Nonautonomous Lyapunov Mechanism

• Nonautonomous Lyapunov mechanism is a framework for analyzing the stability of time-dependent, switched, and nonsmooth systems using generalized derivatives and set-valued methods.
• It deploys Filippov and Krasovskii regularizations along with Clarke gradients to handle system discontinuities and switching behaviors.
• The mechanism generalizes classical Lyapunov and LaSalle–Yoshizawa theorems to ensure robust stability in adaptive control, delay systems, and hybrid models.

A nonautonomous Lyapunov mechanism refers to a rigorous framework for stability and invariance analysis in dynamical systems whose vector fields depend explicitly on time and may exhibit discontinuities, switchings, or nonsmoothness. Unlike autonomous Lyapunov theory, which leverages time-invariant vector fields, the nonautonomous extension accommodates explicit time-dependence and broader classes of systems, including switched, piecewise continuous, state-dependent, and delayed differential inclusions. Central to this theory is the definition of generalized derivatives (e.g., Clarke gradients), the deployment of set-valued dynamics (Filippov or Krasovskii regularizations), and the formulation of Lyapunov inequalities ensuring robustness under time-variance, switching, and nonsmoothness.

1. General Formulation and Set-Valued Regularization

Modern nonautonomous Lyapunov mechanisms are typically set within the context of time-varying switched systems or differential inclusions: $\dot{x}(t) = f_{\rho(x(t), t)}(x(t), t)$ where the switching law $$\rho: \mathbb{R}^n \times [t_0, \infty) \to \mathcal{N}^0 \subset \mathbb{N}$$ picks among a collection of vector fields, $\{f_\sigma\}_{\sigma \in \mathcal{N}^0}$, each measurable in time and locally bounded in $x$.

To treat discontinuities and nonsmoothness, the mechanism applies set-valued extensions:

• Filippov regularization:

$$\mathcal{F}[g](x,t) := \bigcap_{\delta>0} \bigcap_{\mu(N)=0} \mathrm{co}\{g(y,t) \mid y \in B(x,\delta)\setminus N\}$$

• Krasovskii regularization:

$$\mathcal{K}[g](x,t) := \bigcap_{\delta>0} \overline{\mathrm{co}}\{g(y,t)\mid y\in B(x,\delta)\}$$

where $\mu$ is Lebesgue measure and $\mathrm{co}$ its convex hull.

These constructions enable the deployment of generalized solution concepts for possibly discontinuous $f$, with system trajectories described by differential inclusions $\,\dot{x}(t) \in \mathcal{F}(x, t)$ or $\,\mathcal{K}(x, t)$ (Kamalapurkar et al., 2016).

2. Generalized Lyapunov Function and Derivative

A central object is a common candidate Lyapunov function: $$V: \mathbb{R}^n \times [t_0, \infty) \to \mathbb{R}$$ which is locally Lipschitz and positive-definite. Its generalized time-derivative is specified as follows: $$\dot{\overline{V}}_F(x, t) := \max_{p \in \partial V(x, t)} \max_{q \in F(x, t)} \langle p, [q; 1] \rangle$$ where $\partial V(x,t)$ denotes the Clarke gradient at $(x, t)$, and $F$ is a compact-valued set-valued map, such as $\mathcal{F}$ or $\mathcal{K}$. For regular (Clarke regular) $V$, a minimal variant is also defined.

A function $V$ is a non-strict Lyapunov function for the inclusion $\,\dot{x} \in F(x, t)$ if there exist continuous, positive-definite functions $\underline{W}(x)$, $\overline{W}(x)$, and a continuous positive-semidefinite $W(x)$ such that: $$\underline{W}(x) \leq V(x, t) \leq \overline{W}(x), \qquad \dot{\overline{V}}_F(x, t) \leq -W(x)$$ (Kamalapurkar et al., 2016, Fischer et al., 2012).

3. Generalized LaSalle–Yoshizawa Principle

A core result is the invariance-like (generalized LaSalle–Yoshizawa) theorem for switched, possibly nonsmooth, nonautonomous systems:

• If each Filippov (or Krasovskii) subsystem admits the same non-strict Lyapunov function $V$ with the bounds above, and the switching is locally finite, then every solution $x(t)$ to the regularized inclusion is bounded, complete, and

$\lim_{t\to\infty} W(x(t)) = 0$

If $W(x) = 0$ only at $x=0$, then $x(t)\to 0$ (Kamalapurkar et al., 2016, Fischer et al., 2012). The proof leverages that the regularized switched inclusion is contained in the closed convex hull of the regularizations of the subsystems, and convexity extends the Lyapunov monotonicity property.

