Exponential Synchronization of Hyperbolic Complex Spatio-Temporal Networks with Multi-Weights

: This paper deals with the leader-following synchronization of ﬁrst-order, semi-linear, complex spatio-temporal networks. Firstly, two sorts of complex spatio-temporal networks based on hyperbolic partial differential equations (CSTNHPDEs) are built: one with a single weight and the other with multi-weights. Then, a new distributed controller is designed to address CSTNHPDE with a single weight. Sufﬁcient conditions for the synchronization and exponential synchronization of CSTNHPDE are presented by showing the gain ranges. Thirdly, the proposed distributed controller addresses of CSTNHPDE with multi-weights, and gain ranges are obtained for synchronization and exponential synchronization, respectively. Finally, two examples show the effectiveness and good performance of the control methods.

A number of important works discuss the synchronization of complex networks [14][15][16][17][18]. This literature shows node dynamics depending only on time. In practice, the dynamics of all processes are spatio-temporal [19][20][21]. As a consequence, it is necessary to study complex spatiotemporal networks (CSTNs), which is with spatio-temporal characteristics [22]. Wu et al. studied the synchronization of CSTNs with space-independent coefficients and space-dependent coefficients, with or without spatio-temporal disturbance [23]. Huang et al. proposed a fuzzy synchronization method for nonlinear CSTNs with reaction-diffusion terms [24]. Luo et al. studied event-triggered control for the finite-time synchronization of reaction-diffusion CSTNs [25]. Yang et al. studied the boundary control of fractional-order CSTNs [26]. Zheng et al. researched synchronization analysis for fractional-order CSTNs with time delays [27]. Shen et al. studied the H ∞ synchronization of Markov jump CSTNs using an observer-based method [28]. Kabalan et al. studied boundary control for the synchronization of leader-follower CSTNs with in-domain coupling [29].
Most references are modeled by parabolic PDEs, while there are few methods studying hyperbolic PDEs. There are many hyperbolic PDEs systems in practice, including shallow-water systems [30], epidemic models [31], district heating networks [32], heat exchangers [33], and reactor models [34]. Therefore, it is important to study the synchronization of hyperbolic PDEs-based CSTNs (HPDECSTNs).
Chueshov presented invariant manifolds and nonlinear master-slave synchronization hyperbolic and parabolic CSTNs [35]. Li studied the synchronization, exact synchronization and approximate synchronization of HPDECSTNs [36]. Li and Lu researched exactboundary synchronization for a kind of first-order hyperbolic system [37]. Lu proposed a local exact-boundary synchronization for a kind of first-order, quasi-linear hyperbolic system [38]. However, technical difficulties remain regarding the synchronization of a semilinear, first-order HPDECSTNs when the convection coefficient is symmetric semi-negative definite or semi-positive definite, which motivate this paper. Multi-weights exist in many physical networks [39][40][41][42][43]. As a result, HPDECSTN with multi-weights is important and remains challenging.
This paper mainly studies the leader-following synchronization control of a semilinear HPDECSTN with two sorts of boundary conditions in a one-dimensional space. This paper's contributions are as follows: (1) Two sorts of HPDECSTN models are built, one with a single weight and the other with multi-weights. (2) A new distributed controller is designed to address CSTNHPDE with a single weight. Sufficient conditions for the synchronization and exponential synchronization of CSTNHPDE are presented by providing the gain ranges. (3) The proposed distributed controller addresses CSTNHPDE with multiweights and gain ranges, obtained for synchronization and exponential synchronization, respectively. (4) Two examples show the effectiveness and good performance of the control methods.

Problem Formulation
This paper first studies a class of leader-following, semi-linear, hyperbolic PDE-based, complex spatio-temporal networks (HPDECSTNs) with a single weight. The following node is assumed to be where (ζ, t) ∈ [0, L] × [0, ∞) are space and time, respectively. z i (ζ, t) and u i (ζ, t) ∈ R n are the state and control input, respectively. 0 < L ∈ R is a constant. A ∈ R n×n , B ∈ R n×n , and Γ ∈ R n×n are constant matrices. f (·) is a nonlinear function. The coupling strength The leader node is assumed to be where s(ζ, t) ∈ R n is the state. This paper aims to study a distributed controller u i (ζ, t), driving HPDECSTN (1) to the leader node (2), designed as where d i are the control gains to be determined.

Synchronization of HPDECSTNs with a Single Weight
Let the synchronization error be e i (ζ, t) The error system of between HPDECSTN (1) and (2)) yields where and F(e(ζ, t)) where Ψ I N ⊗ A+A T 2 + 0.5I N ⊗ BB T + 0.5χ 2 I Nn + 0.5c(G ⊗ Γ + G T ⊗ Γ T ).
where Ξ I N ⊗ A+A T 2 + 0.5I N ⊗ BB T + 0.5χ 2 I Nn + 0.5c 1 ( Proof. The proof is similar to that of Theorem 1, and so it is omitted.
Proof. The proof is similar to that of Theorem 2, and so it is omitted.

Remark 1.
This paper addresses not only the synchronization of HPDECSTNs, but also the exponential synchronization. Moreover, this paper addresses HPDECSTNs not only with a single weight, but also with multi-weights.

Remark 3. Different from the control design for synchronization of parabolic PDEs-based
CSTNs [44,45], this paper deals with the synchronization of hyperbolic PDEs-based CSTNs.

Remark 4.
Only a few important results discussed the synchronization, exact synchronization and approximate synchronization of HPDECSTNs [36][37][38]. Different from those with a single weight, this paper addresses the case with multi-weights.

Numerical Simulation
Example 1. Consider a single weighted HPDECSTN (1) with random initial conditions and The single weight takes (25) Figure 1 shows that HPDECSTN (1) cannot reach synchronization without control. It is obvious that χ = 1. With Theorem 1, solve (16) by Matlab, the feedback gains d i = 12.04 are obtained. Figure 2 shows that HPDECSTN (1) reaches exponential synchronization under the controller (2) with d i = 12.04. The controller (2) with the feedback gains d i = 12.04 is shown in Figure 3.

Example 2.
Consider multi-weighted HPDECSTN (20) with random initial conditions and the same parameters as those of Example 1, except: The weights take (28) Figure 4 shows that HPDECSTN (20) cannot reach synchronization without control. With Theorem 4, solving (23) using Matlab, the feedback gains d i = 26.21 are obtained. Figure 5 shows that HPDECSTN (20) reaches exponential synchronization under controller (2) with d i = 26.21. The controller (2) with the feedback gains d i = 26.21 is shown in Figure 6.

Conclusions
This paper has dealt with the leader-following synchronization control of two classes of semi-linear HPDECSTNs: one HPDECSTN with a single weight, and the other with multi-weights. To drive HPDECSTNs to synchronization, one new distributed controller was constructed. Dealing with HPDECSTNs with a single weight, sufficient conditions for synchronization and exponential synchronization of CSTNHPDE were presented by providing gain ranges. Furthermore, the proposed distributed controller was used to address CSTNHPDE with multi-weights and gain ranges, which were obtained for synchronization and exponential synchronization, respectively. Two examples illustrated the effectiveness of the developed theoretical results. In future work, the event-triggered control and pinning control of HPDECSTNs will be studied.