Time-Barrier Predefined-Time Stability

• Time-barrier predefined-time stability is a nonlinear system property that ensures state convergence to equilibrium within a designer-specified deadline by embedding time-varying singularities in the control law.
• It utilizes a nonautonomous Lyapunov framework with explicit barrier conditions, which forces convergence independently of initial conditions while enabling tunable rates through parameters like α and β.
• This approach underpins robust control, observer design, and distributed system stabilization, though it may require managing high gains near the convergence deadline.

Time-barrier predefined-time stability is a rigorous nonlinear system property that enforces state convergence to equilibrium within an exact designer-specified time, independent of initial conditions, by embedding time-barrier mechanisms—such as explicit time-varying singularities—directly into the Lyapunov framework or closed-loop dynamics. In contrast to classical fixed-time or predefined-time approaches based on autonomous Lyapunov decay inequalities, time-barrier schemes achieve a hard convergence deadline via the intrinsic divergence of a time-dependent barrier as the prescribed time limit is approached, rather than relying solely on state-dependent gain scaling. This concept distinguishes itself structurally and functionally from traditional Lyapunov-based predefined-time stability theorems, offering transparent deadlines, tunable convergence, and new avenues for robust nonlinear control, estimation, and safety-critical system design.

1. Foundational Concepts and Formal Definition

Time-barrier predefined-time stability was formalized by Bingöl (Bingöl, 28 Dec 2025) and is defined for systems

$\dot{x} = f(x,t),\quad x(0) = x_0,\quad f(0,t) = 0$

with a user-specified convergence deadline $T_c>0$. The equilibrium $x=0$ is time-barrier predefined-time stable if, for all $x_0$, the solution $x(t;x_0)$ exists on $[0,T_c)$ and

$\lim_{t\to T_c^-} x(t;x_0) = 0$

—no solution can survive away from equilibrium past $T_c$.

Unlike autonomous Lyapunov techniques enforcing

$$\dot{V}(x) \le -\Phi(V(x)), \qquad \int_0^{V(x_0)} \frac{dV}{\Phi(V)} \le T_c$$

(time $T_c$ only serves as an upper bound dependent on initial state or parameter gain), time-barrier schemes force convergence structurally by introducing nonautonomous terms that blow up as $t\to T_c$. Classical methods cannot generally replicate this state-independent "barrier divergence" mechanism.

2. Time-Barrier Lyapunov Framework and Sufficient Conditions

The central mechanism utilizes a time-variant singularity: $b(t) = (T_c - t)^\beta, \quad \beta > 0$ with the Lyapunov function scaled as

$W(t) = \frac{V(x(t), t)}{(T_c - t)^{\beta}}$

and dissipation inequality

$$\dot{V}(x,t) \le -\beta\frac{V(x,t)}{T_c-t} - q V(x,t)^\alpha, \quad q > 0, \ \alpha \in (0,1)$$

subject to the structural barrier condition

$\beta(1-\alpha)\ge 1$

This guarantees, via barrier divergence,

$$\int_0^{T_c} (T_c - t)^{-\beta(1-\alpha)} dt = +\infty$$

and hence all trajectories reach the equilibrium within $T_c$, independently of initial conditions.

Unlike autonomous decay, the barrier-enforced rate is state-independent and dictated solely by the prescribed time and singularity exponent, providing an explicit, nonconservative deadline.

3. Structural Distinction vs. Autonomous Predefined-Time Stability

The principal difference is that conventional predefined-time stability is realized via integral upper bounds stemming from state-dependent Lyapunov decay, which may be overly conservative or dependent on large initial states (Xiao et al., 2024, Jiménez-Rodríguez et al., 2019). In time-barrier designs, convergence is controlled by a direct restriction on the remaining time. Autonomous formulations cannot, in general, produce the hard deadline achieved by explicit time-dependent barriers, due to the lack of state-independent divergence.

