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This paper addresses a distributed consensus optimization problem of a first-order multiagent system with time-varying delay. A continuous-time distributed optimization algorithm is proposed. Different from most ways of solving distributed optimization problem, the Lyapunov-Razumikhin theorem is applied to the convergence analysis instead of the Lyapunov-Krasovskii functionals with LMI conditions. A sufficient condition for the control parameters is obtained to make all the agents converge to the optimal solution of the system. Finally, an example is given to validate the effectiveness of our theoretical result.

In recent years, the distributed optimization problem of multiagent systems has been investigated by many researchers; researches on distributed optimization and control theorem have been developing rapidly and have been applied to various fields of industry and defense, like smart grid [

The current literatures about distributed optimization problems are more focused on discrete-time algorithms (see [

On the other hand, it is common that time-delay exists in practical systems because of the finite speeds of information transmission and spreading as well as traffic congestions. Therefore, time-delay should be taken into account in algorithm design of multiagent systems. For time-delay systems modelled by delayed differential equations, an effective way to deal with their convergence and stability analysis is based on the Lyapunov-Krasovskii functionals or Lyapunov-Razumikhin functions. Most of the existing works concentrate on Lyapunov functions combining with Linear Matrix Inequality (LMI) techniques to deal with the consensus problem of multiagent systems with time-delay [

Motivated by the above observations, the distributed consensus optimization problem of continuous-time multiagent systems with time-varying delay is considered. The interconnected graph is assumed to be directed, strongly connected, and weighted-balanced. The Lyapunov-Razumikhin function is used in the stability analysis. The convergence of the proposed algorithm is guaranteed with the model parameters satisfying some conditions. Meanwhile, the conditions can also give an estimate of the upper bound of the time-delay, which can avoid verifying and calculating the complicated LMI conditions. From the results, we can also see clearly the relationship among the parameters in the system.

The outline of this paper is organized as follows. Some basic knowledge on the algebraic graph theory and useful lemmas are presented in Section

Consider a multiagent system consisting of

The next lemmas related to the important properties of Laplace

Laplace matrix

For matrices

For a given real matrix

We consider a multiagent system consisting of

Consider the multiagent optimization problem, in which the goal is to minimize the sum of local cost functions associated with the individual agent. More specially, it can be expressed as

In this paper, our goal is to design a distributed controller for each agent such that the states of all the agents converge to the optimal solution of the optimization problem (

Before proceeding, we give the following assumption on the local cost function

(a) For each

(b) for

Under Assumption

The digraph

From Lemma

When considering the presence of time-varying communication delay among the information transmission, the continuous-time distributed optimization protocol is proposed for agent

Let

Using the transformation

Before analyzing the consensus and optimization problem (

The definition of the stability of the solution

Let

Usually,

Then the main results can be obtained as follows.

Suppose Assumptions

Then, the optimization problem (

Let

Denote

The derivation of

For the second and fourth equalities of system (

By the Leibniz-Newton formula

Thus, we can get

Due to

According to the Lyapunov-Razumikhin Theorem, take

With the transformation

The continuous-time protocol considered in this paper is based on the algorithm proposed in [

In this section, we give an example to validate our theoretical results. In the example, we consider a multiagent system consisting of five agents. Suppose that the interconnected topology is described as in Figure

Connected graph.

Consider the following optimization problem:

Let the initial values

The trajectories of

The trajectories of

We can see that the trajectories

In this paper, the consensus optimization problem of multiagents with communication delays was considered. By a continuous-time algorithm, consensus and optimization under some parameter bound conditions are ensured. Graph theory is used to describe the interconnection topologies. Lyapunov-Razumikhin theory were employed for stability analysis. The connectivity assumption of directed graph plays a key role in the analysis of algorithm convergence. Numerical examples were given to illustrate the theoretical results.

The authors declare that they have no conflicts of interest.

This work is supported in part by Natural Science Foundation of China (61273183 and 61374028).