In this article, the following boundary value problem of fractional differential equation with Riemann-Stieltjes integral boundary conditionis studied, where

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In this paper, a fixed-point theorem is used to establish existence results for fractional Dirichlet boundary value problemwhere 1 < α 2, D α x(t) is the conformable fractional derivative, and f : [0, 1] × R 2 → R is a continuous function. The main condition is sign condition. The method used is based upon the theory of fixed-point index.

Summary In this article, we consider regional consensus problem for a group of identical linear systems represented by a linear differential inclusion over an undirected communication topology. Each vertex system of the linear differential inclusion is represented by a general linear system subject to input saturation, and hence only regional consensus can be achieved. For given saturated distributed linear control protocols, we establish a set of conditions under which these control protocols achieve regional consensus and a level set of a Laplacian quadratic function can be used as an estimate of the domain of consensus. These conditions are given in the form of matrix inequalities and involve the properties of the communication topology. Based on these matrix inequalities, we formulate a linear matrix inequalities based optimization problem for obtaining as large an estimate of the domain of consensus as possible. By viewing the gain matrix in the consensus algorithms as an additional variable, this optimization problem can be adapted for the design of the consensus protocols. Simulation results illustrate the effectiveness of our proposed approach.

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