Existence and stability analysis of optimal control

By constructing a complete metric space and a compact set of admissible control functions, this paper investigates the existence and stability of solutions of optimal control problems with respect to the right‐hand side functions. On the basis of set‐valued mapping theory, by introducing the notion of essential solutions for optimal control problems, some sufficient and necessary criteria guaranteeing the existence and stability of solutions are established. New derived criteria show that the optimal control problems whose solutions are all essential form a dense residual set, and so every optimal control problem can be closely approximated arbitrarily by an essential optimal control problem. The example shows that not all optimal control problems are stable. However, our main result shows that, in the sense of Baire category, most of the optimal control problems are stable.