Lyapunov-Based Small-Gain Theorem

• Lyapunov-Based Small-Gain Theorem is a framework that uses composite Lyapunov functions to systematically certify input-to-state stability in interconnected systems.
• It aggregates local Lyapunov dissipation inequalities with interconnection gains to enable finite-step analysis even when some subsystems are not inherently ISS.
• The theorem extends to hybrid, infinite-dimensional, and exponentially stable networks by employing robustification operators and numerical methods for verification.

A Lyapunov-Based Small-Gain Theorem provides systematic criteria for establishing input-to-state stability (ISS) and its variants (including exponential ISS, ISS with respect to closed sets, and ISS for hybrid or infinite-dimensional systems) in networks composed of multiple—potentially infinitely many—interconnected subsystems. The theorem connects local Lyapunov dissipation inequalities and interconnection gains to the global stability properties of the entire network, replacing classical trajectory-based arguments with composite Lyapunov function constructions that reflect the coupling topology and strength.

1. Foundations: ISS, Lyapunov Functions, and Gain Operators

Input-to-state stability (ISS) for a discrete-time system $x(k+1) = G(x(k),u(k))$ (state $x\in\R^n$, input $u\in\R^m$) is defined by the existence of functions $\beta\in\KL$ and $\gamma\in\K$ such that

$$\|x(k,\xi,u)\| \leq \beta(\|\xi\|,k) + \gamma(\|u\|_\infty) \quad \forall k\in\mathbb{N}.$$

A dissipative ISS Lyapunov function $V:\R^n\to\R_+$ satisfies

$$V(G(\xi, \mu)) \leq \rho(V(\xi)) + \sigma(\|\mu\|),$$

with $\rho,\sigma\in\K$, $\rho<\text{id}$. Finite-step relaxations require the bound after $M\geq 1$ steps.

For an interconnected network with $N$ subsystems, each subsystem $x_i$ has a local Lyapunov-type function $V_i$, and finite-step ISS inequalities are formulated in terms of interconnection gains $\gamma_{ij}$ and external gains $\gamma_{i,u}$:

$$V_{i}(x_{i}(M,\xi,u)) \leq \max_{j=1,\ldots,N} \gamma_{ij}(V_{j}(\xi_{j})) \vee \gamma_{i,u}(\|u\|_\infty).$$

This leads to a vector-valued gain operator:

$\Gamma(s)_i = \max_{j} \gamma_{ij}(s_j),$

where $s\in\R_+^N$.

2. Main Theorem: Relaxed Finite-Step Small-Gain for Discrete-Time Networks

The relaxed finite-step ISS small-gain theorem (Geiselhart et al., 2014) claims:

If for each $i=1,\ldots,N$,

• $V_i$ are proper, positive-definite;
• There exist $M\geq 1$ and gains $\gamma_{ij}, \gamma_{i,u}$ such that $$V_i(x_i(M,\xi,u))\leq \max_j \gamma_{ij}(V_j(\xi_j))\vee \gamma_{i,u}(\|u\|_\infty)$$ for all $(\xi,u)$,
• A diagonal "robustification" $D = \operatorname{diag}(id+\delta_i)$ with $\delta_i\in\K$ is such that $D\circ\Gamma(s)\not\geq s$ for all $s>0$,

then one can construct:

• an $\ell^\infty$-path $\sigma=(\sigma_1,\ldots,\sigma_N)$ with $\sigma_i\in\K$ satisfying $(D_1\circ\Gamma\circ D_2)(\sigma(r))<\sigma(r)$ for all $r>0$ (with $D=D_2\circ D_1$);
• a global finite-step Lyapunov function

$$V(x)=\max_{i=1,\ldots,N} \sigma_i^{-1} \circ D_1^{-1} \circ V_i(x_i)$$

satisfying after $M$ steps:

$$V(x(M,x,u)) \leq \rho(V(x)) + \sigma_u(\|u\|_\infty), \quad \rho<\text{id}.$$

Thus, the network is ISS.

The sum-form variant replaces the maximum with sums and applies a similar condition (Theorem 5.4 in (Geiselhart et al., 2014)).

3. Non-Conservatism in ExpISS Networks

For exponentially ISS networks,

$$\|x(k)\|\leq C\lambda^k \|\xi\| + \gamma(\|u\|_\infty), \qquad 0\leq\lambda<1,$$

any norm $V(x)=\|x\|$ becomes a dissipative finite-step ISS Lyapunov function for sufficiently large $M$ (Theorem 4.6 (Geiselhart et al., 2014)). The relaxed small-gain condition in terms of finite-step gains $\gamma_{ij}<\text{id}$ is both sufficient and necessary for ISS in the exponentially ISS class, making the relaxed theorem non-conservative for expISS networks.

