Necessary Optimality Conditions for Optimal Control Problems with Time-Varying Delays

In this paper, necessary optimality conditions for a class of optimal control problems containing time-varying delays in control and state variables are discussed. There is an important aspect of these problems in that time-varying delays are applied to both state and control variables. Also, the cost functional of problems is influenced by the time-varying delays in state and control. We prove necessary optimality conditions in this study. A key aspect of the proof is calculating the variations of control and state variables when there are time-dependent delays. we make use of appropriate changing variables to derive these variations. In order to illustrate the use of these conditions, several examples are solved and numerical results are presented. At the end, some conclusions are drawn.