# Synchronization of multi-agent hybrid systems with synchronous state-dependent jumps

Alexandre Cellier-Devaux¹, Luca Zaccarian², Daniele Astolfi¹ and Vincent Andrieu¹

**Abstract**— In this paper, we investigate the problem of synchronization of multi-agent systems described by hybrid non-linear input affine dynamics. We consider networks described by undirected connected graphs without leader. We present a set of sufficient conditions based on an LMI approach in order to design a state-feedback distributed control law. Then, by exploiting the properties of the graph incidence matrix we provide an optimization of the tuning parameters. The incidence matrix properties also allow us to construct a Lyapunov function to establish synchronization of the hybrid network.

**Index Terms**— Synchronization, hybrid dynamics, multi-agent systems

## I. INTRODUCTION

SYNCHRONIZATION of multi-agent systems is current a hot topic, due to its importance in many applications such as robotic swarms, autonomous vehicles, or localization. Many notable results have been provided regarding the synchronization problem for continuous-time dynamics; see [1] for linear systems, and for nonlinear ones, see [2], [3], to cite a few. The literature on synchronization of discrete-time multi-agent systems is extensive for linear dynamics (see, e.g. [4], [5]), but fewer results are available for nonlinear dynamics [6]. In this work, we focus on the synchronization of multi-agent systems where the agent's dynamics are described by hybrid equations. Hybrid dynamical systems emerge in many real-world applications that combine continuous-time control for real-time operations and discrete-time control for event handling or decision-making. An example of such systems is a drone fleet used in search and rescue operations. For more examples, we refer for instance to [7]. Existing works on synchronization of hybrid dynamics are very scarce. In [8], the dynamics are continuous-time, but the controller is discrete-time. Similarly, in [9], the communication is hybrid and more precisely time-triggered or event-triggered. In [10], the agent dynamics are hybrid, but quite simple (double integrator). Recent work [11] studies multi-cluster synchronization in linear hybrid systems with time-driven jumps over connected

A. Cellier-Devaux, D. Astolfi and V. Andrieu are with Université Claude Bernard Lyon 1, CNRS, LAGEPP UMR 5007, 43 boulevard du 11 novembre 1918, F-69100, Villeurbanne, France. (name.surname@univ-lyon1.fr).

L. Zaccarian is with the LAAS-CNRS, University of Toulouse, 31400 Toulouse, France and with the University of Trento, Italy.

Work funded by the ANR under grant OLYMPIA ANR-23-CE48-0006 and by the MUR grant STARLIT (CUP n. E53D23001130006).

directed graphs, using separate control laws for the exclusive reaches and the common part of the graph.

We address the synchronization problem in hybrid multi-agent systems with nonlinear input-affine jump and flow dynamics. To this end, we propose a distributed hybrid diffusive coupling strategy and construct a hybrid Lyapunov function based on the framework from [3], where the Lyapunov function is defined as a sum over differences between connected agents. With this construction, we assume an undirected communication graph—an essential condition to be able to exploit the edge Laplacian's properties. Similarly to [11], we define a jump set that enforces synchronous jumps across all agents, unlike the asynchronous approach in [9]. Specifically, we suppose that the jumps are triggered by state-dependent conditions and, whenever one agent jumps, a synchronous jump is enforced for all other agents in the network. We assume that the agents satisfy a hybrid stabilizability-like condition. Specifically, we require that both the continuous-time and discrete-time dynamics are stabilizable under a contraction stabilizability-like condition with a common quadratic Lyapunov function. In the case of linear dynamics, this condition reduces to the existence of a common quadratic solution to the continuous-time and discrete-time algebraic Riccati inequalities. For nonlinear dynamics, this consists in requiring the existence of a Demidovich-like condition for both continuous-time [12] and discrete-time [6]. Finally, a Lyapunov-based approach allows us to establish global asymptotic stability of the synchronization manifold.

The rest of the paper is organized as follows. We state the problem of hybrid multi-agent synchronization in Section II. Then, we state the main result of this article in Section III. An illustration is given in Section IV. Then, we present a Lyapunov-based proof of our main result in Section V. Conclusions are drawn in Section VI.

