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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.

This paper proposes a new delay-depended stability criterion for a class of delayed Lur'e systems with sector and slope restricted nonlinear perturbation. The proposed method employs an improved Wirtinger-type inequality for constructing a new Lyapunov functional with triple integral items. By using the convex expression of the nonlinear perturbation function, the original nonlinear Lur'e system is transformed into a linear uncertain system. Based on the Lyapunov stable theory, some novel delay-depended stability criteria for the researched system are established in terms of linear matrix inequality technique. Three numerical examples are presented to illustrate the validity of the main results.