2020, Dai et al., 2022).

When the class-$V : X \to \mathbb{R}_+$0 bounds are quadratic, strictness upgrades asymptotic stability to exponential stability. One formulation states that if $V : X \to \mathbb{R}_+$1, then

$V : X \to \mathbb{R}_+$2

with suitable constants. Closely related formulations appear in control Lyapunov-value functions (CLVFs), where the admissible control set is defined by

$V : X \to \mathbb{R}_+$3

again making the decrease rate explicit (Grandia et al., 2020, Gong et al., 2024).

This strictness is the conceptual boundary between classical CLFs and relaxed variants. The survey on flexible Lyapunov functions contrasts the classical requirement

$V : X \to \mathbb{R}_+$4

with flexible formulations in which the contraction factor becomes an online decision variable. That work identifies classical strict CLFs with fixed, monotonic decrease and describes flexible CLFs as allowing controlled non-monotonicity, mainly to reduce conservativeness in fast constrained systems (Lazar, 2010).

2. Feedback synthesis from strict CLFs

A strict CLF is not only a certificate; it is also a feedback design primitive. A standard construction is the CLF-QP, which solves

$V : X \to \mathbb{R}_+$5

thereby synthesizing a control that satisfies the strict decrease constraint pointwise in time. In the robotic NMPC work, this strict CLF-QP condition serves as the baseline stabilization mechanism against which receding-horizon variants are compared (Grandia et al., 2020).

A second major construction is Sontag-type feedback. For input-affine systems $V : X \to \mathbb{R}_+$6, one recent study shows that a Sontag-type controller built from a CLF minimizes a CLF-dependent cost. When the CLF is chosen as the LQR value function $V : X \to \mathbb{R}_+$7, the resulting Sontag-type feedback exactly recovers the LQR in a neighborhood of the equilibrium, and the closed-loop region is stated to be at least as large as that generated by the LQR. For globally feedback-linearizable systems, the same paper gives a constructive global CLF in transformed coordinates, yielding global asymptotic stability while preserving local LQR behavior (Bongard et al., 4 Feb 2026).

Strict CLFs also support explicit pointwise-optimal stabilization in decomposed high-dimensional settings. In the CLVF framework, once a reconstructed CLF or CLVF is available, the controller is obtained from the quadratic program

$V : X \to \mathbb{R}_+$8

which enforces strict exponential decay. In exact reconstruction cases, the full-dimensional CLVF is the maximum of subsystem CLVFs; in inexact cases, a sum construction yields a Lipschitz continuous strict CLF on the region where the admissible-control-set intersection remains nonempty (Gong et al., 2024).

The control interpretation of strictness is therefore twofold. First, it produces statewise feasibility conditions for stabilizing inputs. Second, it enables redesigns with performance meaning, including inverse-optimal and LQR-consistent constructions. This suggests that strictness is often valued not merely because it proves convergence, but because it fixes the sign structure needed to embed stabilization into optimization.

3. Strict CLFs in joint safety–stability formulations

A central modern development is the interaction between strict CLFs and control barrier functions (CBFs). In joint safe stabilization, strict compatibility means that the CLF can still be decreased when the state lies on the safety boundary and that the safety condition can be enforced simultaneously. Under the convention $V : X \to \mathbb{R}_+$9, one formulation requires

$\mathcal{K}_\infty$0

and on the boundary $\mathcal{K}_\infty$1,

$\mathcal{K}_\infty$2

Under the opposite boundary orientation used in the converse theorem, the strict controlled-invariance condition appears as $\mathcal{K}_\infty$3 on $\mathcal{K}_\infty$4. The sign change reflects the safe-set convention, not a disagreement about the role of strict inwardness (Liu, 13 Nov 2025, Quartz et al., 15 Sep 2025).

The 2025 converse theorem states that the existence of a strictly compatible pair of CLF and CBF is equivalent to the existence of a single smooth Lyapunov function certifying both asymptotic stability and safety. The resulting control Lyapunov-barrier function (CLBF) satisfies

$\mathcal{K}_\infty$5

and solves a first-order PDE with Dirichlet boundary data,

$\mathcal{K}_\infty$6

That result further states that if such a smooth joint Lyapunov function does not exist, then any pair of CLF/CBF necessarily leads to a conflict and cannot be satisfied simultaneously in a robust sense (Quartz et al., 15 Sep 2025).

Constructive CLBF synthesis is developed in two later papers. One uses a patching formula

$\mathcal{K}_\infty$7

with a smooth bump function $\mathcal{K}_\infty$8 supported on a boundary band $\mathcal{K}_\infty$9, so that $\alpha_1,\alpha_2,\alpha_3$0 in the interior and $\alpha_1,\alpha_2,\alpha_3$1 on and outside the boundary. Another combines a softmax relaxation of a nonsmooth maximum barrier with counterexample-guided half-space refinement before the same kind of smooth patching. Both papers emphasize that strict compatibility is required only on the boundary band, and both report less conservative certified regions than SOS-based compatible CBF-CLF designs in their examples (Liu, 13 Nov 2025, Liu et al., 2 Oct 2025).