*Notation:* $\mathbb{R}$, resp. $\mathbb{N}$, denotes the set of real numbers, resp. non-negative integers. Given $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$, denote $(x, y) = [x^\top y]^\top$. The time derivative of $x$ is denoted $\dot{x}$ and $x^+$ denotes the value of $x$ after a jump. We use $|\cdot|$ for the standard Euclidean norm on $\mathbb{R}^n$ and the induced norm for a matrix in $\mathbb{R}^{n \times n}$. The symbol $\otimes$ denotes the Kroenecker product. Given $M$ square matrices $A_i \in \mathbb{R}^{n_i \times n_i}$ for $i \in \{1, \dots, M\}$, diag$(A_1, \dots, A_M)$ is the square matrix with $A_1, \dots, A_M$ on the main diagonal and zeros everywhere else. Given a matrix $A = (a_{ij})$, we denote by $\ker(A)$ its kernel. $\bar{\lambda}_A$ and $\underline{\lambda}_A$ refer to the maximum and the minimum eigenvalue of a symmetric matrix $A$, respectively. For 2 symmetric matrices $A$ and $B$,
we note $A \succ B$, resp. $A \succeq B$, if $A - B$ is definite positive, resp. semi-definite positive. $I_N$ indicates the identity matrix of dimension $N \times N$. $\mathcal{G} = (\mathcal{V}, \mathcal{E}, A)$ denotes a communication graph, where $\mathcal{V} = \{v_1, v_2, ..., v_N\}$ is the set of the $N \in \mathbb{N}$ nodes (the $N$ agents), $\mathcal{E} = \{(v_{k_1}, v_{l_1}), ..., (v_{k_m}, v_{l_m})\} \subset \mathcal{V} \times \mathcal{V}$ is the set of edges that models the interconnection between the nodes, and $A = (a_{ij}) \in \mathbb{R}^{N \times N}$ is the adjacency matrix defined by the weight $a_{kl} \ge 0$ of each edge from node $k$ to node $l$. For an undirected graph, if $(v_k, v_l) \in \mathcal{E}$, then $(v_l, v_k) \in \mathcal{E}$ and moreover $a_{kl} = a_{lk}$. We denote by $L = (\ell_{ij}) \in \mathbb{R}^{N \times N}$ the Laplacian matrix of the graph, defined as $\ell_{ij} = -a_{ij}$ for $i \ne j$, and $\ell_{ij} = \sum_{k=1}^N a_{ik}$ for $i = j$, Given an undirected and connected graph with $Q$ edges $(\varepsilon_1, ..., \varepsilon_Q)$, by giving an arbitrary orientation for each edge of the graph, we denote by $E = (e_{ij}) \in \mathbb{R}^{N \times Q}$ the incidence matrix defined as $e_{ij} = -1$, resp. $e_{ij} = +1$, resp. $e_{ij} = 0$, if $v_i$ is the initial node of edge, resp. the terminal node, resp. otherwise. We denote by $w_i = a_{kl} > 0$, for each edge $\varepsilon_i = (v_k, v_l)$, the weight of the edge. Thus, we note the weighting matrix $W$ as $W = \text{diag}(w_1, ..., w_Q)$. The Laplacian is related to the incidence and weighting matrices as $L = EWE^\top$ [13].

## II. PRELIMINARIES

This section briefly overviews hybrid systems and formulates the multi-agent synchronization problem.

### A. Hybrid systems

We consider hybrid systems of the form

$$ \mathcal{H} : \begin{cases} \dot{\xi} = F(\xi), & \xi \in \mathcal{C}, \\ \xi^+ = G(\xi), & \xi \in \mathcal{D}, \end{cases} \quad (1) $$

where $\xi \in \mathcal{X} \subseteq \mathbb{R}^{n_\xi}$ is the state, $\mathcal{C} \subseteq \mathcal{X}$ is the flow set, $\mathcal{D} \subseteq \mathcal{X}$ is the jump set, $F$ is the flow map and $G$ is the jump map. We refer the reader to [7] for a detailed analysis and foundation of this class of systems.

**Definition 1:** [7, Def. 3.6] Consider system (1) and a closed set $\mathcal{A}$. Denoting by $|x|_{\mathcal{A}}$ the point-to-set distance from $x$ to $\mathcal{A}$, the set $\mathcal{A}$ is said to be Uniformly Globally pre-Asymptotically Stable (UGpAS) if

* there exists a class-$\mathcal{K}_\infty$ function $\alpha$ such that any solution $\xi(t, j)$ of (1) satisfies $|\xi(t, j)|_{\mathcal{A}} \le \alpha(|\xi(0, 0)|_{\mathcal{A}})$, $\forall (t, j) \in \text{dom}\xi$;

* for every $\varepsilon > 0$ and $r > 0$, there exists $T > 0$ such that every solution $\xi(t, j)$ of (1) with $|\xi(0, 0)|_{\mathcal{A}} \le r$ satisfies $|\xi(t, j)|_{\mathcal{A}} \le \varepsilon$ for all $(t, j) \in \text{dom}\xi$ such that $t + j \ge T$. We recall a stability result for unbounded sets in hybrid systems, as stated in [7, Theorem 3.18].

**Theorem 1:** [7, Thm 3.18] Consider system (1) and a closed set $\mathcal{A}$. If there exist a function $\xi \mapsto V(\xi)$ continuous and differentiable in an open set containing $\mathcal{C}$, functions $\beta_1, \beta_2 \in \mathcal{K}_\infty$ and a continuous positive definite function $\rho$ such that

$$ \beta_1(|\xi|_{\mathcal{A}}) \le V(\xi) \le \beta_2(|\xi|_{\mathcal{A}}), \quad \forall \xi \in \mathcal{C} \cup \mathcal{D} \cup G(\mathcal{D}), \quad (2a) $$

$$ \langle \nabla V(\xi), F(\xi) \rangle \le -\rho(|\xi|_{\mathcal{A}}), \quad \forall \xi \in \mathcal{C}, \quad (2b) $$

$$ V(G(\xi)) - V(\xi) \le -\rho(|\xi|_{\mathcal{A}}), \quad \forall \xi \in \mathcal{D}, \quad (2c) $$

then $\mathcal{A}$ is UGpAS for (1).

### B. The synchronization problem

Consider a connected, undirected network of $N \ge 2$ identical agents with $Q$ edges. Each agent follows the hybrid dynamics

$$ \left\{ \begin{aligned} \dot{x}_i &= f(x_i) + Bu_i \\ x_i^+ &= g(x_i) + Gv_i \end{aligned} \right. \quad i \in \{1, \dots, N\}, \quad (3a) $$

where $x_i \in \mathbb{R}^n$ is the state of agent $i$, $u_i \in \mathbb{R}^{n_u}$ and $v_i \in \mathbb{R}^{n_v}$ are the control action on agent $i$, respectively, for the flow and the jump dynamics, $f: \mathbb{R}^n \to \mathbb{R}^n$ and $g: \mathbb{R}^n \to \mathbb{R}^n$ are $C^1$ globally Lipschitz functions and describe the dynamics with the input matrices $B \in \mathbb{R}^{n \times n_u}$ and $G \in \mathbb{R}^{n \times n_v}$.

