Abstract: In this paper, for the delay control system: x(t)=f[x(t),x(t-1),u(t),t] t∈[t₀,t₁] x(t)=φ(t) t∈[t₀,t₁] with State right endpoint restricled by condition φ_k=[x(t₁)]=0 (k=1,2,…,l) a maximum principle is given. And as a specificexample. this paper gives a maximumprinciple under the condition that partial states right endpoints be completely fixed.Finally , this paper gives an example to explain the application of the main result ofthis paper. All the results are suitable for the control systems with multidelay as well.

Key words: delay control system, restricted state right endpoint, maximum principle