Reaching a nonlinear consensus: polynomial stochastic operators

We provide a general nonlinear protocol for a structured time-varying and synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of a polynomial stochastic operator associated with a multidimensional stochastic hypermatrix. We show that the multi-agent system eventually reaches to a consensus if either one of the following two conditions is satisfied: (i) every member of the group people has a positive subjective opinion on the given task after some revision steps or (ii) all entries of a multidimensional stochastic hypermatrix are positive. Numerical results are also presented

Velocity of Long Gravity Waves in the Ocean

Multi-Agent Systems (MAS) have attracted more and more interest in recent years. Most researches in the study of discrete-time MAS, presented in the past few years, have considered linear cooperative rules. However, local interactions between agents are more likely to be governed by nonlinear rules. In this paper, we investigate the consensus of discrete-time MAS with time invariant nonlinear cooperative rules. Based on our presented nonlinear model, we show a consensus in the discrete-time MAS. Our model generalizes a classical time invariant De Groot model. It seems that, unlike a linear case, a consensus can be easily achieved a nonlinear case

Mathematical models of nonlinear uniform consensus

We consider a nonlinear protocol for a structured time-invariant and synchronous multi-agent system. In the multi-agent system, we present opinion sharing dynamics as a trajectory of a cubic triple stochastic matrix. We provide a criterion for a uniform consensus of the multi-agent system. We show that the multi-agent system eventually reaches a consensus if either one of the following two conditions is satisfied: (i) every member of the group people has a positive subjective opinion on the given task after some revision steps or (ii) all entries of the given cubic triple stochastic matrix are positive

Reaching nonlinear consensus via non-autonomous polynomial stochastic operators

This paper is a continuation of our previous studies on nonlinear consensus which uni�es and generalizes all previous results. We consider a nonlinear protocol for a structured time-varying synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of non-autonomous polynomial stochastic operators associated with multidimensional stochastic hyper-matrices. We show that the multi-agent system eventually reaches to a nonlinear consensus if either one of the following two conditions is satisfied: (i) every member of the group people has a positive subjective distribution on the given task after some revision steps or (ii) all entries of some multidimensional stochastic hyper-matrix are positive

Nonlinear uniform consensus: a polynomial protocol

We provide a general nonlinear protocol for a structured time-invariant and synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of a polynomial stochastic operator associated with a stochastic multidimensional hyper-matrix. We provide a criterion for a uniform consensus in the multi-agent system. Particularly, the uniform consensus is achieved in the multi-agent system if all entries of the stochastic multidimensional hyper-matrix are positive

Mathematical models of nonlinear uniform consensus II

This paper is a continuation of our previous studies on nonlinear consensus. We have considered a nonlinear protocol for a structured time-invariant and synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of a polynomial stochastic operator associated with a stochastic multidimensional hyper-matrix. We provide a criterion for a uniform consensus in the multi-agent system. Particularly, the uniform consensus is achieved in the multi-agent system if all entries of the stochastic multidimensional hyper-matrix are positive. Some numerical results are also presented to support our theoretical results

Mathematical Models of Nonlinear Uniform Consensus II

This paper is a continuation of our previous studies on nonlinear consensus. We have considered a nonlinear protocol for a structured time-invariant and synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of a polynomial stochastic operator associated with a stochastic multidimensional hyper-matrix. We provide a criterion for a uniform consensus in the multi-agent system. Particularly, the uniform consensus is achieved in the multi-agent system if all entries of the stochastic multidimensional hyper-matrix are positive. Some numerical results are also presented to support our theoretical results

Sarymsakov cubic stochastic matrices

The class of Sarymsakov square stochastic matrices is the largest subset of the set of stochastic, indecomposable, aperiodic (SIA) matrices that is closed under matrix multiplication and any infinitely long left-product of the elements from any of its compact subsets converges to a rank-one (stable) matrix. In this paper, we introduce a new class of the so-called Sarymsakov cubic stochastic matrices and study the consensus problem in the multi-agent system in which an opinion sharing dynamics is presented by quadratic stochastic operators associated with Sarymsakov cubic stochastic matrices

Ganikhodjaev\u27s conjecture on mean ergodicity of quadratic stochastic operators

A linear stochastic (Markov) operator is a positive linear contraction which preserves the simplex. A quadratic stochastic (nonlinear Markov) operator is a positive symmetric bilinear operator which preserves the simplex. The ergodic theory studies the long term average behavior of systems evolving in time. The classical mean ergodic theorem asserts that the arithmetic average of the linear stochastic operator always converges to some linear stochastic operator. While studying the evolution of population system, S.Ulam conjectured the mean ergodicity of quadratic stochastic operators. However, M.Zakharevich showed that Ulam\u27s conjecture is false in general. Later, N.Ganikhodjaev and D.Zanin have generalized Zakharevich\u27s example in the class of quadratic stochastic Volterra operators. Afterwards, N.Ganikhodjaev made a conjecture that Ulam\u27s conjecture is true for properly quadratic stochastic non-Volterra operators. In this paper, we provide counterexamples to Ganikhodjaev\u27s conjecture on mean ergodicity of quadratic stochastic operators acting on the higher dimensional simplex

Reaching a consensus: a discrete nonlinear time-varying case

In this paper, we have considered a nonlinear protocol for a structured time-varying and synchronous multi-agent system. By means of cubic triple stochastic matrices, we present an opinion sharing dynamics of the multi-agent system as a trajectory of a non-homogeneous system of cubic triple stochastic matrices. We show that the multi-agent system eventually reaches to a consensus if either of the following two conditions is satisfied: (1) every member of the group people has a positive subjective distribution on the given task after some revision steps or (2) all entries of some cubic triple stochastic matrix are positive