In this work, the dynamics of the agent prioritizes jumps over flows in order to have simultaneous jumps for all the agents of the network. Let $\mathcal{F} \subset \mathbb{R}^n$ and $\mathcal{J} \subset \mathbb{R}^n$ be two different closed sets. By setting $\xi = (x_1, \dots, x_N) \in \mathbb{R}^{n_N}$, if one agent jumps all the other ones jump too. This synchronous jump protocol, where a jump condition met by any single agent triggers a network-wide jump, is adopted to model scenarios requiring coordinated mode switching and for analytical tractability within our proposed framework. Similar synchronous assumptions are common in the study of some hybrid observer designs [14] or [15]. As a consequence, we define the jump and flow set as

$$ \mathcal{D} := \{\xi \in \mathbb{R}^{n_N} : \exists i, x_i \in \mathcal{J}\}, \quad \mathcal{C} := \{\xi \in \mathbb{R}^{n_N} : \forall i, x_i \in \mathcal{F}\}. \quad (3b) $$

These assumptions, along with the definitions of $\mathcal{C}$ and $\mathcal{D}$, satisfy a simplified version of the hybrid basic conditions from [7, Assumption 6.5], guaranteeing the existence of solutions.

The main objective is to find distributed control laws $u_i$ and $v_i$ achieving synchronization of the $N$ agents of the network. To this end, we define the synchronization set

$$ \mathcal{A} = \{\xi \in \mathbb{R}^{n_N} : x_i = x_j \quad \forall i, j \in \{1, \dots, N\} \} \quad (4) $$

and the distance to this set $|\xi|_{\mathcal{A}} = \inf_{y \in \mathcal{A}} |\xi - y|$. Then our objective is to ensure stability and convergence to $\mathcal{A}$ which corresponds to synchronization of the agents. To this end, we will present a set of sufficient conditions for a distributed feedback of the form

$$ u_i = \sum_{j=1}^{N} \ell_{ij} \phi_F(x_j), \quad v_i = \sum_{j=1}^{N} \ell_{ij} \phi_J(x_j), \quad (5) $$

to solve the synchronization problem, for a suitable choice of the functions $\phi_F, \phi_J$.

## III. MAIN RESULT

### A. A hybrid factorization of the incidence matrix

The following lemma, based on [3], stems from the analysis of the relationship between the incidence and Laplacian matrices.

**Lemma 1:** Consider an undirected graph with $N$ agents and $Q$ edges. Let $W, E, L = EWE^\top$, and $L_e = E^\top E W$ be the weight matrix, the incidence matrix, the Laplacian matrix, and the edge Laplacian matrix, respectively. Given any
diagonal matrix $\Lambda \succ 0$, there exist a matrix $\Upsilon \in \mathbb{R}^{Q \times Q}$, and
real numbers $\underline{\alpha}, \bar{\alpha} > 0$ verifying

$$E^{\top}\Lambda L = \Upsilon E^{\top} \qquad (6a)$$

$$W\Upsilon + \Upsilon^{\top}W \succeq 2\underline{\alpha}W \qquad (6b)$$

$$\Upsilon^{\top}W\Upsilon \preceq (\bar{\alpha})^2W \qquad (6c)$$

*Proof:* First note that with $\ker(E) = \emptyset$ we have $E^{\top}\Lambda E \succ 0$ and $\Upsilon = E^{\top}\Lambda EW$ satisfy directly the three equations. Consider now the case $\ker(E) \neq \emptyset$ and denote its dimension by $\ell$. Let $(v_1, \dots, v_\ell)$ in $\mathbb{R}^Q$ be an orthonormal basis of $\ker(E)$. Let $\mu$ be a real number, then let

$$\Upsilon = E^{\top}\Lambda EW + \mu \sum_{i=1}^{\ell} v_i v_i^{\top}. \qquad (7)$$

Note that (6a) is satisfied with choice (7) of $\Upsilon$, indeed we obtain $\Upsilon E^{\top} = E^{\top}\Lambda EW E^{\top} + \mu \sum_{i=1}^{\ell} v_i v_i^{\top} E^{\top} = E^{\top}\Lambda L$. Then, we get $W\Upsilon + \Upsilon^{\top}W = Q_1 + \mu Q_2$ where $Q_1 = 2WE^{\top}\Lambda EW$ and $Q_2 = W \sum_{i=1}^{\ell} v_i v_i^{\top} + \sum_{i=1}^{\ell} v_i v_i^{\top} W$. Let $\omega$ in $\mathbb{R}^Q$ be such that $\omega^{\top}Q_1\omega = 0$ and $\omega \neq 0$, which implies $EW\omega = 0$ and therefore $W\omega = \sum_{k=1}^{\ell} a_k v_k$ for $\ell$ real numbers $a_k$ not all identically zero. Therefore we get $\omega^{\top}Q_2\omega = (W\omega)^{\top} \sum_{i=1}^{\ell} v_i v_i^{\top} \omega + \omega^{\top} \sum_{i=1}^{\ell} v_i v_i^{\top} (W\omega)$ which gives $\omega^{\top}Q_2\omega = 2(\sum_{k=1}^{\ell} a_k v_k^{\top})W^{-1}(\sum_{k=1}^{\ell} a_k v_k) > 0$ as $W$ is diagonal with positive elements. To sum up, we have that $(\omega^{\top}Q_1\omega = 0, \omega \neq 0) \Rightarrow \omega^{\top}Q_2\omega > 0$. Hence, by Finsler's lemma, there exists $\mu$ such that $Q_1 + \mu Q_2 = W\Upsilon + \Upsilon^{\top}W \succ 0$. Therefore there exists a sufficiently small scalar $\underline{\alpha} > 0$ such that $W\Upsilon + \Upsilon^{\top}W \succeq 2\underline{\alpha}W$ which shows that (6b) is satisfied with the previous choice (7) of matrix $\Upsilon$. Finally, note that with the same approach, using Finsler's lemma with $Q_1 = \Upsilon^{\top}W\Upsilon$ and $Q_2 = W$, there exists a sufficiently large scalar $\bar{\alpha} > 0$ such that $\Upsilon^{\top}W\Upsilon \preceq (\bar{\alpha})^2W$, concluding the proof of the second case $\ker(E) \neq \emptyset$. ■

