Multiple Lyapunov Functions for Stability Analysis

Updated 29 June 2026

Multiple Lyapunov functions are mode-specific certificates that verify the stability of switched, hybrid, and networked systems by enforcing inter-mode decrease conditions.
Path-complete Lyapunov frameworks use directed graphs to encode decrease conditions across modes, ensuring stability even with complex switching rules.
Computational methods such as SDP/SOS relaxations and neural network parameterizations enable efficient synthesis of Lyapunov certificates for high-dimensional nonlinear systems.

Multiple Lyapunov functions (MLFs) provide a foundational framework for certifying the stability of dynamical systems with multiple operational modes, such as switched, hybrid, or networked systems. Unlike the classical common Lyapunov function (CLF) approach, which seeks a single function decreasing along all trajectories, the MLF methodology allows the assignment of a distinct Lyapunov function to each mode or subsystem and encodes switching or coupling relations through inter-function inequalities. This algebraic versatility enables the analysis of systems exhibiting intricate switching rules, mode-dependent behaviors, or complex interconnections, which frequently thwart single-function strategies. Theoretical developments and computational realizations of MLFs span continuous-time, discrete-time, nonlinear, and hybrid settings, and leverage tools ranging from sum-of-squares (SOS) programming to recent neural-network training frameworks (Huang et al., 2 Jan 2026).

1. Formal Definition and Classical Stability Results

Consider a switched system with modes $i\in Q=\{1,\dots,N\}$ , continuous-time dynamics $\dot{x} = f_i(x)$ , or discrete-time dynamics $x_{k+1} = f_i(x_k)$ . A collection of continuously differentiable functions $\{V_i:\R^n\rightarrow\R_{\ge 0}\}_{i\in Q}$ constitutes a multiple Lyapunov function if there exist class- $\mathcal{K}_\infty$ functions $\alpha_i,\lambda_{ij}$ such that for all $x\neq 0$ and each mode $i$ :

(i) Positivity: $V_i(x) > 0,\; V_i(0)=0$ ,
(ii) Mode-wise decrease:
- Continuous-time: $\dot{V}_i(x) = \nabla V_i(x)^{\top} f_i(x) \leq -\alpha_i(\|x\|)$ ,
- Discrete-time: $\dot{x} = f_i(x)$ 0,
(iii) Switch-induced decrease: When switching from mode $\dot{x} = f_i(x)$ 1 to $\dot{x} = f_i(x)$ 2 at state $\dot{x} = f_i(x)$ 3, $\dot{x} = f_i(x)$ 4.

The classical Branicky theorem states: if an MLF exists satisfying these conditions on a forward-invariant $\dot{x} = f_i(x)$ 5, then the switched system is globally asymptotically stable for all switching signals consistent with these inequalities, ensuring strict decrease of the composite Lyapunov value along every trajectory (Huang et al., 2 Jan 2026).

2. Graph-Based and Path-Complete Lyapunov Frameworks

The path-complete Lyapunov function (PCLF) methodology generalizes the MLF philosophy by encoding inter-function inequalities via a directed, edge-labeled graph $\dot{x} = f_i(x)$ 6 (Angeli et al., 2016, Ahmadi et al., 2011, Philippe et al., 2017, Jongeneel et al., 23 Mar 2025). Each node $\dot{x} = f_i(x)$ 7 is assigned a candidate function $\dot{x} = f_i(x)$ 8, and each edge $\dot{x} = f_i(x)$ 9 encodes the decrease condition $x_{k+1} = f_i(x_k)$ 0. Path-completeness of the graph ensures that every legal switching sequence is covered by a path in the graph, guaranteeing that the set of inequalities collectively certifies uniform stability under all admissible switching.

A key result is that any PCLF induces a single common Lyapunov function by a finite min-max composition over the "observer graph" associated with $x_{k+1} = f_i(x_k)$ 1. Explicitly,

$x_{k+1} = f_i(x_k)$ 2

where $x_{k+1} = f_i(x_k)$ 3 runs over the core strongly-connected component of the observer graph (Angeli et al., 2016, Philippe et al., 2017). This construction tightly couples the combinatorial properties of $x_{k+1} = f_i(x_k)$ 4 with the analytic properties of the coupled Lyapunov functions.

