Distributed Lyapunov Control Framework

• Distributed Lyapunov-Based Control Framework is a method that uses local Lyapunov functions to assess stability and enable synchronization in large-scale chaotic networks.
• The framework employs nonlinear coupling driven by local state differences, ensuring exponential convergence and robustness against delays, switching topologies, and noise.
• A key advantage is its localized computation, reducing overhead by eliminating the need for global LMIs and enabling real-time performance in resource-constrained environments.

A distributed Lyapunov-based control framework structures control synthesis and stability assessment for large-scale dynamical networks using local Lyapunov arguments and minimal inter-agent communication. In the context of leader-follower synchronization of chaotic multi-agent systems, such as those composed of Roessler, Lü, or Chen oscillators, the framework enables both complete and phase synchronization through nonlinear coupling mechanisms validated by explicit Lyapunov and matrix measure analyses. This approach avoids the computational overhead of global LMI constraints or adaptive control laws and delivers conditions that require only local state and neighbor information, robustly handling delays, switching topologies, and noise.

1. System Architecture and Communication Topology

The framework targets directed multi-agent networks in which agents exhibit identical chaotic dynamics of the form

$\dot x_i(t) = F(x_i(t)) = L x_i(t) + G(x_i(t)),$

where $L \in \mathbb{R}^{n \times n}$ is linear and $G : \mathbb{R}^n \to \mathbb{R}^n$ is a nonlinear map. Agents are organized such that node $1$ acts as a leader and nodes $i=2,\dots,N$ are followers. The communication graph $\mathcal G = (\mathcal V, \mathcal E)$, with vertex set $\mathcal V$ and directed edge set $\mathcal E$, supports neighbor-based information exchange through its adjacency matrix $A = [a_{ij}]$, where $a_{ij} > 0$ iff node $i$ receives from $j$.

A crucial structural requirement is that $\mathcal G$ possesses at least one directed spanning tree rooted at the leader. Each follower's control and stability check uses only the local state and the nonlinear evaluation $G(x)$ as transmitted by its immediate in-neighbors.

2. Nonlinear Distributed Coupling

The distinctive feature of the control law is the exclusive use of the nonlinear part in the coupling term: $$u_i(t) = \alpha \sum_{j \in \mathcal N_i} a_{ij} (G(x_j(t)) - G(x_i(t))), \quad \alpha > 0,$$ where $\mathcal N_i$ denotes the set of neighbors supplying information to agent $i$. This coupling operates on the nonlinear evaluation of the states and does not require knowledge of the global Laplacian or network topology. The closed-loop dynamics for followers become

$\dot x_i = L x_i + G(x_i) + u_i,$

where $u_i$ encapsulates only locally gathered data.

3. Lyapunov-Based Synchronization Conditions

Let the synchronization error for follower $i$ be $e_i = x_i - x_1$. The error dynamics admit linearization via the Jacobians of $G$: $$\dot e_i \approx (L + (1-\alpha)J_G(x_i)) e_i + \alpha \sum_{j \in \mathcal N_i} a_{ij} (J_G(x_j) e_j - J_G(x_i) e_i),$$ with $J_G(x) = \partial G / \partial x$. The Lyapunov candidate for each follower is $V_i(e_i) = e_i^\top P e_i$, $P = P^\top > 0$.

Complete synchronization is certified if $\exists~P > 0$, $\alpha>0$ such that

$$(L + (1-\alpha) J_G(x))^\top P + P (L + (1-\alpha) J_G(x)) \leq -Q, \quad Q = Q^\top > 0,$$

for all $x$ on the compact operating set $\Omega$. Under this, $e_i(t) \to 0$ exponentially for all followers.

Phase synchronization arises in rank-deficient $L$, with the matrix measure condition: $$\max_{x \in \Omega} \mu_p(L + (1-\alpha) J_G(x)) < 0$$ yielding bounded constant offsets between follower and leader states as $t \to \infty$.

4. Robustness to Delays and Topology Switching

Delay robustness is achieved via a Lyapunov-Krasovskii functional

$$V_i(t) = e_i^\top P e_i + \int_{t-\tau}^t e_i(s)^\top R e_i(s) ds, \quad R > 0,$$

and requiring

$$(L + (1-\alpha)J_G)^\top P + P(L + (1-\alpha)J_G) + \tau R < 0$$

for synchronization persistence under constant communication delay $\tau$.

Synchronizability under switching topologies is maintained if each active graph maintains a leader-rooted spanning tree and the dwell time is sufficiently large. The synchronization conditions remain locally checkable and unaffected by global changes.

5. Computational and Implementation Characteristics

A key computational advantage is the local verification and synthesis: each agent only solves a $n \times n$ LMI locally, requiring knowledge of its own adjacency weights and neighbor Jacobians. This scalability drastically reduces calculation times—e.g. 3.8 s for $N=5$ agents—versus orders of magnitude slower centralized checks.

No adaptive laws or high-dimensional neural decision variables are present, and no global topology information or Laplacian is needed. The approach is robust to Gaussian noise (errors < $3\times10^{-3}$ even for $\sigma^2=0.1$) and switching graphs, as demonstrated empirically.

6. Validation on Chaotic Multi-Agent Networks

Extensive simulation confirms the framework's efficacy:

• Lü system: $N=5$, directed tree, $\alpha=0.95$, complete synchronization in 12 s ($\lim_t E(t) \approx 2.1\times10^{-4}$).
• Rössler system: feedback and delays ($\tau=0.5$ s, $\alpha=1.2$), synchronization within 25 s in delayed topology ($\lim_t E(t) \approx 6.5\times10^{-3}$).
• Chen system: phase synchronization with constant offset, matching theoretical predictions.

Both convergence rate and steady-state error are superior to adaptive-neural or conventional LMI-based schemes.

7. Context and Comparative Impact

Compared to global LMI or adaptive schemes—where feasibility and verification require full-network information and heavy computation—the distributed Lyapunov-based control framework guarantees exponential convergence (or phase locking), exact delay and switching robustness, and vastly reduced communication, computational, and informational requirements (Abarghoei et al., 22 Nov 2025). The approach generalizes beyond chaotic synchronization to other classes of coupled nonlinear networks, by utilizing explicit Lyapunov and matrix measure conditions derived from local subsystem data and neighbor communication.

This methodology provides a scalable, theoretically rigorous route to distributed stabilization and synchronization in complex networks, dispensing with adaptation or neural approximations and enabling parallel, real-time execution even in resource-constrained scenarios.