Abstract
In this article, we consider a retarded linear fractional differential system with distributed delays and Caputo type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches—by the use of the Gronwall’s inequality and by the use of integral representations of the solutions are obtained. As application of the obtained estimates, different sufficient conditions which guaranty finite-time stability of the solutions are established. A comparison of the obtained different conditions in respect to the used estimates and norms is made.
Highlights
System with Distributed Delays andAs a highly applicable mathematical tool to study models of real-world phenomena, fractional calculus theory attracts a lot of attention
It is clear that the estimations (32) and (33) will be essentially used, but to obtain a practical applicable estimation we need to solve two problems: (a) First, we need to clarify which impact is leading for the process, the impact hereditary of the process expressed by kΦk, the impact of the outer perturbations expressed by k F ( T )k, or the complex of both factors expressed by the ratio CΦ = kΦk−1 k F ( T )k
In this work we set out some considerations illustrating our point of view concerning the different sources of the impacts of the finite-time stability
Summary
As a highly applicable mathematical tool to study models of real-world phenomena, fractional calculus theory attracts a lot of attention. The wide appearance of the aftereffect to regard it as a universal property of the surrounding world, is a serious reason to consider mathematical models with delay and fractional derivatives This explains why a lot of papers are devoted to different aspects of this problem. As an application, we apply these estimates to investigate the finite-time stability of fractional differential systems with Caputo type derivatives in the case of incommensurate fractional orders and distributed delays.
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