Structure of Optimal Control Problems

system inputs such as temperature versus time relation in a batch reaction, 

system properties such as diffusivity versus concentration relation in a gas-polymer system, and 

entities generated in the system, e. g., the rate of heat generation versus 

In most optimal control problems, it is not possible to obtain optimal control laws, i. e., optimal controls as explicit functions of system state. Note that system state is the set of system properties such as temperature, pressure, and concentration. They are subject to change with independent variables like time and space. In the absence of an optimal control law, the optimal control needs to be determined all over again if the initial system state changes. 

The controls that are not given by optimal control laws are often called open-loop controls. They simply are functions of independent variables and specific to the initial system state. The application of open-loop controls is termed open loop control, which is the subject matter of this book. 

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The above examples help us identify the structure of optimal control problems. As shown in Figure 1.12, an optimal control problem involves one or more controls and other inputs to a system under change. The optimal controls provide the desired output from the system. 

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