A maximum principle for time-lag control problems with bounded state

A maximum principle is obtained for control problems involving a constant time lag τ in both the control and state variables. The problem considered is that of minimizing $$I(x) = \int_{t^0 }^{t^1 } {L (t,x(t), x(t - \tau ), u(t), u(t - \tau )) dt} $$ subject to the constraints 1 $$\begin{gathered} \dot x(t) = f(t,x(t),x(t - \tau ),u(t),u(t - \tau )), \hfill \\ x(t) = \phi (t), u(t) = \eta (t), t^0 - \tau \leqslant t \leqslant t^0 , \hfill \\ \end{gathered} $$ 1 $$\psi _\alpha (t,x(t),x(t - \tau )) \leqslant 0,\alpha = 1, \ldots ,m,$$ 1 $$x^i (t^1 ) = X^i ,i = 1, \ldots ,n$$ . The results are obtained using the method of Hestenes.