Application to Simplest Optimal Control Problem

Consider the simplest optimal control problem in which it is desired to find a continuous control function u t) that optimizes the objective functional 

Equation (3.5) is called the state equation since it describes the state of the system through the state variable y as a function of the independent variable t. It is assumed that F and g have continuous partial derivatives with respect to y and u. Note that this problem is autonomous in the sense that the independent variable does not appear explicitly in F or y. When it does, the problem is easily convertible to the autonomous form (see Exercise 3.4). 

Observe that the objective functional / in Equation (3.4) is influenced by the control u directly as well as indirectly. While the direct influence is 

to solve the problem, we need to first obtain an explicit solution y = y u) and then substitute it in the expression of F. However, such solutions do not exist for most optimal control problems, which are typically constrained by highly non-linear state equations. 

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