The three equations in (6) are essential for the upcoming Lyapunov analysis. Equation (6b) will enable us to establish conditions for the continuous-time system, whereas equation (6c) is critical for the discrete-time jump analysis.

## B. A Riccati-based hybrid synchronization result

Building upon the hybrid multi-agent dynamics (3)-(5), we introduce sufficient conditions for designing distributed control laws that guarantee synchronization.

**Assumption 1:** The graph $\mathcal{G}$ is undirected and connected.

Extending this approach to directed graphs remains an open problem, as preserving the symmetry and essential properties of the edge Laplacian from [3] is still unresolved. To ensure global synchronization, we assume the graph is connected. If not, the network decomposes into connected clusters, to which our approach can be applied independently, resulting in multi-cluster synchronization.

**Assumption 2:** The set $\mathcal{F}$ and $\mathcal{F} \cup \mathcal{J}$ are convex and there exist positive definite symmetric matrices $P, R_K, R_F \succ 0$ and

real numbers $k_F, \varepsilon_F, k_J, \varepsilon_J > 0$ satisfying

$$\frac{\partial f}{\partial x}(x)^{\top}P + P\frac{\partial f}{\partial x}(x) - k_F PBR_F^{-1}B^{\top}P \preceq -\varepsilon_F P, \forall x \in \mathcal{F}, \qquad (8a)$$

$$\frac{\partial g}{\partial x}(x)^{\top}\Omega\frac{\partial g}{\partial x}(x) - P \preceq -\varepsilon_J P, \quad \forall x \in \mathcal{F} \cup \mathcal{J}, \qquad (8b)$$

where $\Omega := P - k_J PG(R_J + G^{\top}PG)^{-1}G^{\top}P$.

Each inequality in (8) corresponds to a classical controllability-like assumption used in the contraction framework. If the functions $f, g$ are linear, (8) coincides with standard Algebraic Riccati Inequalities for continuous-time and discrete-time systems.

In Assumption 2, however, we ask for a “common solution $P$” satisfying both the continuous-time and the discrete-time conditions. While requiring such a common $P$ can be conservative, this choice is instrumental for constructing a single quadratic Lyapunov function (as in Eq. (14)) to analyze the stability of the overall hybrid system using Theorem 1. This approach provides a tractable path to LMI-based conditions and is often adopted in the study of hybrid and switched systems when a unified stability proof is sought (e.g., in reset systems or for homogeneous hybrid systems where common quadratic Lyapunov functions are frequently employed). Moreover, the discrete-time condition (8b) has to hold not only in the set $\mathcal{J}$, but also in the set $\mathcal{F}$. This is needed because, due to the definition of the jump set $\mathcal{D}$ in (3b), jumps may triggered for an agent even though its state is not in the set $\mathcal{J}$. Finally, convexity in needed in order to apply those conditions to any pair of vectors $x_i, x_j \in \mathcal{F} \cup \mathcal{J}$.

Based on conditions (8), we introduce as a degree of freedom two diagonal matrices $\Lambda_F = (\lambda_1^F, \dots, \lambda_N^F) \in \mathbb{R}^N$ and $\Lambda_J = (\lambda_1^J, \dots, \lambda_N^J) \in \mathbb{R}^N$, with $\Lambda_F \succ 0$ and $\Lambda_J \succ 0$ and we propose a feedback of the following form

$$u_i = -\gamma_F R_F^{-1} B^{\top} P \sum_{j=1}^{N} \lambda_i^F \ell_{ij} x_j \qquad (9a)$$

$$v_i = -\gamma_J (R_J + G^{\top}PG)^{-1}G^{\top}P \sum_{j=1}^{N} \lambda_i^J \ell_{ij} g(x_j) \qquad (9b)$$

where $\ell_{ij}$ are the elements of the Laplacian matrix $L$, $\gamma_F$ and $\gamma_J$ are positive scalars to be chosen in a specific set, and $P, R_F, R_J$ selected to satisfy Assumption 2 (some guidelines about this are given in Section III-C). We highlight that $\Lambda_F$ and $\Lambda_J$ are normalizing factors for the Laplacian matrix. While typically for the continuous time feedback-law one can simply pick $\Lambda_F = I_N$, for the discrete-time counterpart, the use of normalized Laplacian matrices is often needed in order to design stabilizing feedback [4]. As a consequence, $\Lambda_J$ can be chosen as a diagonal matrix which can be optimized in order to increase the possible choices for $\gamma_J$. For instance, one can pick $\Lambda_J = [I_N + \text{diag}(L1_N)]^{-1}$. In doing so, it can be verified that $\tilde{L} = \Lambda_J L$ is a normalized Laplacian matrix (see, e.g. [4]).

