Time-Varying Barrier Lyapunov Functions

• TVBLF is a time-varying Lyapunov function that remains finite inside a moving safe set and diverges as the state nears the boundary.
• It is used in adaptive and robust control of Euler–Lagrange systems, MRAC, and robotic manipulators to enforce moving state or error envelopes.
• TVBLF methods offer offline feasibility certificates and guarantee forward invariance, ensuring system stability and constraint satisfaction.

A Time-Varying Barrier Lyapunov Function (TVBLF) is a scalar Lyapunov-type function defined on a time-varying safe set and constructed so that it becomes unbounded as the state approaches the moving constraint boundary. In the formulation used for constrained adaptive control, if $$h:\mathbb{R}^n\times\mathbb{R}_{\ge 0}\to\mathbb{R}$$ defines $\Omega_x(t)=\{x\in\mathbb{R}^n\mid h(x,t)>0\}$, then a TVBLF is positive definite in $x$, continuously differentiable in $x$ and $t$, and satisfies $V(x,t)\to\infty$ as $x$ approaches the boundary of $\Omega_x(t)$ (Ghosh et al., 9 Mar 2026). This mechanism is used to enforce moving state envelopes in uncertain Euler–Lagrange systems, model reference adaptive control, and robot manipulators, and it is closely related to broader time-varying barrier constructions in converse safety theory and predefined-time stability (Ghosh et al., 9 Mar 2026, Ghosh et al., 29 Aug 2025, Gupta et al., 29 May 2026, Maghenem et al., 2022, Bingöl, 28 Dec 2025).

1. Formal definition and canonical constructions

In the Euler–Lagrange formulation, the defining feature of a TVBLF is divergence at the boundary of a time-varying admissible region. For norm-bounded constraints of the form $\|x\|<\phi_x(t)$, the logarithmic barrier used in constrained adaptive control is

$V_1=\log\frac{\phi_x(t)^2}{\phi_x(t)^2-x^{\top}x},$

which tends to $\Omega_x(t)=\{x\in\mathbb{R}^n\mid h(x,t)>0\}$0 as $\Omega_x(t)=\{x\in\mathbb{R}^n\mid h(x,t)>0\}$1 (Ghosh et al., 9 Mar 2026). Because $\Omega_x(t)=\{x\in\mathbb{R}^n\mid h(x,t)>0\}$2 is explicitly time-varying, the barrier wall itself moves, so the admissible set may expand or shrink over time.

A closely related construction appears in constrained MRAC, where the tracking-error barrier is written in a weighted quadratic form,

$\Omega_x(t)=\{x\in\mathbb{R}^n\mid h(x,t)>0\}$3

on the set

$\Omega_x(t)=\{x\in\mathbb{R}^n\mid h(x,t)>0\}$4

As $\Omega_x(t)=\{x\in\mathbb{R}^n\mid h(x,t)>0\}$5, the denominator vanishes and the barrier diverges (Ghosh et al., 29 Aug 2025).

In adaptive artificial time-delay control for Euler–Lagrange robots, the barrier is constructed componentwise for both position and velocity errors: $\Omega_x(t)=\{x\in\mathbb{R}^n\mid h(x,t)>0\}$6 Here the logarithmic terms again blow up as $\Omega_x(t)=\{x\in\mathbb{R}^n\mid h(x,t)>0\}$7 or $\Omega_x(t)=\{x\in\mathbb{R}^n\mid h(x,t)>0\}$8, embedding the constraints directly into the Lyapunov candidate (Gupta et al., 29 May 2026).

These constructions share the same structural principle: the TVBLF is finite strictly inside the moving admissible set and singular at its boundary. That singularity is the core analytical device used to prove forward invariance of the constrained region.

2. Constraint envelopes and error-coordinate reformulation

A standard design step in TVBLF-based control is to convert plant-level state constraints into constraints on tracking errors. For uncertain Euler–Lagrange systems,

$\Omega_x(t)=\{x\in\mathbb{R}^n\mid h(x,t)>0\}$9

the prescribed constraints are

$x$0

together with

$x$1

If the desired trajectory satisfies $x$2 and $x$3, then the plant constraints induce the error envelopes

$x$4

with $x$5 and $x$6 (Ghosh et al., 9 Mar 2026).