This principle generalizes classical smooth, autonomous LaSalle–Yoshizawa theorems to encompass time-variance, nonsmoothness, set-valued dynamics, and arbitrary switching, provided only a "mild" local finiteness hypothesis on the switching.

4. Specialized Mechanisms and Variants

Numerous specialized nonautonomous Lyapunov mechanisms have been developed:

• Clarke-Razumikhin for delay systems: Lyapunov–Razumikhin functions coupled with pointwise (nonuniform, nonautonomous) delay inequalities and auxiliary ODE construction establish stability for state-dependent delay equations (Humphries et al., 2015).
• Time-barrier Lyapunov condition: Enforces hard, nonautonomous, predefined-time deadlines by embedding divergence in a time-dependent term, $-\beta V/(T_c-t)$, in the Lyapunov inequality, distinct from state-only conditions in classical fixed-time methods (Bingöl, 28 Dec 2025).
• Divergence-based mechanisms: Encodes stability and feedback design via integral and differential continuity-equation inequalities for Lyapunov densities, coupling the flow's divergence to the Lyapunov decay (Furtat, 2020).
• Canonical-form methodologies (Poincaré approach): Employs a canonizing transformation and bundle topology to construct Lyapunov functions for broad classes of parametric inclusions, reducing the nonautonomous analysis to a standard quadratic form in suitable coordinates (Sparavalo, 2014).

5. Applications: Switched Adaptive Control and Nonlinear Models

The nonautonomous Lyapunov mechanism accommodates:

• Adaptive switched control: Generic switched formulations with unknown parameters, adaptive feedback, and externally bounded disturbances are unified via a common, nonsmooth Lyapunov candidate (often quadratic in error coordinates). The analysis proceeds through Filippov/Krasovskii calculus establishing derivative bounds and invariance (Kamalapurkar et al., 2016).
• Nonsmooth systems: Sliding-mode, relay, or variable-structure controllers with state or time-dependent switching admit analysis through nonautonomous, set-valued Lyapunov derivatives (Fischer et al., 2012).
• Nonautonomous epidemiological models: Stability of time-dependent, treatment-affected compartmental systems is established by global Lyapunov functions using nonautonomous inequalities that bind the time-varying reproduction number and ensure convergence to disease-free or endemic equilibria (Alsammani, 2023).

6. Structural Features, Limitations, and Significance

The nonautonomous Lyapunov mechanism is characterized by:

• Universality: Robustness to nonsmoothness, arbitrary time-variance, switching, and discontinuity, given only a common (possibly nondifferentiable) Lyapunov function and suitable set-valued derivative bounds.
• Invariance-like results: Asymptotic attractivity of invariant sets or equilibria, forward invariance of sublevel sets, and convergence without requiring strict dissipation.
• Flexibility: Unification of analysis across deterministic, switched, stochastic, delayed, and parameter-dependent systems, embedding control and estimation as special cases.

A plausible implication is that these mechanisms, by relying on generalized derivatives and set-valued inclusions, provide the essential tools for modern stability and invariance analysis in broad classes of engineering and physical systems subject to hybrid, time-varying, and uncertain environments.

7. Illustrative Table: Main Nonautonomous Lyapunov Mechanisms

Paper / Author	System Class	Key Technical Mechanism
(Kamalapurkar et al., 2016) Kamalapurkar et al.	Nonsmooth switched systems	Filippov/Krasovskii regularization, Clarke derivative, generalized LaSalle–Yoshizawa
(Fischer et al., 2012) Andrews & Orlov	Discontinuous nonautonomous	Filippov inclusion, Clarke gradient, set-valued Lyapunov derivative
(Bingöl, 28 Dec 2025) Guseinov & Mironchenko	Nonlinear, hard deadline	Time-barrier Lyapunov function, divergent nonautonomous term
(Humphries et al., 2015) Humphries & Magpantay	State-dependent delay DDEs	Nonautonomous Lyapunov–Razumikhin, auxiliary ODE method
(Furtat, 2020) Furtat	General nonautonomous systems	Divergence/continuity equation, Lyapunov density
(Sparavalo, 2014) Sparavalo	Parametric differential inclusions	Poincaré strategy, canonical form reduction

These paradigms demonstrate the breadth and power of nonautonomous Lyapunov mechanisms for guaranteeing robust stability, attractivity, and convergence in time-varying, hybrid, and nonsmooth dynamical systems.