Next, given $\Lambda_F$ and $\Lambda_J$, employing Lemma 1 with $\Lambda = \Lambda_F$ and $\Lambda = \Lambda_J$, respectively, yields two matrices $\Upsilon_F$ and $\Upsilon_J$, and the corresponding values $\underline{\alpha}_F, \bar{\alpha}_F, \underline{\alpha}_J, \bar{\alpha}_J$, that satisfy
conditions (6a)-(6c). Based on these scalars, and also based on the gains $\kappa_F$, $\kappa_J$ appearing in (8), we introduce the following subsets $\mathcal{I}_F$ and $\mathcal{I}_J$ of the real line

$$ \mathcal{I}_F := \left[ \frac{k_F}{2\alpha_F}, +\infty \right), \quad \kappa := \frac{\alpha_J}{\bar{\alpha}_J}, \\ \mathcal{I}_J := \left[ \frac{\kappa - \sqrt{\kappa^2 - k_J}}{\bar{\alpha}_J}; \frac{\kappa + \sqrt{\kappa^2 - k_J}}{\bar{\alpha}_J} \right], \quad (10) $$

The conditions for these sets to be well-defined (i.e., for $\mathcal{I}_J$ to contain real numbers) are established in Theorem 2, which is the main result of this work pertaining hybrid synchronization for multi-agent systems. Its proof is given in Section V.

**Theorem 2:** Consider a network described by $N$ agents (3) and suppose that Assumptions 1-2 hold with $k_J \le \kappa^2$, given in (10). Then, the distributed feedback control law (9) solves the synchronization problem, namely, the set $\mathcal{A}$ defined in (4) is pre-asymptotically stable for any $(\gamma_F, \gamma_J) \in \mathcal{I}_F \times \mathcal{I}_J$, defined as in (10).

We remark that if the closed-loop solutions of the collective dynamics (3)-(9) are such that bounded sets of initial conditions of all the agents lead to bounded synchronizing responses, then the stability result of Theorem 2 is semiglobally robust in the sense of [7, ch. 7]. In particular, leveraging [7, Thm 7.21] we obtain that for each (arbitrarily large) compact set of initial conditions, there exists a nontrivial perturbation function such that the perturbed dynamics (as characterized in [7, Def 6.27]) enjoys the same global asymptotic stability result.

### C. Convex formulation of the design conditions

In this section equations (6) and (8), necessary to design the matrices $\Upsilon_F$, $\Upsilon_J$ and $P$, are reformulated as a Linear Matrix Inequality (LMI) problem.

Previously in Theorem 2 and equation (9), we introduced the two gains $\gamma_F$ and $\gamma_J$ of the control law for the flow and the jump part. The choice of these gains leading to the synchronization of the agents is restricted to the two set defined in (10). To ensure the largest possible choice of gains, we can optimize the choice of $\Upsilon_F$, $\Upsilon_J$, $\Lambda_F$ and $\Lambda_J$ so that both sets are as large as possible. Such maximization is desirable as it offers greater flexibility for controller tuning concerning performance aspects (such as convergence rate or control effort, though not explicitly optimized here) and can contribute to improved robustness margins against model uncertainties.

**Remark 1:** To maximize $\mathcal{I}_F$ in (10), we need to minimize the term $\frac{k_F}{2\alpha_F}$, which amounts to finding the largest possible $\bar{\alpha}_F$ for $\Lambda_F$ satisfying Lemma 1.

**Remark 2:** To maximize the set $\mathcal{I}_J$ in (10) we need to maximize the term $\frac{2}{\bar{\alpha}_J}\sqrt{\kappa^2 - k_J}$. Due to homogeneity of our convexified conditions, without loss of generalities, we can assume that $\alpha_J = 1$, then we want to select $\frac{2}{\bar{\alpha}_J}\sqrt{1 - k_J\bar{\alpha}_J^2}$ as large as possible. To this end, we need to minimize $\bar{\alpha}_J$ for any $\Upsilon_J$ satisfying (6) in Lemma 1.

The following lemma presents a convex formulation of the graph-related conditions in (6).

**Lemma 2:** Let $E, L$ and $W$ be the incidence, the Laplacian and the weighting matrices. Then, the following LMIs in the decision variables $\Upsilon_F, \Lambda_F > 0, \alpha_F > 0, \Upsilon_J, \Lambda_J > 0$, and $\alpha_J > 0$

$$ 2\alpha_F W - W\Upsilon_F - \Upsilon_F^\top W \le 0, \quad \Upsilon_F E^\top = E^\top \Lambda_F L, \quad (11) $$

$$ \begin{gathered} 2W - W\Upsilon_J - \Upsilon_J^\top W \le 0, \quad \Upsilon_J E^\top = E^\top \Lambda_J L, \\ \begin{pmatrix} \bar{\alpha}_J W & \Upsilon_J^\top W \\ W\Upsilon_J & \bar{\alpha}_J W \end{pmatrix} \succeq 0 \end{gathered} \quad (12) $$

hold if and only if conditions (6a)-(6c) from Lemma 1 hold with $\Lambda = \Lambda_J$ and $\alpha_J = 1$, and also with $\Lambda = \Lambda_F$ and for some $\bar{\alpha}_F$ sufficiently large.

