Stability Analysis Of Delay VOLTERRA Integro-Differential Systems

The stability theory for solution of delay Volterra integro-differential equation is an important part of integro-differential equation theory. Any advances in research of this field will give impetus to other disciplines greatly. So far, this theory has been widely applied to many research areas, such as network reservoir, storage systems, matter accumulation, industrial process, physics, astronomy, and so on. In particular, the modeling of integro-differential equation and its solving are dependent on this stability theory.Generally speaking, for a given differential equation and its non-negative initial condition, if the solution of this equation is also non-negative, then we call this system a positive definite one. The positive definite theory of differential equation develops from non-negative matrix theory. Also, research on stability theory of differential equation has achieved some important theory achievements.Delay Volterra integro-differential equation model was firstly put forward by Volterra, who is a famous scholar. And then, Canchy, Fredhol and Hilvert conducted some researches on this subject from different perspectives and got some results. However, to some extent, there are still some difficulties in solving the judgment rules for stability of delay Volterra integro-differential equation exponent. Therefore, on this problem, some meaningful researches about are conducted in this paper.Firstly, some basic or related theories about Volterra integro-differential equation are introduced. And then based on some properties of Metzler matrix, the positive definite problem of delay Volterra integro-differential equation is discussed in intensity continuous semi-group. For the judgment rules of positive definiteness, a necessary and sufficient condition is deduced to judge whether or not the solution of differential equation is positive definite. Furthermore, the stability of common delay integro-differential equation is discussed. The sufficient condition for stability of Volterra integro-differential equation exponent that meets certain initial conditions is given. The stability of positive definite delay Volterra integro-differential equation exponent is discussed in the final chapter that is the most important part of this paper. Finally, a sufficient condition used for judging the stability of delay Volterra integro-differential equation exponent meeting a certain initial conditions is concluded.