This chapter presents the conditions related to the optimality of a functional. We derive the necessary conditions for optimal control to exist and apply them to optimal control problems. We also present sufficient conditions assuring the optimum under certain conditions. Readers are encouraged to review the logic of the conditional statement in Section 9.25 (p. 282).
3.1 Necessary Condition for Optimality
Let I be a functional of a function y. If I is minimum at y, then the values of I for all other admissible functions in the vicinity of y cannot be lower than the value of I at y. Precisely
is the necessary condition for the minimum of I where y is any admissible function in the vicinity ...