**Proof:** The two first part of (11) and (12) are equivalent to (6b) and (6a). The last part of (12) is equivalent to (6c) using a Schur complement. ■

The next lemma presents a convex formulation of the conditions in Assumption 2.

**Lemma 3:** Let $R_F \in \mathbb{R}^{n \times n_u}$ and $R_J \in \mathbb{R}^{n \times n_v}$ be two positive definite matrices, let $k_F > 0$ and $k_J \in (0, 1]$. The following LMIs in the decision variables $S > 0$, and $\Pi > 0$

$$ S \frac{\partial f}{\partial x}(x)^\top + \frac{\partial f}{\partial x}(x)S - k_F B R_F^{-1} B^\top \preceq 0 \quad (13a) $$

$$ \left( R_J^{-1} - \Pi \quad \sqrt{1 - k_J} R_J^{-1} G^\top \right) \succeq 0 \quad (13b) $$

$$ \left( S \quad S \frac{\partial g}{\partial x}(x)^\top \\ \frac{\partial g}{\partial x}(x)S \quad (S + k_J G \Pi G^\top) \right) > 0 \quad (13c) $$

hold for all $x \in \mathcal{F} \cup \mathcal{J}$ if and only if conditions (8) hold with $P = S^{-1}$. Moreover, (13) is a generalized eigenvalue problem<fnref n="1" /> in $(R_F, R_J, k_F, k_J)$.

**Remark 3:** Note that the conditions (13a)-(13c) represent an infinite set of LMIs due to their dependence on the state variable $x \in \mathcal{F} \cup \mathcal{J}$. However, in many practical cases, these conditions can be reduced to a finite (though potentially large) set of LMIs using techniques like sector conditions, polytopic embeddings, or sum-of-squares (SOS) methods. This allows for the practical implementation of these conditions using standard LMI solvers.

**Proof:** The equivalence of (13a) with (8a) is direct by pre and post multiplying by $P = S^{-1}$. For the jump part, the equivalence between the pair of inequalities (13b)-(13c) and condition (8b) is established as follows. Using the Schur complement and the matrix inversion lemma, one can show that (13b) is equivalent to the bound $\Pi \preceq (R_J + (1 - k_J)G^\top PG)^{-1}$. Similarly, (13c) is equivalent to

$$ \frac{\partial g}{\partial x}(x)^\top (P - PG((k_J \Pi)^{-1} + G^\top PG)^{-1} G^\top P) \frac{\partial g}{\partial x}(x) < P. $$

Substituting the upper bound for $\Pi$ (or equivalently, the lower bound for $\Pi^{-1}$) from the first condition into the second one retrieves exactly inequality (8b) after standard algebraic manipulations. ■

<footnote n="1">¹In other words, if the LMI problem is feasible for $(\bar{R}_F, \bar{R}_J, \bar{k}_F, \bar{k}_J)$, then it is feasible for any $(R_F, R_J, k_F, k_J)$ such that $R_F \preceq \bar{R}_F, R_J \preceq \bar{R}_J, k_F \ge \bar{k}_F$ and $k_J \ge \bar{k}_J$, and vice-versa for infeasibility.</footnote>
**Algorithm 1** Hybrid Synchronization Control Design Procedure

1. **Input:** Dynamics $f, g, B, G$; graph defining $E, W$; sets $\mathcal{F}, \mathcal{J}$.

2. **Output:** Control gains $\gamma_F, \gamma_J$; matrix $P$.

3. Construct graph Laplacian $L = EWE^\top$.

4. Choose normalization matrices $\Lambda_F \succ 0$ and $\Lambda_J \succ 0$ (e.g., $\Lambda_F = I_N$, $\Lambda_J = (I_N + \text{diag}(L1_N))^{-1}$ or $I_N$).

5. Solve LMIs in Lemma 2 (Eqs. (11), (12)) to find matrices $\Upsilon_F, \Upsilon_J$ and scalars $\underline{\alpha}_F > 0, \overline{\alpha}_J > 0$ (optimizing as per Remarks 1 and 2, e.g., maximize $\underline{\alpha}_F$, minimize $\overline{\alpha}_J$ with $\underline{\alpha}_J = 1$ for (12)).

6. Choose positive definite design matrices $R_F$ and $R_J$.

7. Solve LMIs in Lemma 3 (Eqs. (13a)-(13c)) for $S \succ 0$ (then $P = S^{-1}$) and $\Pi \succ 0$, to find scalars $k_F > 0, \varepsilon_F > 0, k_J \in (0,1], \varepsilon_J > 0$ satisfying Assumption 2. This requires verifying the conditions for all $x \in \mathcal{F} \cup \mathcal{J}$ (see Remark 3).

8. Calculate $\kappa = \underline{\alpha}_J/\overline{\alpha}_J$. Verify $k_J \le \kappa^2$. If not satisfied, adjust choices in Steps 2, 4 or revisit LMI solutions.

9. Determine feasible gain intervals $\mathcal{I}_F := [\frac{k_F}{2\alpha_F}, +\infty)$ and $\mathcal{I}_J := [\frac{\kappa-\sqrt{\kappa^2-k_J}}{\alpha_J}; \frac{\kappa+\sqrt{\kappa^2-k_J}}{\alpha_J}]$ using Eq. (10).