An important caveat arises: not every max-of- $x_{k+1} = f_i(x_k)$ 5-quadratics Lyapunov function admits representation as a PCLF over a $x_{k+1} = f_i(x_k)$ 6-node path-complete graph, highlighting inherent limitations of the LMI-based graph approach versus the more general BMI-based max/min constructions (Angeli et al., 2016, Philippe et al., 2017).

3. Computational Methods: From SOS to Neural-Parameterized MLFs

MLFs and their graph-theoretic relatives admit algorithmic search via convex optimization, particularly when the candidate functions are restricted to specific templates (e.g., quadratics, piecewise-affine, or SOS polynomials):

SDP/SOS relaxations: For linear or polynomial dynamics, the search for candidate functions $x_{k+1} = f_i(x_k)$ 7 is formulated as a semidefinite program (SDP), with constraints induced by the path-complete graph or automaton structure. Notable hierarchies (e.g., De Bruijn graphs) achieve explicit approximation guarantees for the joint spectral radius, with conservatism shrinking as the number of templates grows (Ahmadi et al., 2011).
Sum-of-squares in nonlinear networks: The vector Lyapunov function and comparison system approach constructs local SOS Lyapunov functions $x_{k+1} = f_i(x_k)$ 8 for subsystems or neighborhoods, coordinating them via comparison systems (Metzler matrices) or aggregated decay inequalities. Multiple-comparison-systems methods adaptively partition the state space and yield scalable, distributed certificates (Kundu et al., 2016, Wang et al., 25 Jun 2025).
Neural MLFs: For high-dimensional and highly nonlinear switched systems, neural-network parametrizations are used for each $x_{k+1} = f_i(x_k)$ 9, trained jointly to satisfy mode-wise and switch-induced decrease constraints via loss functions. Counterexample-guided inductive synthesis (CEGIS) enforces global validity by searching for violations and iteratively augmenting the training set. This approach enables practical synthesis of Lyapunov certificates for problems where symbolic methods are intractable (Huang et al., 2 Jan 2026).

4. Extensions: Hybrid Systems, Dwell-Time, and Stochastic/Data-Driven Settings

MLFs provide the analytic substrate to address a spectrum of system-theoretic challenges beyond asymptotic stability under arbitrary switching:

Hybrid and finite-time stability: MLFs are adapted to hybrid systems with both continuous flows and discrete jumps, allowing modes in which the Lyapunov function may (temporarily) increase, as long as a global decrease over persistently stabilizing modes or sufficient dwell-time is ensured. Sufficient conditions involve bounding the sum of increases via class- $\{V_i:\R^n\rightarrow\R_{\ge 0}\}_{i\in Q}$ 0 functions and enforcing minimal activation durations of finite-time stable modes (Garg et al., 2019).
Average dwell-time analysis: MLFs underpin the derivation of explicit lower bounds on admissible average dwell times for stability in switched systems. Optimization-based procedures (including LPs over piecewise-affine templates) yield tight bounds and outperform classical quadratic-based criteria in several benchmark scenarios (Hafstein et al., 2023).
Constrained switching and data-driven MLFs: When the switching signal is governed by an automaton or sofic shift, stability is characterized by the constrained joint spectral radius. Multiple (quadratic) Lyapunov functions indexed by graph nodes yield tight upper and lower bounds on the spectral radius, and data-driven scenario optimization techniques enable probabilistic certification in black-box settings, with explicit sample complexity and confidence levels (Banse et al., 2022, Rossa et al., 2023).

5. Distributed and Networked System Applications

Distributed stability verification in large-scale nonlinear and networked systems leverages multiple Lyapunov functions localized to subsystems or neighborhoods. The vector Lyapunov function approach, in conjunction with comparison systems and sum-of-squares programming, enables the decomposition of global verification into local, parallelizable tasks. Composite certificates, formed by weighted combinations of partial Lyapunov functions, offer scalable and accurate approximations of regions of attraction, including non-convex morphologies induced by multi-stability (Wang et al., 25 Jun 2025, Kundu et al., 2016).

