Unified Lyapunov Framework

• Unified Lyapunov framework is a comprehensive method that constructs energy-like functions to certify stability, convergence, and performance in diverse dynamical systems.
• It systematically integrates composite, piecewise, categorical, and infinite-dimensional approaches to overcome challenges in nonlinear, hybrid, and uncertain environments.
• The framework incorporates data-driven techniques like neural networks and Koopman operator methods to provide robust and verifiable Lyapunov certificates.

A unified Lyapunov framework encompasses a broad class of mathematical constructions and analysis techniques that systematize stability, convergence, and qualitative behavior in dynamical systems across nonlinear control, optimization, hybrid dynamics, stochastic processes, and infinite-dimensional analysis. Modern research—drawing on control theory, variational analysis, ergodic theory, and spectral methods—has extended classical Lyapunov approaches into rigorous, algorithmically tractable frameworks that capture distinct notions of stability and address technical challenges posed by discontinuities, switching, uncertainty, and high-dimensionality.

1. Foundational Principles of Unified Lyapunov Theory

At the core of the unified Lyapunov approach is the systematic construction of scalar (or vector-valued) functions whose evolution along the trajectories of a dynamical system directly encodes stability properties. The underlying principles are:

• Monotonic Decrease Condition: A Lyapunov function $V(x)$ serves as an energy-like certificate, satisfying positivity ($V(x)=0$ at equilibrium, $V(x)>0$ elsewhere) and strictly decreasing along solutions: $\dot V(x)\le 0$.
• Composite and Piecewise Structures: Modern frameworks exploit composite Lyapunov functions or piecewise-smooth constructions to handle switching behavior, nonsmooth dynamics, and convergence to sets.
• Parametric and Functional Generality: Lyapunov functions are parameterized over general dictionaries (polynomial, neural, Koopman, or indicator functions), enabling systematic synthesis across application domains.
• Invariant and Attracting Sets: The function decreases strictly except possibly on the invariant set (equilibrium, limit set, or critical manifold), which becomes the attractor under suitable regularity and compactness conditions.

These principles are unified by the requirement that the Lyapunov decrease is algorithmically or constructively imposed, facilitating both analytical proofs and data-driven verification (Zhou et al., 3 Dec 2024, Zhou et al., 2022, Lu et al., 2015).

2. Frameworks for Nonlinear and Hybrid Systems

2.1 Piecewise Smooth Control-Lyapunov Functions for Switched Systems

For general switched nonlinear systems: $$\dot x(t) = f_{\sigma(t)}(x(t)),\quad \sigma(t)\in\{1,\dots,M\}$$ the piecewise smooth control-Lyapunov function (PSCLF) approach provides a unifying structure for stabilization under arbitrary switching, including the influence of sliding motions (Lu et al., 2015). A PSCLF is continuous, radially unbounded, and $C^1$ on each cell of a finite open partition, with the key condition: $$\min_{i}\ D V(x;f_i(x)) \le -W(x) \quad \forall x\in\mathbb R^n$$ where $D V(x;\xi)$ denotes the directional derivative.

The constructive switching law $\nu(x)$ selects at each point the index $i$ minimizing the directional derivative, with explicit treatment of boundary and sliding phenomena via Filippov solutions and boundary index sets. The main theorems ensure global asymptotic stability, including sliding, under checkable boundary conditions.

Quadratic and piecewise-quadratic PSCLFs recover and strengthen classical linear switched system results, with boundary-matching and S-procedure constraints implemented as bilinear matrix inequalities (BMIs). Composite constructions (pointwise minimum, maximum, or convex hulls of $C^1$ functions) are systematically accommodated without conservative ad hoc boundary conditions.

2.2 Symbolic Memory and Multiple Lyapunov Frameworks for Hybrid Dynamics

Hybrid and constrained switched systems parameterized over symbolic sequences or sofic shifts are characterized via sequence- or block-dependent single Lyapunov functions $V(x,w)$, encoding memory or anticipation of past/future symbols (Rossa et al., 2023). Classical multiple Lyapunov criteria for individual subsystems are shown to be equivalent to a single sequence-dependent functional, whose decrease condition over sofic coverings translates to finite semidefinite programs. This approach yields a strict unification, algorithmic tractability, and systematically tighter certificates for joint spectral radius and uniform exponential stability.

3. Composite Lyapunov Frameworks for Convergence and Optimization Dynamics

In continuously evolving nonlinear systems—especially optimization-driven dynamics—a composite Lyapunov approach is used to establish convergence to sets or equilibria (Saoud, 9 Oct 2025). By constructing a pair of Lyapunov-type functions $V_1,V_2$ with dissipation observables $N_1,N_2$, and coupling them via a function $h(N_1)$: $\dot V_1 \le -N_1,\quad \dot V_2 \le -N_2 + h(N_1)$ the framework produces a composite function $W=V_1+\delta V_2$ with strict monotonic decrease for sufficiently small $\delta>0$.

