source of the problemIn any control system with feedback devices time lag always exists.Takea simple example as shown in the following block diagram:(?)the differential equation of which is usually written as(?)However,strictly speaking,this diagram should be represented by the fol-lowing difference-differential equation(?)where τ>0 may be constant or function of time t,since even electromagneticwaves need time to propagate.In ordinary technical literatures(2)is replaced by(1)on the ground thatr is very small.This replacement requires mathematical justification,sinc,efor example,theequation(?)possesses stable trivial solution y≡0,whereas the trivial solution of the equation(?)isunstable,no matter how smali the positive number τ may be.This paper presents systematic results of the equivalence of the differentialequation and the difference-differential equation in the theory of stability.Ⅱ.The equivalence of linear equationsTheorem 1.Given a differential-difference equation(?)where the constants p and q satisfy the condition(?) there exists a number Δ=Δ(p,q)>0 such that,under the condition 0<δ<Δ,the trivial solution of (3) is asymptotically stable.In other words,from the fact that the trivial solution of the equation(?)is asymptotically stable we can deduce the result that the same statement holdsfor the equation(3),provided 0<δ<Δ.In this theorem Δ may be taken as π/8(|p|+|q|)in general case.Theproof is based on the results of Hayes.Theorem 2.Let the condition(4)in theorem 1 be replaced by the condition(?)p+q<0,(4)'then the trivial solution of the equation(3)is unstable for any δ>0.In other words,from the fact that the trivial solution of(5)is unstablewe can deduce the result that the same statement holds for the equation(3),no matter how the positive number δ may be taken.Actually we proved that there exists a real positive function s=s(δ),for δ≥0,such that exp(st)is a particular solution of equation(3).Theorem 3.Let the condition(4)in theorem 1 be replaced by the conditionp+q=0,(4)″there exists a number Δ=Δ(p)>0,such that,under the condition 0<δ<Δ,the trivial solution of the equation (3) is stable.In this theorem Δ may be taken as 1/|p| for |p|>0,and Δ cannot be +∞in general,since we have counter example:u(t)=at is a particular solutionfor the equation (3),in the casep+q=0,p>0,δ=1/p.Ⅲ.The equivalence of non-linear equationsBased on theorem 1 and a result of Bellman,we getTheorem 4.Given an equation(?)where the following conditions are fulfilled(i)p+q0,such that,under the condition O<δ<Δ,thetrivial solution of the equation(6)is asymptotically stable, In other words,from the fact that the trivial solution of the equation(?)is asymptotically stable (p+q≠0) we can deduce the result that the samestatement holds for the equation(6),provided0<δ<Δ.Theorem 5.Let the condition(4)in theorem 4 be replaced by the condition(4)′,then the trivial solution of the equation(6)is unstable for anyδ>0.In other words,from the fact that the trivial solution of the equation(7)is unstable (p+q≠0)we can deduce the result that the same statement holdsfor the equation(6),no matter how the positive number δmay be taken.Twocases are separated in the proof. For q≤0,we use the equation(?)for comparison,whereas for q>0,we take a suitable equation of theorem 2for comparison.Theorem 6.Given an equation(?)where p≠0,δ>0,we can always find suitable function F_2(u(t),u(t-δ))suchthat the trivial solution of(8)is either unstable for any δ>0,or stable for0<δ<Δ,as we wished.For unstable case,two subcases are separated in the proof. For example,for p>0,F_2=pu~2(t),and for p<0,F_2=-pu~2(t-δ).Ⅳ.Summary of the resultsThe instability of (6) and(7)is equivalent for any δ>0,(p+q≠0),the asymptotical stability of(6) and(7)is equivalent (p+q≠0) provided0<δ<Δ(=π/8(|p|+|q|),say).In the critical case (p+q=0) the linearequations are both stable provided 0<δ<Δ(=1/|p|,say),whereas the stablebehaviour may be changed by adding terms of higher orders for both cases.