The same paper introduces the filtered error

$x$7

and proves that if $x$8 remains inside a time-varying tube $x$9, and if

$x$0

together with

$x$1

then automatically

$x$2

This establishes a constraint-conversion mechanism: enforcing a TVBLF on $x$3 implies the original state constraints (Ghosh et al., 9 Mar 2026).

In constrained MRAC, the same pattern appears in linear form. For $x$4 with reference model $x$5, the state constraint $x$6 is converted into an error constraint

$x$7

where $x$8 and $x$9 (Ghosh et al., 29 Aug 2025).

In robotic TVBLF designs, the envelopes themselves are often prescribed-performance functions. One reported choice is the exponentially decaying form

$t$0

which enforces

$t$1

This is explicitly described as a shrinking-funnel constraint design (Gupta et al., 29 May 2026).

3. Controller synthesis patterns

TVBLFs are typically embedded into adaptive or robust controllers through gains that become large as the state approaches the moving boundary. In uncertain Euler–Lagrange systems, the auxiliary control law is

$t$2

followed by the saturated input

$t$3

The saturation guarantees $t$4, while the projection-based adaptive law confines the parameter estimate to a known compact set (Ghosh et al., 9 Mar 2026).

Constrained MRAC uses an analogous decomposition. The actual control is a saturation of the auxiliary input

$t$5

with

$t$6

The term $t$7 appears specifically because the safe set is time-varying, and the adaptive update law contains the barrier denominator $t$8, which increases adaptation intensity near the boundary (Ghosh et al., 29 Aug 2025).

For Euler–Lagrange robots with adaptive time-delay estimation, the nominal control takes the form

$t$9

with state-dependent diagonal gains

$V(x,t)\to\infty$0

These gains become large near the constraints and create repulsive feedback. The final torque law combines time-delay estimation, the BLF nominal term, and an adaptive robust term $V(x,t)\to\infty$1 (Gupta et al., 29 May 2026).

A related but distinct design appears in perturbed integrator chains. There, a prescribed-time reaching phase is followed by a barrier-function phase with robustifying gain

$V(x,t)\to\infty$2

which blows up as $V(x,t)\to\infty$3 and keeps the state inside the invariant set $V(x,t)\to\infty$4. The source explicitly states that this second phase is “very much in the TVBLF spirit,” although the paper does not use the exact term throughout (Estrada et al., 2024).

4. Feasibility, invariance, and boundedness

A notable feature of several TVBLF-based designs is the presence of an offline verifiable feasibility condition. For the Euler–Lagrange adaptive design with state and input envelopes, the condition is

$V(x,t)\to\infty$5

which certifies that the input bound is large enough to sustain the worst-case control effort required to keep the filtered error inside its time-varying barrier envelope (Ghosh et al., 9 Mar 2026).

The constrained MRAC formulation provides an analogous certificate: $V(x,t)\to\infty$6 or equivalently

$V(x,t)\to\infty$7

This is explicitly described as a certificate of existence of a feasible constrained adaptive policy and as an offline test that avoids online optimization or feasibility monitoring (Ghosh et al., 29 Aug 2025).

The proof mechanism is similar across these formulations. A Lyapunov candidate combines a barrier term with adaptive-parameter error terms. Its derivative is shown to be nonpositive or ultimately bounded under the controller and adaptive law. If the trajectory were to reach the moving constraint boundary, the TVBLF term would diverge to $V(x,t)\to\infty$8, contradicting boundedness of the Lyapunov function. This yields forward invariance of the admissible set and, consequently, strict constraint satisfaction for all $V(x,t)\to\infty$9 (Ghosh et al., 9 Mar 2026, Ghosh et al., 29 Aug 2025, Gupta et al., 29 May 2026).

The reported guarantees are strong. In the Euler–Lagrange adaptive design, the main theorem states closed-loop stability, $x$0, $x$1, $x$2, and boundedness of all closed-loop signals. If the saturation error vanishes after some time $x$3, then asymptotic tracking follows: $x$4 A bounded-disturbance extension preserves the theorem under a modified feasibility condition with $x$5 (Ghosh et al., 9 Mar 2026).

For adaptive TDE with barrier constraints, the theorem states

$x$6

and derives a bound of the form

$x$7

which implies boundedness of all closed-loop signals (Gupta et al., 29 May 2026).

