Many phenomena occurring in biology, mechanical, chemical, electricity, electronic, medicine, aircraft industry, etc. can be described in terms of mathematical expressions, such as partial differential equations (PDE), or ordinary differential equations (ODE). For many different kinds of problems, the natural question that arises is how to intervene to produce the "best" outcome, as measured by some predetermined goals. One approach to this is via optimal control theory. In many applications, when using ordinary differential equations or partial differential equations, one always assumes that a system is governed by the principle of causality, namely, the future state of the system is determined by the current state only, while the past has no impact on the future with the present of the current state. However, this is not logically reasonable in many processes. When taking into account the impact of the past state, this includes the time delays. Mathematical models with time delays have been used to understand the dynamic of many systems. However, optimal control problem with time delays has not been well studied in the literature. So, it is important to develop some new results for delayed optimal control problems and to apply it.
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