Opuscula Math. 39, no. 1 (2019), 77-89
https://doi.org/10.7494/OpMath.2019.39.1.77

Opuscula Mathematica

# The existence of consensus of a leader-following problem with Caputo fractional derivative

Ewa Schmeidel

Abstract. In this paper, consensus of a leader-following problem is investigated. The leader-following problem describes a dynamics of the leader and a number of agents. The trajectory of the leader is given. The dynamics of each agent depends on the leader trajectory and others agents' inputs. Here, the leader-following problem is modeled by a system of nonlinear equations with Caputo fractional derivative, which can be rewritten as a system of Volterra equations. The main tools in the investigation are the properties of the resolvent kernel corresponding to the Volterra equations, and Schauder fixed point theorem. By study of the existence of an asymptotically stable solution of a suitable Volterra system, the sufficient conditions for consensus of the leader-following problem are obtained.

Keywords: leader-following problem, Caputo fractional differential equation, consensus, nonlinear system, Schauder fixed point theorem.

Mathematics Subject Classification: 26A33, 34K20, 45D05.

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• Communicated by Josef Diblík.
• Revised: 2018-03-11.
• Accepted: 2018-03-30.
• Published online: 2018-08-07.

Ewa Schmeidel, The existence of consensus of a leader-following problem with Caputo fractional derivative, Opuscula Math. 39, no. 1 (2019), 77-89, https://doi.org/10.7494/OpMath.2019.39.1.77

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