10. Select control gains $\gamma_F \in \mathcal{I}_F$ and $\gamma_J \in \mathcal{I}_J$.

11. Implement distributed control laws $u_i$ and $v_i$ as in (9).

<figure><img src="image_1.png" /><figcaption>Fig. 1. Connected network with 5 nodes. The numbers above arrows denote the weights of the edge between any two agents.</figcaption></figure>

## IV. NUMERICAL EXAMPLE

Consider the undirected and connected network with $N = 5$ agents represented in Figure 1. Each agent $i = 1, \dots, N$, is described by the following dynamics

$$ 
\left\{ 
\begin{aligned}
\dot{x}_{i,1} &= x_{i,2} + \frac{1}{2}\sin(x_{i,1}) + u_i \\
\dot{x}_{i,2} &= -x_{i,1} + \frac{1}{2}x_{i,2}\cos(x_{i,1}) + \frac{1}{4}\cos(x_{i,1})\sin(x_{i,1}) + 2u_i \\
x_{i,1}^{+} &= \frac{1.4x_{i,1}^2}{1+x_{i,1}^2} + v_i \\
x_{i,2}^{+} &= \frac{x_{i,1}^2}{1+x_{i,1}^2+x_{i,2}^2} - 2v_i.
\end{aligned}
\right. 
$$

In this simulation, the flow set is $\mathcal{F} = [-3, 3]^2$ such that each time a state of any agent reach the value 3, all the agents jump. Then the jump set is $\mathcal{J} = \{x \in [-3; 3]^2 : \max|x| = 3\}$. The agents are randomly initialized within $\mathcal{F}$, and the procedure from Algorithm 1 is followed. In Step 4, we set $\Lambda_F = \Lambda_J = I_5$, and in Step 6, $R_F = R_J = I_2$. Solving the LMIs with MATLAB's Robust Control Toolbox yields feasible gain intervals, from which we select $\gamma_F = 7.86$ and $\gamma_J = 0.027$. The simulation shown in Figure 2, run using MATLAB's Hybrid Equations Toolbox [16], confirms that the distributed control law (9) synchronizes all the agents.

<figure><img src="image_2.png" /><figcaption>Fig. 2. Evolution of the state $(x_1, x_2)$ of agents $a_i, i = 1, \dots, N$.</figcaption></figure>

## V. LYAPUNOV-BASED PROOF OF THEOREM 2

### A. Lyapunov function and its properties

We rely on Theorem 1 in order to prove Theorem 2 concerning the synchronization problem for system (3), (9) under Assumptions 1, 2. To this end, we construct a Lyapunov function satisfying the conditions of Theorem 1, with $\mathcal{A}$ defined as in (4).

Let $e \in \mathbb{R}^{nQ}$ be the vector $e := (E^\top \otimes I_n)\xi$, with $E \in \mathbb{R}^{N \times Q}$ the incidence matrix. We set $V(\xi) := \xi^\top (L \otimes P)\xi$, so

$$ V(\xi) = e^{\top}(W \otimes P)e = \sum_{i=1}^{Q} w_i(x_{k_i} - x_{l_i})^{\top} P(x_{k_i} - x_{l_i}), \quad (14) $$

where $L$ is the Laplacian matrix, $W$ is the diagonal weight matrix and $P$ is the positive definite matrix verifying conditions (8). The Lyapunov function (14) satisfies the conditions of Theorem 1 as stated in the following lemma.

**Lemma 4:** Consider a network described by $N$ agents (3) and suppose that Assumptions 1-2 hold. Then, the Lyapunov function (14) satisfies (2a)-(2c) with $\mathcal{A}$ defined in (4).

The proof of Theorem 2 follows from direct application of Theorem 1. Since $\mathcal{A}$ is UGpAS, global asymptotic synchronization of the agents of the network is guaranteed.

### B. Proof of Lemma 4

Inequality (2a) follows from (14) and the positive definiteness of $P$. To show the flow condition (2b), the dynamics of the vector $e$ are written using $\Upsilon_F$ from Lemma 1 as:

$$ \dot{e} = (\phi_f(\xi, e, z) - \gamma_F(\Upsilon_F \otimes BR_F^{-1}B^\top P))e, \quad (15) $$

$$ \phi_f(\xi, e, z) = \int_0^1 \underbrace{\text{diag}\left\{\frac{\partial f}{\partial x}(x_{l_i} + ze_i)\right\}}_{D_f(\xi, e, z)} \mathrm{d}z \in \mathbb{R}^{nQ \times nQ}. $$

The directional derivative of (14) along (15) is computed as

$$ 
\begin{aligned}
\dot{V}(\xi) = \int_{0}^{1} e^{\top} \left( D_{f}(\cdot)^{\top} (W \otimes P) + (W \otimes P) D_{f}(\cdot) \right) e \mathrm{d}z \\
- e^{\top} \left( \gamma_{F}(W \Upsilon_{F} + \Upsilon_{F}^{\top} W) \otimes PBR_{F}^{-1} B^{\top} P \right) e
\end{aligned}
$$

where we may apply (8a), satisfied at all points between $x_i$ and $x_{l_i} + e_i$ due to convexity of $\mathcal{F}$. We obtain

$$ 
\begin{aligned}
\dot{V}(\xi) \le -\varepsilon_F V(\xi) \\
- e^{\top} ([\gamma_F(W\Upsilon_F + \Upsilon_F^\top W) - k_F W] \otimes PBR_F^{-1} B^\top P) e.
\end{aligned}
$$

