Switched and Composite SVFs

• Switched and composite SVFs are structured Lyapunov functions that certify stability in switched systems using multiple candidate functions organized by graph structures.
• They employ composition lifts and transitive closures to refine stability certificates, often improving performance bounds by significant margins such as a 40% enhancement in joint spectral radius approximation.
• The framework supports large-scale networked systems via compositional methods, enabling automated controller synthesis and safe abstraction-based verification.

Switched and composite storage–Lyapunov functions (SVFs) constitute a foundational framework for analyzing, certifying, and refining dynamical properties within networks of switched systems. These tools leverage multiple candidate Lyapunov-type functions, often indexed by graphs or structured via compositional techniques, to accommodate the complexity and heterogeneity inherent in switched, interconnected, or abstracted systems. The theory and practical algorithms around SVFs have seen rapid evolution, with core contributions relating to graphical orderings, combinatorial lifts, compositional small-gain guarantees, and simulation/abstraction interfaces.

1. Path-Complete Lyapunov Functions and Graph Structures

The concept of a path-complete Lyapunov function (PCLF) organizes multiple Lyapunov candidates through a labeled directed graph $\G = (S, E)$, where $S$ is a finite set of nodes, $E \subset S \times S \times \Sigma$ is a set of edges labeled by the switching alphabet $\Sigma$, and each node $s \in S$ is assigned a function $V_s$ from a fixed template $\mathcal V$ of positive-definite, radially unbounded functions. The defining property of path-completeness is: every finite word $w = i_1 i_2 \cdots i_k \in \Sigma^k$ must label a walk through $\G$. The PCLF is valid if for all edges $(a,b,i) \in E$,

$S$0

where the maps $S$1 define the switched system $S$2. For $S$3, the PCLF certifies uniform global asymptotic stability (UGAS) of the origin.

Preorders between graphs are introduced to compare PCLF expressivity. Given two path-complete graphs $S$4, one writes $S$5 if any stability certificate in the form of a PCLF on $S$6 for dynamics class $S$7 and template $S$8 can be lifted to a certificate (with the same $S$9) on $E \subset S \times S \times \Sigma$0 (Jongeneel et al., 23 Mar 2025).

2. Composition Lifts and Graph Refinement Operations

Closure under composition—a key structural property of $E \subset S \times S \times \Sigma$1—enables the pivotal composition-lift operations. For each template function $E \subset S \times S \times \Sigma$2, if $E \subset S \times S \times \Sigma$3 for all $E \subset S \times S \times \Sigma$4, one can systematically derive new Lyapunov inequalities by composing both sides with available system maps. The $E \subset S \times S \times \Sigma$5-forward composition lift constructs the node set

$E \subset S \times S \times \Sigma$6

and edges mimicking the passage of switching words, iterated over all $E \subset S \times S \times \Sigma$7. The full composition lift is the union $E \subset S \times S \times \Sigma$8. At the inequality level, this produces "lifted" Lyapunov inequalities relating such composed functions.

For comparing PCLFs or constructing stronger certificates, these lifts enable refinement of path-complete graphs, systematically incorporating compositional implications that might be nonlocal in the base graph structure (Jongeneel et al., 23 Mar 2025).

3. Completeness of Composition Lifts and the Transitive Closure

It had been conjectured that the full composition lift $E \subset S \times S \times \Sigma$9 would always simulate any less-expressive graph in the preorder, under reasonable assumptions (invertible dynamics, composition-closed templates): that is, $\Sigma$0 would characterize $\Sigma$1 iff $\Sigma$2 simulates $\Sigma$3. However, the minimal counterexample (with $\Sigma$4, and explicit graphs $\Sigma$5, $\Sigma$6 as constructed in (Jongeneel et al., 23 Mar 2025)) demonstrates that while $\Sigma$7, there does not exist a node map from $\Sigma$8 simulating $\Sigma$9.