A parallel line of work studies compatibility without constructing a single patched function. An SOS-based verification framework derives exact necessary and sufficient conditions for compatibility independent of any nominal controller and uses Farkas’ Lemma to reduce pointwise feasibility of the joint CLF–CBF inequalities to an emptiness condition over dual variables. This nominal-controller-free viewpoint is motivated by the observation that runtime CLF-CBF-QP controllers can fail if compatibility is ignored (Dai et al., 2024).

4. Computational synthesis and verification

Strict CLFs are computationally difficult because their defining condition contains a statewise existential quantifier over controls. Several works recast that difficulty into convex or formally checkable forms.

For polynomial control-affine systems with polyhedral input constraints, one approach reformulates the strict CLF condition

$\alpha_1,\alpha_2,\alpha_3$2

into a contrapositive statement that is universal in $\alpha_1,\alpha_2,\alpha_3$3. Because $\alpha_1,\alpha_2,\alpha_3$4 is linear in $\alpha_1,\alpha_2,\alpha_3$5, checking the inequality for all $\alpha_1,\alpha_2,\alpha_3$6 in a polytope is equivalent to checking it at the polytope vertices. The resulting Positivstellensatz certificate is necessary and sufficient for verification, while an SOS S-procedure gives a convex sufficient condition for synthesis. The same paper uses sequential convex optimization and inscribed ellipsoid maximization to enlarge the certified stabilizable region (Dai et al., 2022).

Compatibility verification with bounded controls has also been reduced to quantifier-free inequalities suitable for satisfiability modulo theories (SMT) solvers. For norm-bounded inputs, one sufficient condition for a boundary barrier inequality is

$\alpha_1,\alpha_2,\alpha_3$7

with analogous expressions for CLFs and for box-constrained controls. The same framework extends Farkas-type duality to the strict compatibility constraints and then uses SMT to verify the resulting formulas on the exact nonlinear safe set. The companion softmax-relaxation paper uses the SMT counterexample to add half-space cuts until the strict barrier condition becomes verifiable (Liu, 13 Nov 2025, Liu et al., 2 Oct 2025).

Switched systems introduce an additional computational issue: strict decrease under arbitrary switching can produce Zeno behavior. The switched-systems synthesis paper therefore introduces non-Zeno CLFs, requiring for each mode a strict descent condition of the form

$\alpha_1,\alpha_2,\alpha_3$8

together with boundedness conditions on $\alpha_1,\alpha_2,\alpha_3$9 and $$\alpha_1(\|x\|) \leq V(x) \leq \alpha_2(\|x\|), \qquad \inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|), \quad \forall x \in X.$$0. Under the associated switching law, the closed loop has guaranteed minimum dwell time, positive invariance of a sublevel region, and asymptotic stability. Template-based polynomial synthesis is then handled by counterexample-guided inductive synthesis (CEGIS) combined with LMI relaxations (Ravanbakhsh et al., 2015).

These methods show that strict CLF synthesis has split into at least three computational regimes: convex polynomial certificates, formal SMT-based verification on exact sets, and iterative witness-based synthesis for quantified problems. A plausible implication is that strictness itself is not the primary computational obstacle; the obstacle is the quantified feasibility geometry induced by strictness.

5. Receding-horizon, high-dimensional, and data-driven regimes

Strict CLFs have been incorporated into receding-horizon control to separate stability from performance. In the Segway study on nonlinear MPC, the CLF decrease condition is imposed as an explicit NMPC constraint rather than being delegated to terminal penalties. Several variants are introduced, including CLF-0, CLF-All, LLS-N, and LLS-All, where the first enforces the strict CLF decrease only at the first control sample and the others impose horizon-wide or level-set constraints such as

$$\alpha_1(\|x\|) \leq V(x) \leq \alpha_2(\|x\|), \qquad \inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|), \quad \forall x \in X.$$1

The paper states that the addition of a prediction horizon provides a performance advantage over CLF-based controllers, while explicit stability constraints remove the need for difficult cost-function and parameter tuning required by traditional NMPC (Grandia et al., 2020).

High-dimensionality motivates different constructions. The exit-time optimal-control approach builds a global CLF by concatenating a local CLF with an exit-time value function defined relative to a sublevel set of that local CLF. Outside the local region, the value function solves an exit-time HJB equation in viscosity sense; its values can be computed at selected states through finite-dimensional optimal control problems, which the paper presents as a curse-of-dimensionality-free approach, while also cautioning that the curse of complexity can remain severe in practice (Yegorov et al., 2019). The decomposition-based CLVF framework attacks dimensionality structurally by solving lower-dimensional HJ problems on subsystems and reconstructing either an exact max-type CLVF or a sum-type Lipschitz strict CLF, depending on admissible-control-set intersections (Gong et al., 2024).

