Some Stability Questions About Degenerate Differential Equations With Delay

From the late 70s, more and more people found degradation generally consisted in the practical problems of engineering, social, economy and biology, and many other aspects. Due to the special structure of degenerate systems, studies on degenerate systems are much more complex than that on classical linear systems, which attracted many scholars in interest and importance. In recent years, research in the degenerate systems has also been great progress, there have been many achievements. Degradation and delay are common phenomenon in the objective world and engineering practice. For many practical systems, it is necessary to consider the impact of delay and degradation when accurate description, further design, analysis and application. Therefore, the study for the degenerate differential systems with delay has important practical significanceStability theory of ordinary differential equations is quite perfect, but the degenerate differential systems and the degenerate differential systems with delay are inadequate. In this paper, several types of the stability problems of the degenerate differential systems with delay were researched and drawn the corresponding conclusions.First of all, a type of the stability problems of the degenerate differential systems with delay was analyzed. And through making use of the associated conclusions of the characteristic equation and characteristic roots of the degenerate differential systems with delay, sufficient conditions for stability of global time-delay systems were discussed when rank(E)=1,rank(E)=2, rank(E)=3Secondly, we discussed the stability of a kind of neutral delayed linear differential equation with single delay and multiple delays. Using Lyapunov-Krasovskii V functional and fundamental inequality, the derivative of the V along system V to zoom processing could reduce the conservative delay caused by this amplification to some extent.Finally, the stability of a type of time-varying differential system with multi-delay was discussed. Sufficient conditions for stability of time-varying differential system with multi-delay were obtained through limited the general solution by making use of variable formula of time-varying differential system with multi-delay and combining with Gronwall-Bellman integral inequality. Meanwhile, the stability conclusions of this kind of system were further promoted.