Table: Comparison of Autonomous vs. Time-Barrier Predefined-Time Stability

Feature	Autonomous Lyapunov (Xiao et al., 2024)	Time-Barrier (Bingöl, 28 Dec 2025)
Mechanism	State-dependent decay $\Phi(V)$	Time-dependent singularity
Deadline guarantee	Upper bound, adjustable	Hard deadline, exact
Dependence on initial state	Typically implicit, conservative	Removed (state-independent)
Implementational caveats	May need large gains	Barrier singularity near $T_c$

4. Design Methodologies and Implementation

The barrier Lyapunov approach requires selecting exponents $\alpha,\beta$ satisfying barrier conditions and enforcing the singular Lyapunov decay. Control or estimation algorithms must embed the time-dependent singularity in the closed-loop derivative, typically with a term proportional to $1/(T_c-t)$. For example, the scalar nonlinear system: $$\dot{x} = -\beta \frac{x}{T_c - t} - q|x|^\alpha \operatorname{sign}(x)$$ ensures convergence in time $T_c$, even for arbitrary initial $x(0)$ (Bingöl, 28 Dec 2025).

For higher-order systems and robust control under uncertain dynamics or disturbances, time-barrier approaches can be integrated into sliding-mode frameworks, nonlinear observers, or distributed controllers (Xiao et al., 2024, Kokolakis et al., 19 Nov 2025).

Practical design guidelines:

• Select $\alpha \in (0,1)$, $\beta>0$ so that $\beta(1-\alpha) \ge 1$
• Implement singular decay term in control law or system dynamics
• Ensure boundedness or manage high-gain behavior near $t \to T_c$
• For sampled-data systems or differentiators, similar time-scaling and implicit Euler discretizations can preserve predefined-time barriers (Aldana-López et al., 2021, Jiménez-Rodríguez et al., 2020)

5. Cascades, Interconnections, and Extensions

Recent results extend time-barrier stability to interconnected and cascaded systems (Krishnamurthy et al., 2024), leveraging time-varying blow-up functions for stabilization within a prescribed interval: $\varphi(t) = c(T/(T-t))^k, \quad k \ge 1$ Lyapunov certificates constructed with such blow-up rates ensure prescribed-time exponential convergence for each subsystem. Feedback and cascade interconnections with proper small-gain conditions retain the overall time-barrier stability.

Further, robust physics-informed machine learning approaches utilize time-barrier Lyapunov and Hamilton-Jacobi-Isaacs constraints to guarantee safe stabilization in adversarial settings, with joint enforcement of safety and hard convergence deadline (Kokolakis et al., 19 Nov 2025).

6. Strengths, Limitations, and Practical Aspects

Advantages:

• Guarantees convergence by the exact specified time $T_c$, regardless of initial conditions
• Transparent tuning: only the barrier exponents link $T_c$ to rate constants
• Compatible with nonlinear, stochastic, or distributed contexts

Limitations:

• Control/observer gains may become extremely large near the time deadline $T_c$, requiring careful saturation or boundary-layer design to suppress noise amplification (Bingöl, 28 Dec 2025, Aldana-López et al., 2023)
• Extensions to output-feedback, digital implementation, or unbounded disturbances require further research

Table: Implementation Caveats

Issue	Practical Impact
High gain as $t \to T_c$	Noise amplification, actuator limits
Robustness to uncertainty	Requires barrier Lyapunov adaptation
Sampled-data discretization	Implicit Euler preserves time-barrier

7. Applications and Research Directions

Time-barrier predefined-time stability is central to safety-critical control, real-time estimation, optimal nonlinear stabilization under adversarial and uncertain environments, and distributed systems requiring strict deadlines. Robust sliding-mode controllers, Lyapunov-based observers, differentiators with tight settling-time bounds, and networked controllers for interconnected nonlinear systems are among the primary applications (Xiao et al., 2024, Kokolakis et al., 19 Nov 2025, Sui et al., 10 Apr 2025, Aldana-López et al., 2023).

Current research focuses on:

• Construction of barrier Lyapunov functions for complex nonlinear systems amenable to machine learning-based synthesis
• Integration with stochastic and sampled-data systems ensuring mean or almost-sure convergence within time-barrier bounds (Zhang et al., 2022)
• Extensions to distributed, multi-agent, and networked control with joint prescribed-time stability guarantees

Time-barrier predefined-time stability thus constitutes a new paradigm in nonlinear control theory, offering structurally enforced, initial-free hard convergence deadlines with wide applicability in safety, robotics, estimation, and cyber-physical systems.