This yields simultaneously: vanishing of dissipation, quantitative convergence rates (including sublinear and exponential rates under quadratic growth), and semistability or pointwise asymptotic stability of invariant sets. Applications include inertial gradient systems and primal-dual gradient flows, where explicit Lyapunov structures recover known or improved convergence results for nonconvex and constrained optimization problems.

4. Unified Categorical, Metric, and Infinite-Dimensional Theories

4.1 Categorical Equivalence

A categorical perspective constructs a direct equivalence between Lyapunov functions and Lyapunov equilibria, via level-set morphisms and functorial diagrams (Mattenet et al., 2023). For any deterministic forward dynamical system, the existence of either a Lyapunov function (monovariant) or a stability certificate in terms of level-sets is shown to be categorically equivalent, unifying all direct and converse Lyapunov theorems for discrete, continuous, and topological dynamical systems.

4.2 Metric-Space and Compact Set Stability

In locally compact metric spaces, compact invariant sets are stabilized by Lyapunov functions formed by averaging the maximal distance to the set over the future flow, ensuring strict decrease and continuity (Hadadi, 10 Dec 2024). This approach generalizes to attraction to limit cycles and arbitrary compact sets. The construction recovers both necessary and sufficient criteria for asymptotic stability under minimal assumptions, bridging abstract topological dynamical systems and classical ODE theory.

4.3 Infinite-Dimensional and PDE Extensions

For PDEs and infinite-dimensional flows, a unified indirect Lyapunov method identifies conditions under which Fréchet differentiability of the nonlinear semigroup at an equilibrium ensures that linearized stability properties (via the derivative semigroup) transfer to the nonlinear system (Jamal et al., 2015). This framework subsumes existing results for quasilinear, semilinear, and non-quasilinear equations, and leads to new sufficient conditions for stability in systems where classical approaches may fail (e.g., Kuramoto–Sivashinsky PDEs).

5. Data-Driven, Neural, and Operator-Based Lyapunov Synthesis

Recent advances achieve unification at the algorithmic level for systems with unknown or high-dimensional dynamics using neural representations or Koopman operator methods (Zhou et al., 3 Dec 2024, Zhou et al., 2022).

• Koopman-Lyapunov Learning: Simultaneously learns the vector field, Lyapunov function, and region of attraction using low-sample data, convex optimization over observable dictionaries, physics-informed loss, and formal region-of-attraction certification by satisfiability modulo theories (SMT) solvers. This approach leverages operator-theoretic embedding to bridge data-driven identification with formal stability guarantees.
• Neural Lyapunov Certificates: Neural networks parameterize both the closed-loop system and Lyapunov function, with loss functions and SMT-based verification enforcing positivity and decrease conditions. Certified ROAs greatly exceed those from linearization, with robustness to approximation errors rigorously accounted for.

6. Applications in Optimization, Control, and Stochastic Systems

The unified Lyapunov perspective streamlines convergence proofs and algorithm design in first-order and accelerated optimization methods via strong Lyapunov conditions (Chen et al., 2021, Toyoda et al., 4 Apr 2024). Unified Euler–Lagrange/Lyapunov systems generate all classical accelerated ODEs as instances, with convergence rates deduced from a family of parameterized Lyapunov functionals and explicit scaling constraints.

In stochastic systems and uncertainty quantification, the unified framework aggregates finite-time Lyapunov exponents and stochastic sensitivity into a single covariance structure, capturing the interplay of initial state and model uncertainty (Blake et al., 8 Mar 2024). Applications range from robust geophysical diagnostics to data assimilation and assessment of control predictability.

Table: Illustrative Unified Lyapunov Constructions

Domain	Unified Lyapunov Concept	Reference
Nonlinear switching (PSCLF)	Piecewise smooth CLF, directional derivatives	(Lu et al., 2015)
Composite convergence	Multi-term Lyapunov and dissipation	(Saoud, 9 Oct 2025)
Categorical equivalence	Functorial level-set morphisms	(Mattenet et al., 2023)
Optimization	Strong Lyapunov/time-scaling, Euler–Lagrange	(Chen et al., 2021, Toyoda et al., 4 Apr 2024)
Data-driven	NN/Koopman parameterized Lyapunov + SMT	(Zhou et al., 3 Dec 2024, Zhou et al., 2022)
Metric space	Distance-averaged Lyapunov compact sets	(Hadadi, 10 Dec 2024)
Hybrid/symbolic	Sequence-dependent Lyapunov, graph LMI	(Rossa et al., 2023)

7. Impact, Extensions, and Generalization

Unified Lyapunov frameworks have advanced the field by:

• Rendering stability analysis tractable for complex, high-dimensional, and hybrid systems.
• Enabling algorithmic, data-driven certificate synthesis that interfaces naturally with convex optimization, neural descent, and formal methods.
• Systematically incorporating uncertainties, nonsmoothness, or switching, with explicit certification of invariance and attraction for nonlinear and stochastic systems.
• Providing general, modular theory that reduces to known results under specialization and extends to previously intractable regimes.

Current research emphasizes further generalization to non-Euclidean and infinite-dimensional settings, integration with learning under uncertainty, and the development of scalable computational tools—embedding these unified Lyapunov constructions as core primitives within the analysis and design of modern dynamical and control systems.