A representative workflow includes:

Construction of local or partial Lyapunov functions $\{V_i:\R^n\rightarrow\R_{\ge 0}\}_{i\in Q}$ 1 via SOS programs.
Coordination through comparison inequalities or augmented comparison lemmas that account for coupling residuals.
Aggregation into a convex composite Lyapunov function that certifies decay rate via an LMI in the aggregation weights.

Such architectures have been demonstrated on van der Pol and Ising oscillator networks, consistently outperforming centralized SOS approaches in high-dimensional regimes (Wang et al., 25 Jun 2025).

6. Theoretical Extensions and Graph-Based Ordering

The formal structure of MLFs in terms of path-complete graphs has catalyzed a theory of preorders to compare and refine stability criteria. Algebraic and combinatorial conditions (e.g., simulation, composition lifts, transitive closure in composition graphs) enable the systematic refinement of certificates and reveal the structural relationship between candidate graphs and their associated Lyapunov inequalities (Jongeneel et al., 23 Mar 2025, Philippe et al., 2017).

Ordering relations $\{V_i:\R^n\rightarrow\R_{\ge 0}\}_{i\in Q}$ 2 are defined so that any feasible PCLF on $\{V_i:\R^n\rightarrow\R_{\ge 0}\}_{i\in Q}$ 3 yields a feasible PCLF on $\{V_i:\R^n\rightarrow\R_{\ge 0}\}_{i\in Q}$ 4, with algorithmic tests (such as linear programs in incidence matrices) available for some notions of preorder. Iterative refinement via composition lifts generates systematically richer path-complete graphs and, therefore, tighter Lyapunov certificates—subject to the template being closed under appropriate composition operations.

7. Practical Application Domains and Emerging Directions

MLFs have been foundational in:

Stability certification of constrained or state-dependent switched systems beyond the reach of CLFs.
Decentralized goal assignment and collision-avoidance in multi-agent robotics, with Lyapunov-like barrier functions employed for provably safe task-switching (Panagou et al., 2014).
Probabilistic and black-box stability verification for complex or partially unknown systems, using scenario-based optimization and automaton-induced multinorm frameworks (Banse et al., 2022).
Neural and distributed implementations for certified control of high-dimensional or strongly nonlinear networks (Huang et al., 2 Jan 2026, Wang et al., 25 Jun 2025).

Active areas of research include improvements in SMT-based verification scalability, automated co-design of controllers and Lyapunov functions in neural and SOS frameworks, language-constrained switching analysis, and exactness/tightness trade-offs across graph- and template-based families.

Summary Table: Main MLF Concepts and Approaches

Concept/Class	Characterization	Example Source
Classical MLF	Mode-wise $\{V_i:\R^n\rightarrow\R_{\ge 0}\}_{i\in Q}$ 5, inter-mode inequalities	(Huang et al., 2 Jan 2026)
Path-complete Lyapunov graph	Node-wise $\{V_i:\R^n\rightarrow\R_{\ge 0}\}_{i\in Q}$ 6, edge-wise inequalities, graph completeness	(Angeli et al., 2016)
Sum-of-squares (SOS) MLF	Polynomial $\{V_i:\R^n\rightarrow\R_{\ge 0}\}_{i\in Q}$ 7, certified by SOS/SDP	(Kundu et al., 2016)
Neural MLF	Neural $\{V_i:\R^n\rightarrow\R_{\ge 0}\}_{i\in Q}$ 8, trained with CEGIS	(Huang et al., 2 Jan 2026)
Finite-time/hybrid MLF	GLF with global sum-increase bounds	(Garg et al., 2019)
Data-driven/automaton-constrained MLF	Scenario-optimal quadratic multinorms, LMIs via samples	(Banse et al., 2022)

This spectrum of methodologies illustrates the central and unifying role played by multiple Lyapunov functions in the stability theory of complex systems across continuous, discrete, hybrid, and data-driven contexts.