In data-driven Koopman-based bilinear systems, the situation is more restrictive. For lifted bilinear dynamics

$$\alpha_1(\|x\|) \leq V(x) \leq \alpha_2(\|x\|), \qquad \inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|), \quad \forall x \in X.$$2

quadratic candidates $$\alpha_1(\|x\|) \leq V(x) \leq \alpha_2(\|x\|), \qquad \inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|), \quad \forall x \in X.$$3 are characterized exactly by a QCQP over the set where the control terms vanish. A semidefinite relaxation yields a sufficient condition, and for single-input systems the relaxation is exact. The main theorem states that, under SDP exactness, a quadratic CLF exists if and only if the bilinear system is stabilizable by a constant input. The paper presents this as evidence that quadratic CLFs are highly restrictive in high-dimensional Koopman models (Hanna et al., 10 Apr 2026).

Taken together, these works show that strict CLFs remain central in advanced regimes, but the ambient representation matters. In decomposition and exit-time constructions, strictness can be preserved while avoiding global gridding. In Koopman liftings, by contrast, even the existence of a simple quadratic strict CLF can collapse to an unexpectedly strong structural property.

6. Specialized constructions, applications, and recurrent limitations

Several papers develop strict CLFs that are strongly shaped by application structure. For discrete-time quantum systems with QND measurements, a systematic method based on graph theory and inversion of Laplacian matrices constructs a strict control-Lyapunov function of the form $$\alpha_1(\|x\|) \leq V(x) \leq \alpha_2(\|x\|), \qquad \inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|), \quad \forall x \in X.$$4 and a feedback law that maximizes it after each measurement; the closed-loop state preparation becomes deterministic and globally stabilizing toward a chosen target state (Amini et al., 2011). For predator–prey dynamics with positive states and positive controls, explicit non-separable Volterra-style strict CLFs are constructed, including

$$\alpha_1(\|x\|) \leq V(x) \leq \alpha_2(\|x\|), \qquad \inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|), \quad \forall x \in X.$$5

with negative-definite derivative under positive feedback laws such as $$\alpha_1(\|x\|) \leq V(x) \leq \alpha_2(\|x\|), \qquad \inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|), \quad \forall x \in X.$$6 or $$\alpha_1(\|x\|) \leq V(x) \leq \alpha_2(\|x\|), \qquad \inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|), \quad \forall x \in X.$$7 (Krstic, 25 Feb 2026). For the unicycle in polar coordinates, a modular framework combines the distance CLF $$\alpha_1(\|x\|) \leq V(x) \leq \alpha_2(\|x\|), \qquad \inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|), \quad \forall x \in X.$$8 with angular strict CLFs $$\alpha_1(\|x\|) \leq V(x) \leq \alpha_2(\|x\|), \qquad \inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|), \quad \forall x \in X.$$9 through composite functions $\inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|)$0, yielding global asymptotic or almost-global barrier-based stabilization and bidirectional parking maneuvers (Todorovski et al., 29 Sep 2025).

Strict CLFs also appear in learning-based control. The SAC-CLF framework constructs a task-specific quadratic CLF from local linearization and an LQR Riccati equation, then enforces

$\inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|)$1

inside a QP layer, with adaptive tuning of $\inf_{u \in U} \dot{V}(x,u) \leq -\alpha_3(\|x\|)$2 to handle unmodeled dynamics and a smoothness term in the objective. The paper presents this as a way to maintain robustness while filtering RL actions through a stability constraint (Chen et al., 18 Jan 2025). A different learning paper treats Lyapunov conditions as inductive biases, using a neural CLF architecture that enforces positivity and zero at equilibrium by construction and optimizes primarily the strict derivative condition; it reports higher convergence rate and larger region of attraction than previous neural CLF learners (Lu et al., 3 Nov 2025).

At the same time, the literature repeatedly identifies limitations of optimization-based safety–stability combinations. One study proves that undesirable equilibrium points occur for most systems in the CLF-CBF-QP framework with multiple CBFs and that their stability depends on CLF and CBF geometrical properties; it introduces compatibility as the condition that the CLF minimum be the only stable equilibrium (Reis et al., 2024). Another shows, through a tumor-dynamics example, that the standard approach of imposing both CLF and CBF inequalities in one QP can lead to slow convergence and oscillating inputs; it proposes instead a QP that penalizes deviation from Sontag’s formula under only a CBF constraint and adds a hybrid law near the equilibrium (Gemert et al., 2024).

The resulting picture is not that strict CLFs are uniformly conservative, nor that they are universally easy to integrate with constraints. Rather, strict CLFs provide the most direct route to quantified convergence and constructive feedback, but their effectiveness depends on compatibility with safety geometry, control bounds, representation choice, and computational architecture. The contrast with flexible CLFs is instructive: flexible formulations reduce conservativeness by allowing online variation of the Lyapunov cone, whereas strict CLFs preserve a fixed sign structure that is stronger analytically but more demanding computationally and geometrically (Lazar, 2010).