## Optimal control problem

Consider finding an admissible control u0pt which causes the system

to follow an admissible trajectory, x*, that minimizes the criteria function

so that u0Vt is called the optimal control. In (18), the term k is a weighting factor which is determined in search of the minimum J.

Let criteria function be formulated as follows:

J = J L dt where the scalar function L is written as

In the calculus of variations, it is well known that a necessary condition for x* to be an optimal of the functional given by (19) is (Naidu, 2003)

This equation is called Euler equation. Application of Euler equation to (19) yields the optimal control law:

3kXuuu | u | 3Kppplpl 2T

with

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