Some optimal control results for differential-difference systems

A brief development of necessary conditions for an extremal path in the case where the system equations and the performance index contain a time delay in both the state and the control variables is given. An analytic solution is also presented for a linear system with a quadratic performance index in the case where the control variable appears in the system equations evaluated both at the present time and at a previous time. Unspecified constants in the solution are obtained by the inversion of a specified matrix. Results for some examples are illustrated.