This failure is rooted in the lack of multi-step closure: $s \in S$0 encodes only atomic (single-step) compositions, not inferred implications achieved by chain reasoning on the graph. This motivates the introduction of the transitive composition-lift, denoted $s \in S$1, which extends $s \in S$2 by the transitive closure on inequalities: whenever both $s \in S$3 and $s \in S$4 are present, $s \in S$5 is added. A key result is that $s \in S$6 is complete: for any path-complete graphs $s \in S$7, $s \in S$8 simulates $s \in S$9 if and only if $V_s$0, and this closure is always finite (Jongeneel et al., 23 Mar 2025).

4. Iterative Refinement and Combinatorial Algorithms

The transitive composition-lift enables a principled iterative refinement procedure:

1. Given $V_s$1, form $V_s$2.
2. Apply the transitive closure to obtain $V_s$3.
3. Terminate when $V_s$4.

This loop produces a refined graph $V_s$5 fixed under the transitive composition-lift. The process is provably finite, with complexity bounded by $V_s$6. All operations are purely combinatorial, allowing automation and efficient use in practice for moderate graph and alphabet sizes. Empirically, a single refinement can significantly tighten bounds, e.g., for joint spectral radius approximation, with up to 40% improvement observed on random matrix pairs (Jongeneel et al., 23 Mar 2025).

5. Switched and Composite Storage Functions in Large Networks

In the broader context of switched, interconnected, or abstracted systems, compositional methods generalize SVF design. The compositional construction leverages local simulation (or storage) functions $V_s$7 for individual subsystems, composed into a global function

$V_s$8

under the satisfaction of small-gain and coupling conditions (Sharifi et al., 2021, Swikir et al., 2019). Key notions include:

• Switched simulation functions: Families $V_s$9 satisfying mode-dependent decay and output bounds, enabling quantifiable error guarantees between a concrete network and its abstraction.
• Small-gain compositionality: With diagonal decay matrix $\mathcal V$0 and gain matrix $\mathcal V$1, requiring the composite operator $\mathcal V$2 to have spectral radius $\mathcal V$3 ensures uniform decay for the composite $\mathcal V$4.
• LMI synthesis for linear systems: Quadratic local simulation/storage functions and corresponding interfaces are computable by solving LMIs tailored to the system and abstraction interrelation. This yields explicit decay rates and coupling gains (Sharifi et al., 2021).

6. Symbolic and Abstraction-Based SVF Design

For networks of switched systems, finite or symbolic abstractions facilitate controller synthesis and analysis. This requires storage (or augmented-storage) functions $\mathcal V$5 that quantify output mismatch between concrete and abstracted systems. Dissipativity-type (incremental passivity) conditions on each constant-mode subsystem ensure the existence of such storage functions. The sum of local augmented-storage functions can serve as a global alternating simulation function, supporting contractively quantifiable output error bounds (see AltSF-I, AltSF-II in (Swikir et al., 2019)). These results underpin the safety and reliability of abstraction-based controller refinement for large-scale or infinite networks.

Example syntheses include:

• Power grid frequency control using switched simulation functions with explicit LMIs for each bus (Sharifi et al., 2021).
• Traffic network symbolic model construction leveraging incremental passivity and dissipative composition to guarantee output error and enable safe controller refinement (Swikir et al., 2019).

7. Theoretical and Computational Implications

The transitive composition-lift and combinatorial graph refinement create a unifying, tractable, and fully combinatorial framework for ordering, comparing, and strengthening families of switched or composite Lyapunov/storage functions. The approach removes algebraic intractability by reducing expressivity comparison to finite graph simulation—a significant advance both for theoretical understanding and scalable practice in stability analysis and abstraction-based control. Furthermore, the generality of these methods allows transfer to analogous settings, including storage and supply function design for switched PDEs and networked control (Jongeneel et al., 23 Mar 2025, Sharifi et al., 2021, Swikir et al., 2019).