5. Relation to standard BLFs, time-varying barrier functions, and control barrier functions

The most immediate comparison is with the standard barrier Lyapunov function. In the constrained Euler–Lagrange formulation, the distinction is explicit: a standard BLF protects a fixed boundary, whereas a TVBLF protects a boundary that changes with time. The central advantage is therefore direct accommodation of expanding or shrinking safe tubes without online optimization (Ghosh et al., 9 Mar 2026).

The converse-safety literature broadens this viewpoint. For differential inclusions $x$8, time-varying barrier functions are shown to be necessary as well as sufficient for safety under mild regularity conditions, while autonomous continuous barriers can fail even for a safe system with a smooth right-hand side. The proposed converse construction is

$x$9

where $\Omega_x(t)$0 is a finite-horizon reachable-set map. The associated safe set is

$\Omega_x(t)$1

This result places time variation at the level of safety certification itself, not only controller synthesis (Maghenem et al., 2022).

A different theoretical extension appears in predefined-time stability. The nonautonomous dissipation condition

$\Omega_x(t)$2

uses a time-singular barrier to force convergence before the prescribed deadline. The barrier-scaled transformation

$\Omega_x(t)$3

is explicitly described as closely related in spirit to TVBLFs, but the role is different: the barrier restricts remaining time rather than state magnitude within a moving envelope (Bingöl, 28 Dec 2025).

Time-varying control barrier functions form another adjacent literature. Robust time-varying CBFs define a safe set $\Omega_x(t)$4 and enforce forward invariance through an SOCP-based safety filter under sector-bounded input nonlinearities. The source explicitly states that this framework is conceptually close to TVBLFs but is built for set invariance-based safety filtering, not performance-tracking-based Lyapunov regulation (Chun et al., 12 Nov 2025).

A common misconception is therefore that all time-varying barrier methods are interchangeable. The reported literature separates at least three roles: TVBLFs for constrained tracking and transient shaping, time-varying barrier functions for converse safety certification, and time-varying CBFs for safety filtering (Maghenem et al., 2022, Chun et al., 12 Nov 2025).

6. Experimental validation and reported applications

TVBLF-based and TVBLF-style methods have been validated on several hardware platforms. In uncertain Euler–Lagrange control with time-varying state and input constraints, real-time experiments were conducted on a Quanser 2-DoF helicopter with generalized coordinates pitch $\Omega_x(t)$5 and yaw $\Omega_x(t)$6. The reported outcomes are that pitch and yaw track the desired trajectories, the position and velocity errors remain inside their prescribed envelopes, and the control input stays within the time-varying input limit (Ghosh et al., 9 Mar 2026).

In adaptive artificial time-delay control with barrier constraints, experiments were performed on a 5-DoF UFactory xArm-5 manipulator running on an NVIDIA Jetson AGX Xavier with a custom end-effector. The task was constrained drawing and erasing: drawing concentric semicircles of radii 20 cm, 22 cm, and 24 cm, then erasing the middle semicircle while staying inside a narrow corridor between the inner and outer paths. Desired trajectories were generated using Ruckig. The reported findings are that the proposed controller tracked the semicircular paths more accurately, preserved the inner and outer semicircles during erasing, kept both position and velocity errors within prescribed bounds throughout, and achieved the lowest RMS errors across all joints in both position and velocity relative to the ABLF and ATDC baselines (Gupta et al., 29 May 2026).

The perturbed-integrator-chain study provides hardware evidence for the broader barrier-preserving principle on a Furuta pendulum. There, changing the prescribed time $\Omega_x(t)$7 changes how fast the state is brought to the target vicinity, while changing $\Omega_x(t)$8 changes the size of the invariant neighborhood. The reported figures show $\Omega_x(t)$9 entering the $\|x\|<\phi_x(t)$0 region before $\|x\|<\phi_x(t)$1 and then remaining below $\|x\|<\phi_x(t)$2, which the source identifies as the experimental signature of a barrier-function-based invariant set (Estrada et al., 2024).

Taken together, these studies show a consistent empirical pattern: time-varying barrier constructions are being used not only as proof devices but as implementable mechanisms for enforcing moving error or safety envelopes in real time. A plausible implication is that TVBLF methodology is increasingly serving as an alternative to online optimization when the constraints can be encoded directly into Lyapunov-barrier structure (Ghosh et al., 9 Mar 2026, Ghosh et al., 29 Aug 2025).