Therefore, with $\gamma_F \ge \gamma_F^* = \frac{k_F}{2\alpha_F}$, the second term is non-positive and we obtain $\dot{V}(\xi) \le -\varepsilon_F V(\xi)$, implying (2b).
Finally, we need to show inequality (2c). Using control law (9b) and recalling $H = (R_J+G^\top PG)^{-1}$, and using the matrix $\Upsilon_J$ satisfying Lemma 1 we obtain

$$e^+ = (I_{nQ} - \gamma_J(\Upsilon_J \otimes GHG^\top P)) \phi_g(\xi, e, z)e,$$

where $\phi_g(\xi, e, z) = \int_0^1 \underbrace{\operatorname{diag}\left(\frac{\partial g}{\partial x}(x_{l_i} + ze_i)\right)}_{D_g(\xi, e, z)} \mathrm{d}z.$

By definition of the jump set $\mathcal{D}$ in (3b), the jump analysis must hold for any $x_i \in \mathcal{F} \cup \mathcal{J}$. Some computations yield:

$$\Delta V(\xi) = - \int_0^1 \tilde{e}^\top \tilde{P} \tilde{e} \mathrm{d}z, \quad \tilde{P} = \begin{pmatrix} W \otimes P & \tilde{P}_{12} \\ \tilde{P}_{12}^\top & W \otimes P \end{pmatrix},$$

with $\tilde{e} = (e, e^+)$, and

$\tilde{P}_{12} = D_g(\xi, e, z)^\top (I_{nQ} - \gamma_J(\Upsilon_J^\top \otimes PGHG^\top)) (W \otimes P)$. Then with a Schur complement from $(W \otimes P) \succ 0$ we get that,

$$\begin{aligned} \Delta V(\xi) < 0 &\Leftrightarrow \tilde{P} \succ 0 \\ &\Leftrightarrow \tilde{P}_{\text{Schur}} = (W \otimes P) - \tilde{P}_{12}(W \otimes P)^{-1}(\tilde{P}_{12})^\top \succ 0 \end{aligned}$$

We compute the expression of $\tilde{P}_{\text{Schur}}$,

$$\tilde{P}_{\text{Schur}} = (W \otimes P) - D_g(\cdot)^\top (W \otimes P) D_g(\cdot) \\ + \gamma_J D_g(\cdot)^\top ((W \Upsilon_J \otimes + \Upsilon_J^\top W) \otimes PGHG^\top P) D_g(\cdot) \\ - D_g(\cdot)^\top (\gamma_J^2 [\Upsilon_J^\top W \Upsilon_J \otimes PGHG^\top PGHG^\top P]) D_g(\cdot).$$

Using inequality (8b), satisfied at all points between $x_i$ and $x_{l_i} + e_i$ by convexity of the set $\mathcal{F} \cup \mathcal{J}$, we obtain

$$D_g(\xi, e, z)^\top (W \otimes P) D_g(\xi, e, z) \\ \preceq (1 - \varepsilon_J)(W \otimes P) + k_J D_g(\cdot)^\top (W \otimes PGHG^\top P) D_g(\cdot).$$

Combining the last two equations we get

$$\tilde{P}_{\text{Schur}} \succeq \varepsilon_J(W \otimes P) + D_g(\xi, e, z)^\top M D_g(\xi, e, z)$$

with $M = ([\gamma_J(W\Upsilon_J + \Upsilon_J^\top W) - k_J W] \otimes PGHG^\top P - \gamma_J^2[\Upsilon_J^\top W\Upsilon_J \otimes PGHG^\top PGHG^\top P])$. Summarizing, we have $\Delta V(\xi) < 0 \Leftrightarrow \tilde{P}_{\text{Schur}} \succ 0$ which is satisfied if $M \succeq 0$. To show $M \succeq 0$ since $R_J$ is positive definite we obtain $PGHG^\top PGHG^\top P \prec PGHG^\top P$, and we know from Lemma 1 that there exist $\alpha_J, \bar{\alpha}_J$ such that

$$M \succ (-(\bar{\alpha}_J)^2 \gamma_J^2 + 2\alpha_J \gamma_J - k_J) [W \otimes PGHG^\top P] \\ = \Psi(\gamma_J) [W \otimes PGHG^\top P]$$

From this last inequality, we conclude that $M \succeq 0$ if $\Psi(\gamma_J) \succeq 0$. By studying the polynomial function $\Psi(\gamma_J)$, using the assumption $k_J \le \kappa^2$, we obtain that for any $\gamma_J \in \mathcal{I}_J$, with $\mathcal{I}_J$ defined as in (10), we have $\Psi(\gamma_J) \ge 0$, namely $\Delta V(\xi) < 0$ for any $x_i \in \mathcal{F} \cup \mathcal{J}$, that is $\xi \in \mathcal{D}$ as defined in (3b). In other words, the jump condition (2c) is satisfied. ■

# VI. CONCLUSION

Based on a new result regarding the incidence matrix properties of a graph, we have proven that the existing distributed feedback design for continuous-time dynamics, combined with an adapted distributed feedback design for discrete-time dynamics, can address the synchronization problem for a network of input-affine hybrid systems connected through a weighted undirected graph. Our proof relies on constructing a specific quadratic Lyapunov function that encompasses the energy of the network's edges. We also propose an optimization-based tuning of the feedback gains using linear matrix inequalities. Additionally, networks can play a central role in distributed filtering, as shown in [17]. Interesting direction for future research include exploring how hybrid dynamics can be integrated into such distributed estimation frameworks, in addition to studying asynchronous jumps in hybrid networks.

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