Linear Multi-to Lag The Amount Rdde Stability Problem Algebraic Criterion

Consider the following equation x′(t)+ax(t)+bx(t-τ)+cx(t-2τ)=0 (E)It's characteristic equationλ+a+be^-τλ+ce^-2τλ=0where a,b,c andτare constants, andτ＞0, abc≠0.In the present paper, we study first order linear delay equations with two delays,which are theoretically important and practically often used. The problem of analyzingthe zero points distribution of exponential functions in the complex plane, which hasreceived a large lot of attention, since it often occurs in many fields of both puremathematics and applied mathematics. In the discussion of stability, it is of great valueto be able to predict stability or instability directly from the coefficients of the givenequation. The above mentioned methods are of great value in this kind of appliedproblems in automatic control theory.In this paper, we introduce several new functions and describe their proper lities.By using Liapunov Theorem, we establish necessary and sufficient conditions for all theroots of the characteristic equation of (E) to have negative parts, and furthermore, obtainnecessary and sufficient conditions for the zero solution of(E)to be asymptoticallystable, which can predict the stability or instability directly from the coefficients of thegiven equation. The above mentioned conditions are entire and easy to verify and apply.We also get a series of new views and methods in solving the problems.