Degenerate Delay Differential Equations Characteristic Root Distribution And Its Solution Ceased Behavior

Since delays are often encountered in engineering systems , ecology systems, electricity systems, management systems and control systems such as the turbojet engine, microwave oscillator, nuclear reactor, controlability of systems and optimal control, etc. The phenomena of delays are investigated by more and more scientists. These papers aim to investigate the distribute of RFDE with delays and the asymptotic behavior of its solutions.In [1], the author investigated RF here and NFDE with delayswhere detCm The author has established theorems of their eigenvalue distribution. In chapter 1, we generalize theorems to degenaration. We investigate NFDE:where T > 0, t ≥ 0, rankE = q < n, A, B,C∈Rn×n, x(t)∈Rn,t > 0, detC≠ 0 andthere exists integral number k∈ [1, m] satisfies detCk≠ 0. RFDE is a special case of NFDE , we generalize the theorem of paper [1] .In paper [2] , the author dealed with the pole placement of degenerate systemsIn chapter 2 , we have investegated the pole placement of a special case of degenerate system with delayOn special conditions , we can place pole at any way, and we resolve the unperturbed problem of these kinds systems.In chapter 3, we 've investigated the retarded Lienard-type equationwhere r is a nonnegative constant, f1,f2,f3, and g on continuous functions on R. It can be reduced to If , it can be reduced to paper the equation of [3]it can be reduced to the well known retarded Lienard-type equationIf f2(x) = 0, it can be reduced to the equation of paper [4]Our results extend and generalize the theorem of paper [4] ?In addition to, we've deal with the boundedness of solution of Lienard-type equationthe results generalize the theorem of paper [5] .In chapter 4, we've deal with the stability of the system with time-varying delayswhere 0≤hi(t)≤H , The results are manageable in practice.In this paper, delays are clue of the four chapters, and reflect the feature of functional differential equations.