4. Small-Gain Condition: Classical, Strong, and Path-Based Forms

The classical small-gain requirement for the gain operator $\Gamma$ is absence of nontrivial fixed points:

$\Gamma(s) \not\geq s, \quad \forall s>0.$

For potentially unstable subsystems, a "strong" small-gain condition is imposed,

$D\circ\Gamma(s)\not\geq s,$

with $D=\operatorname{diag}(id+\delta_i)$ for suitably chosen $\delta_i\in\K$.

Construction of the $\ell^\infty$-path $\sigma$ solving $(D_1\circ\Gamma\circ D_2)(\sigma(r))<\sigma(r)$ is enabled by this stronger condition. The path-of-decay approach translates the vector-valued interconnection into a scalar contraction suitable for global Lyapunov aggregation.

5. Implications and Extensions: Hybrid Systems, Infinite Networks, and Computational Approaches

• The same aggregation methodology applies to hybrid systems, impulsive systems, and systems where not all subsystems are ISS. In those cases, clocks (average dwell-time or reverse dwell-time) modify the Lyapunov functions to establish the necessary rates for small-gain arguments (Mironchenko et al., 2016).
• In infinite networks, the gain operator generalizes to infinite dimensions, with small-gain conditions typically expressed in terms of the spectral radius of the (possibly linear) operator built from interconnection gains. If the spectral radius is less than one, then the infinite network admits a coercive exponential ISS Lyapunov function as a weighted sum of the local Lyapunov functions (Noroozi et al., 2020, Mironchenko et al., 2021, Kawan et al., 2019).

Theorem Variant	System Type	Small-Gain Operator	Key Lyapunov Form
Relaxed finite-step	Discrete-time	Max or sum type, finite N	$V(x)=\max_{i}\sigma_i^{-1}V_i(x_i)$
ExpISS characterization	Discrete-time	Linear gains, finite N	$V(x)=\|x\|$, $M\gg 1$
Infinite network ISS	Infinite-dimensional	Linear operator $\Psi$	$V(x)=\sum_{i}\mu_iV_i(x_i)$
Hybrid extension	Hybrid/impulsive	General monotone	$V(x)=\max_{i}\sigma_i^{-1}V_i(x_i)$

6. Proof Structure and Construction Methods

The proof involves:

• Establishing local finite-step dissipation inequalities in terms of gains.
• Aggregating these inequalities through the monotonicity of the gain operator, usually after $M$ steps.
• Constructing the scalar Lyapunov function with contraction property via path $\sigma$ after robustification.
• Demonstrating that the composite Lyapunov decays along trajectories, up to bounded input terms, and thus certifies ISS.

For practical computation, numerical homotopy methods and fixed-point algorithms to find decay points for the gain operator have been developed (Geiselhart et al., 2011), enabling direct verification and construction of the global Lyapunov function for large or nonlinear networks.

7. Representative Example and Applications

A canonical application is the nonlinear two-subsystem network:

$$\begin{aligned} x_1(k+1) &= x_1(k) - 0.3 x_2(k) + u(k), \ x_2(k+1) &= x_1(k) + 0.3 x_2(k) - (x_1(k) + 0.3 x_2(k))^3 + u(k). \end{aligned}$$

Despite the decoupled $x_1$ subsystem being not 0-input stable, the interconnected network is shown ISS by constructing $M=3$, $V_i=|x_i|$, computing finite-step gains $\gamma_{ij}<1$, checking $\Gamma(s)\not\geq s$, and synthesizing the global Lyapunov $V(x)=\max_i \sigma_i^{-1} V_i(x_i)$ (Geiselhart et al., 2014).

The framework applies to large-scale networks, synchronization problems, distributed observers, hybrid systems, and time-delay systems. The relaxed theorem is particularly potent where classical assumptions of ISS subsystems are either violated or overly conservative.

---

References

• "Relaxed ISS Small-Gain Theorems for Discrete-Time Systems" (Geiselhart et al., 2014)
• "Lyapunov small-gain theorems for networks of not necessarily ISS hybrid systems" (Mironchenko et al., 2016)
• "Small-gain theorem for stability, cooperative control and distributed observation of infinite networks" (Noroozi et al., 2020)
• "ISS small-gain criteria for infinite networks with linear gain functions" (Mironchenko et al., 2021)
• "Numerical Construction of LISS Lyapunov Functions under a Small Gain Condition" (Geiselhart et al., 2011)