## 1 Introduction
In the difference of the discrete Lyapunov-Krasovskii functional, there is a positive quadratic term and its coefficient is the time-delay. Therefore, only if the time-delay is bounded, conditions that makes the difference negative definite exist. In fact, the time delay may be unbounded. When we use a small number of nodes for search and rescue work in a large area, the communication distance between agents keeps increasing. For the multi-agent system with increasing communication distances, since the information transmission time and the communication distance are positively correlated, the time delay tends to infinity as time goes on. Hence, the multi-agent system with increasing communication distances is described as the multi-agent system with unbounded time-delays. Due to the unboundedness of delays, the above-mentioned method no longer applies. With unbounded time-delays, a finite-dimensional extended system cannot describe the multi-agent system with unbounded time-delays at all time. Hence, methods used in [15; 18] are ineffective. Then, let us consider the Lyapunov-Krasovskii functional with double integral such as [14; 16]. With unbounded time-delays, the positive quadratic term whose coefficient is the time-delay tends to positive infinity as time goes on. Therefore, the derivative of the Lyapunov-Krasovskii functional can not be negative definite. Furthermore, under the discrete-time multi-agent system, the discrete Lyapunov-Krasovskii functional with double summation faces the same problem. Hence, the dynamical study of unbounded delay system is recognized as being very difficult. The unbounded delay systems that have received attention are summarized into two categories. The first category is the multi-agent system with distributed infinite delays. Many practical systems, such as the traffic flow (see [19; 20]) and biological networks (see [21]), can be modeled as the system. In this category, the weighted delay information is used to control the system, and the greater the weight of the delay information is, the smaller the delay is. By using the algebraic graph theory and frequency domain analysis, Reference [19] shows that with the gamma distribution weight, the system can achieve a consensus if the topology is strongly connected. Further, under a weaker topological condition that the topology contains a spanning tree, Reference [20] shows that with the exponential distribution weight, the system can achieve a consensus by using a low gain controller. It can be noticed that, in this category, agents need to know the accurate delay of each received information. However, it is difficult to achieve in practical systems. In order to avoid this weakness, the second category is considered. The second category is the multi-agent system with non-distributed unbounded time-delays. Some systems with a spacial nature, such as neural networks (see [22; 23; 24]), can be modeled as the system. Different from the first category, in the second category of unbounded delay system, the delay of each received information is unknown to any agent. For the second category, many results are about the stability of the system (see [22; 23; 24]). In [22], the authors introduced generalized Halanay inequalities to study the stability of the system. In [23], the method, which is based on upper bounding of the state vector by a decreasing function, is presented to analyse the stability of the system. In [24], based on an impulsive differential delay inequality, several novel delay-dependent inequalities are obtained to ensure the global stability of the system. However, only a few results are about the consensus of the system (see [25; 26]). When the adjacency matrix of the topology is impartible, by using the algebraic graph theory and time domain analysis, the system achieves a consensus (see [25]). Furthermore, under the topology that satisfies the spanning tree assumption and the no-cycle assumption, Reference [26] solves the consensus problem of the system.